BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Si Ying Lee (Harvard University) DTSTART:20200408T140000Z DTEND:20200408T153000Z DTSTAMP:20260404T094309Z UID:STAGE/1 DESCRIPTION:Title: Modular forms on Hilbert modular varieties\nby Si Ying Lee (Harva rd University) as part of STAGE\n\n\nAbstract\nWe will give an overview of Katz's paper on the construction of $p$-adic L-functions for CM fields. A key input in this paper is the consideration of modular forms on Hilbert modular varieties. We will discuss some key properties of Hilbert-Blumenth al abelian varieties\, and the associated moduli spaces. We will also defi ne modular forms on Hilbert modular varieties\, and prove a $q$-expansion principle.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Petrov DTSTART:20200415T140000Z DTEND:20200415T153000Z DTSTAMP:20260404T094309Z UID:STAGE/2 DESCRIPTION:Title: $p$-adic modular forms on Hilbert modular varieties\nby Alexander Petrov as part of STAGE\n\n\nAbstract\nWe will define $p$-adic Hilbert mo dular forms via level $p^{\\infty}$ formal Hilbert modular schemes and stu dy the relative de Rham cohomology over that scheme using the Frobenius en domorphism.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng DTSTART:20200422T140000Z DTEND:20200422T153000Z DTSTAMP:20260404T094309Z UID:STAGE/3 DESCRIPTION:Title: Differential operators on modular forms\nby Tony Feng as part of STAGE\n\n\nAbstract\nI will cover Section 2 of Katz's paper\, constructing (analytic and p-adic) differential operators on modular forms.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Ziquan Yang (Harvard University) DTSTART:20200429T140000Z DTEND:20200429T153000Z DTSTAMP:20260404T094309Z UID:STAGE/4 DESCRIPTION:Title: $p$-adic Eisenstein series\nby Ziquan Yang (Harvard University) a s part of STAGE\n\n\nAbstract\nI will cover Section 3 of Katz's paper on $ p$-adic Eisenstein series.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Zijian Yao (Harvard) DTSTART:20200506T140000Z DTEND:20200506T153000Z DTSTAMP:20260404T094309Z UID:STAGE/5 DESCRIPTION:Title: CM Hilbert-Blumenthal abelian varieties\nby Zijian Yao (Harvard) as part of STAGE\n\n\nAbstract\nKatz 1978\, Sections 5.0 and 5.1\, and the statements of 5.2.26 and 5.2.29.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kriz (Massachusetts Institute of Technology) DTSTART:20200513T140000Z DTEND:20200513T153000Z DTSTAMP:20260404T094309Z UID:STAGE/6 DESCRIPTION:Title: Construction of Katz $p$-adic $L$-functions\nby Daniel Kriz (Mass achusetts Institute of Technology) as part of STAGE\n\n\nAbstract\nWe will describe Katz's construction of a $p$-adic measure on the $p^{\\infty}$ r ay class group of CM fields\, whose Mellin transform is a $p$-adic $L$-fun ction interpolating critical values of Hecke $L$-functions. First\, we wil l recall some basics of measures and the construction of the $p$-adic modu lar form-valued Eisenstein measure. Next\, we will obtain Katz's measure b y evaluating the Eisenstein measure at CM points. Finally\, we will recove r the aforementioned interpolation via Katz's insight that the values of t he $p$-adic and complex differential operators at CM points coincide\, whi ch follows from the moduli-theoretic definitions of these operators.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Danielle Wang (MIT) DTSTART:20200907T190000Z DTEND:20200907T203000Z DTSTAMP:20260404T094309Z UID:STAGE/7 DESCRIPTION:Title: Statements of the Weil conjectures\, proof for curves via the Hodge i ndex theorem\nby Danielle Wang (MIT) as part of STAGE\n\n\nAbstract\nR eferences: Poone n\, Rational points on varieties\, Chapter 7 up to Section 7.5.1\; Milne\, The Riemann Hypo thesis over Finite Fields: from Weil to the present day\, pages 8-10.\ n\nThe Weil conjectures concern the zeta functions of varieties over a fin ite field\, which for a smooth proper variety are rational functions that satisfy a functional equation and the Riemann hypothesis. The conjectures led to the development of étale cohomology by Grothendieck and Artin. In this talk\, we will state the Weil conjectures and prove the Riemann hypot hesis for curves using the Hodge index theorem.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20200914T190000Z DTEND:20200914T203000Z DTSTAMP:20260404T094309Z UID:STAGE/8 DESCRIPTION:Title: Smooth and étale morphisms\nby Niven Achenjang (MIT) as part of STAGE\n\n\nAbstract\nReferences: Mumford\, The red book of varieties and schemes\, III.5 and III.10\; or Poonen\, Rational points on varieties\, Section 3.5.\n\nSmooth variet ies give an algebraic analogue of (smooth) manifolds from differential geo metry\, while smooth and étale morphisms give algebraic analogues of subm ersions and local isomorphisms. In addition to translating important notio ns from differential geometry into the algebraic setting\, maps of these t ypes play an important role in later development of étale cohomology. In this talk\, we will introduce the definitions and basic properties of smoo th and étale morphisms with an emphasis on providing intuition for thinki ng about them.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthew Hase-Liu (Harvard) DTSTART:20200921T190000Z DTEND:20200921T203000Z DTSTAMP:20260404T094309Z UID:STAGE/9 DESCRIPTION:Title: Introduction to étale cohomology\nby Matthew Hase-Liu (Harvard) as part of STAGE\n\n\nAbstract\nReferences: Poonen\, Rational \npoints on varieties\, Chapter 6\; or M ilne\, Lectures on é\;tale cohomology.\n\nA crash course on éta le cohomology covering the following: sites and cohomology\, the étale si te and operations on étale sheaves\, Frobenius action\, stalks of étale sheaves\, cohomology with compact support\, and important theorems/necessi ty of torsion coefficients.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuel Marks (Harvard) DTSTART:20200928T190000Z DTEND:20200928T203000Z DTSTAMP:20260404T094309Z UID:STAGE/10 DESCRIPTION:Title: Rationality and functional equation of the zeta function\nby Sam uel Marks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a variety $X/\\m athbb{F}_q$\, the étale cohomology groups $H^i(X_{\\overline{\\mathbb{F}_ q}}\,\\mathbb{Q}_\\ell)$ come equipped with an action of $\\mathrm{Gal}(\\ overline{\\mathbb{F}_q}/\\mathbb{F}_q)$\, and in particular with an action of the $q$-power Frobenius. This Frobenius action can also be described a s coming from the Frobenius morphism $\\mathrm{Fr}:X\\rightarrow X$. By us ing these two perspectives on the Frobenius and some abstract cohomologica l inputs\, we deduce the rationality and functional equation of $Z(X\,T)$ for nice varieties $X$.\n\nReference: Jannsen\, Deligne's proof o f the Weil-conjecture (course notes)\, Section 1.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/10/ END:VEVENT BEGIN:VEVENT SUMMARY:David Roe (MIT) DTSTART:20201005T190000Z DTEND:20201005T203000Z DTSTAMP:20260404T094309Z UID:STAGE/11 DESCRIPTION:Title: Constructible sheaves\nby David Roe (MIT) as part of STAGE\n\n\n Abstract\nConstructible sheaves are built from locally constant sheaves an d serve as the coefficients for étale cohomology. We will discuss the mo tivation behind their definition\, examples and some basic properties.\n\n Reference: Jannsen\, Deligne's proof of the Weil-conjecture (cour se notes)\, Section 2.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Sanath Devalapurkar (Harvard) DTSTART:20201012T190000Z DTEND:20201012T203000Z DTSTAMP:20260404T094309Z UID:STAGE/12 DESCRIPTION:Title: Étale fundamental groups\nby Sanath Devalapurkar (Harvard) as p art of STAGE\n\n\nAbstract\nMotivated by topological considerations\, one can define an algebraic analogue of the fundamental group\, called the eta le fundamental group. We will give a definition (via the abstract theory o f Galois categories from SGA)\, and review some basic calculations.\n\nRef erences: Milne\, Lectures on étale cohomology\, Chapter 3\; and/or Poonen\, Rational points on variet ies\, Sections 3.5.9 and 3.5.11.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Francesc Fité DTSTART:20201019T190000Z DTEND:20201019T203000Z DTSTAMP:20260404T094309Z UID:STAGE/13 DESCRIPTION:Title: Deligne's version of the Rankin method\nby Francesc Fité as par t of STAGE\n\n\nAbstract\nWe will present a proof of the Riemann hypothesi s for smooth and projective curves defined over a finite field due to Katz . The proof reduces the general case to the case of Fermat curves via a de formation argument (the "connect by curves lemma") and the use of Deligne' s version of the Rankin method. For the case of Fermat curves\, we will re call how the Riemann hypothesis amounts to a classical well-known result a bout the size of Jacobi sums.\n\nReference: Katz\, A note on Riemann hypothesis for curves and hypersu rfaces over finite fields\, Sections 1-4.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Ziquan Yang (Harvard University) DTSTART:20201026T190000Z DTEND:20201026T203000Z DTSTAMP:20260404T094309Z UID:STAGE/14 DESCRIPTION:Title: The Riemann hypothesis for hypersurfaces\nby Ziquan Yang (Harvar d University) as part of STAGE\n\n\nAbstract\nI will talk about Katz' meth od of proving the Riemann hypothesis (RH) for hypersurfaces. The basic ide a is very similar to what we saw last time: We reduce to showing RH for a particular hypersurface. Then we show RH for this particular hypersurface by a point-counting argument. \n\nReference: Katz\, A note on Riemann hypothesis for curves and hyper surfaces over finite fields\, Sections 5-8.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Hyuk Jun Kweon (MIT) DTSTART:20201102T200000Z DTEND:20201102T213000Z DTSTAMP:20260404T094309Z UID:STAGE/15 DESCRIPTION:Title: Alterations\nby Hyuk Jun Kweon (MIT) as part of STAGE\n\n\nAbstr act\nIn 1964\, Hironaka proved that over a field of characteristic zero\, every algebraic variety admits a resolution of singularities. However\, th e problem of resolution of singularities is still open in positive charact eristic. As a weaker result\, de Jong proved that every algebraic variety admits regular alterations. We will discuss background\, main statements a nd some applications for de Jong's result. If time allows\, we will discus s a very rough sketch of the proof.\n\n\nReference: Notes from Conrad's le ctures on alternations\, Section 1. The goal is to understand the sta tement of the main theorem on alterations.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Petrov (Harvard University) DTSTART:20201109T200000Z DTEND:20201109T213000Z DTSTAMP:20260404T094309Z UID:STAGE/16 DESCRIPTION:Title: Weights and monodromy\nby Alexander Petrov (Harvard University) as part of STAGE\n\n\nAbstract\nReference: Scholl\, Hypersurfaces and the Weil conjectures\, Secti ons 1 and 2.\n\nWe will discuss the relationship between the action of loc al monodromy around a singular fiber of a proper family and the Frobenius action\, proving Deligne's weight monodromy theorem in equal characteristi c.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhiyu Zhang (MIT) DTSTART:20201116T200000Z DTEND:20201116T213000Z DTSTAMP:20260404T094309Z UID:STAGE/17 DESCRIPTION:Title: Vanishing cycles and deformation to hypersurfaces\nby Zhiyu Zhan g (MIT) as part of STAGE\n\n\nAbstract\nFirstly\, we give a very brief rev iew of Weil conjecture. Following works of Scholl and Katz\, we then outli ne a "10-line" proof of the Weil conjecture by deformation to smooth hyper surfaces and induction on the dimension. In particular\, we will explain t he last step i.e how to derive RH of the special fiber from the (equal cha racteristic) weight-monodromy conjecture of the generic fiber\, using the weight spectral sequence as an input.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond van Bommel (MIT) DTSTART:20201130T200000Z DTEND:20201130T213000Z DTSTAMP:20260404T094309Z UID:STAGE/18 DESCRIPTION:Title: The Bombieri-Stepanov approach to the Riemann hypothesis for curves over finite fields\nby Raymond van Bommel (MIT) as part of STAGE\n\n\n Abstract\nIn this talk\, we will discuss an elementary proof for the Riema nn hypothesis for curves over finite fields due to Bombieri\, based on pre vious work by Stepanov and Schmidt. It uses a method which we would now ca ll the polynomial method\, and the Riemann Roch theorem to prove an upper bound for the number of rational points on a curve.\n\nThe slides for the talk will be available on Monday 30 November.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kriz (MIT) DTSTART:20201207T200000Z DTEND:20201207T213000Z DTSTAMP:20260404T094309Z UID:STAGE/19 DESCRIPTION:Title: Dwork's $p$-adic proof of rationality\nby Daniel Kriz (MIT) as p art of STAGE\n\n\nAbstract\nIn 1959\, ex-electrical engineer Bernard Dwork shocked the mathematical world by proving the first Weil conjecture on th e rationality of the zeta function. Dwork's proof introduced striking new $p$-adic methods\, and defied the expectation that the Weil conjectures co uld only be solved by developing a suitable Weil cohomology theory (later found to be $l$-adic etale cohomology). In this talk we will outline Dwork 's proof and begin the initial part of the argument\, introducing Dwork's general notion of "splitting functions"\, the Artin-Hasse exponential and Dwork's lemma. \n\n\nReference: Koblitz\, p-adic numbers\, p-adic analysis\, and zeta-functions\, pp. 92-95 and then Section V.2 to the end of the book \, some of which may be covered in a second lecture.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kriz (MIT) DTSTART:20201214T200000Z DTEND:20201214T213000Z DTSTAMP:20260404T094309Z UID:STAGE/20 DESCRIPTION:Title: Dwork's $p$-adic proof of rationality\, continued\nby Daniel Kri z (MIT) as part of STAGE\n\n\nAbstract\nWe will go over the main steps of Dwork's argument in detail. First\, we will construct a splitting function for the standard additive character and show it has good convergence prop erties using Dwork's lemma. Next we will establish the "analytic Lefschetz fixed point formula" by studying the trace of this splitting function act ing on $p$-adic Banach spaces of power series. Finally\, we will show this analytic fixed point formula implies the zeta-function is the ratio of tw o entire functions\, and conclude with a general rationality criterion for $p$-adic power series that implies the zeta-function is rational. \n\n\nR eference: Koblitz\, p-adic numbers\, p-adic analysis\, and zeta-functions\, w hatever remains of Chapter V after the first lecture.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Ishan Levy (MIT) DTSTART:20210219T180000Z DTEND:20210219T193000Z DTSTAMP:20260404T094309Z UID:STAGE/21 DESCRIPTION:Title: The infinitesimal site and algebraic de Rham cohomology\nby Isha n Levy (MIT) as part of STAGE\n\n\nAbstract\nThe de Rham cohomology of the analytification of a smooth projective\nvariety over $\\mathbb{C}$ can be computed via an algebraic de Rham complex.\nUnfortunately\, the algebraic de Rham complex is somewhat poorly behaved in\npositive characteristic. To solve this problem\, Grothendieck\nshowed first how to reinterpret de R ham cohomology in characteristic 0\nas cohomology on a site (the infinites imal site)\, and second\nhow to modify the infinitesimal site to obtain a site\nthat works well also in characteristic p (the crystalline site).\n\n In this talk\, we will explain algebraic de Rham cohomology\nand define th e infinitesimal and stratifying sites. \nWe also will define the notion of a classical Weil cohomology theory\,\nwhich de Rham cohomology (char 0) a nd crystalline cohomology give examples of.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Naomi Sweeting (Harvard) DTSTART:20210226T180000Z DTEND:20210226T193000Z DTSTAMP:20260404T094309Z UID:STAGE/22 DESCRIPTION:Title: Crystalline cohomology\nby Naomi Sweeting (Harvard) as part of S TAGE\n\n\nAbstract\nThis talk will provide an overview of key concepts in crystalline cohomology. We will begin with Grothendieck's heuristic argum ent that\, because de Rham cohomology is independent of choice of smooth l ift\, an intrinsic characteristic zero-valued cohomology should exist for schemes in characteristic p. We will then discuss divided power structur es and the crystalline site. After stating the key theorems\, we will des cribe a relative setup in which the general theory of topoi plays a more p rominent role. We will conclude with sketches of crucial ideas in the com parison isomorphisms\, and a glimpse of the relationship between crystals and connections.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/22/ END:VEVENT BEGIN:VEVENT SUMMARY:James Myer (The CUNY Graduate Center) DTSTART:20210305T180000Z DTEND:20210305T193000Z DTSTAMP:20260404T094309Z UID:STAGE/23 DESCRIPTION:Title: Introduction to prismatic cohomology\nby James Myer (The CUNY Gr aduate Center) as part of STAGE\n\n\nAbstract\nThe study of the cohomology of algebraic varieties is depicted by Peter Scholze as a “plane worth ” of pairs of primes $(p\,\\ell)$\, each indexing a cohomology theory fo r varieties over $\\mathbb{F}_p$ with coefficients in $\\mathbb{F}_{\\ell} $. The singular cohomology occupies a vertical line over $\\infty$\; the étale cohomology dances around\, avoiding the pairs $(p\,p)$\; the analyt ic de Rham cohomology occupies the top right corner\, intersecting the sin gular cohomology @ $(\\infty\,\\infty)$\, symbolizing the classical de Rha m comparison theorem\, while the diagonal is picked off by the algebraic d e Rham cohomology. Zooming in on a point on the diagonal\, we begin to won der whether there is a cohomology theory interpolating between the étale to the crystalline (and de Rham). In fact\, the depiction of the plane of pairs of primes is striated by lines from each of the various cohomology t heories\, but no cohomology theory seems to “wash over” any 2-dimensio nal part of the picture and “phase in and out” between any one or the other. The prismatic cohomology theory is this “2-dimensional” theory interpolating between the étale and crystalline (and de Rham) theories.\n \nThe classical de Rham comparison theorem between the (dual of the) analy tic de Rham cohomology and the singular homology offers a geometric interp retation of a (co)homology class as an obstruction to (globally) integrati ng a differential form. This geometric interpretation loses steam when fac ed with torsion classes because the integral over a torsion class is alway s zero. It is also worthwhile to note the relative ease with which we may calculate the de Rham cohomology of a variety (this can be done by machine \, e.g. Macaulay2) as opposed to the singular cohomology of a variety. So\ , how do we detect these torsion cycles algebraically? We will see via a c alculation applying the universal coefficients theorem that the hypothesis of equality of dimensions of the analytic and algebraic de Rham cohomolog y groups of a variety implies lack of torsion in singular cohomology. Some what conversely\, we’ll see that the presence of torsion in the singular cohomology of the analytic space associated to a variety forces the algeb raic de Rham cohomology group to be larger than expected. This interplay b etween the various cohomology theories for varieties\, e.g. singular\, ét ale\, analytic de Rham\, algebraic de Rham\, crystalline\, is facilitated by a (specialization of a sequence of) remarkable theorem(s) whose proof d epends on the existence of\, and motivates the construction of\, the prism atic cohomology theory. \n\nFollowing this introduction\, we will venture into some detail\, set up some notation for the next speaker\, and elabora te a bit more on the story to come.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (Harvard) DTSTART:20210312T180000Z DTEND:20210312T193000Z DTSTAMP:20260404T094309Z UID:STAGE/24 DESCRIPTION:Title: Delta rings\nby Mikayel Mkrtchyan (Harvard) as part of STAGE\n\n \nAbstract\nThis talk will explain some basic properties of $\\delta$-ring s\, following Bhatt-Scholze. This will include examples\, categorical prop erties of delta-rings\, Witt vector considerations\, and\, time permitting \, a connection with pd-envelopes.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Tobias Shin (Stony Brook) DTSTART:20210319T170000Z DTEND:20210319T183000Z DTSTAMP:20260404T094309Z UID:STAGE/25 DESCRIPTION:Title: Derived categories for the working graduate student\nby Tobias S hin (Stony Brook) as part of STAGE\n\n\nAbstract\nWe give a brief review o f derived categories\, then discuss derived tensor products and derived co mpletions.\n\nReferences: The Stacks project section on derived completion and the reference s listed there.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuel Marks (Harvard) DTSTART:20210326T170000Z DTEND:20210326T183000Z DTSTAMP:20260404T094309Z UID:STAGE/26 DESCRIPTION:Title: Distinguished elements and prisms\nby Samuel Marks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a divided power ring $(A\,I)$\, the cry stalline site is defined using divided power thickenings over $(A\,I)$. An alogously\, given a prism $(A\,I)$\, the prismatic site is defined using "prismatic thickenings" over $(A\,I)$. The goal of this talk is to d efine prisms and develop their basic properties.\n\nReferences: Lecture II I of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Sanath Devalapurkar DTSTART:20210402T170000Z DTEND:20210402T183000Z DTSTAMP:20260404T094309Z UID:STAGE/27 DESCRIPTION:Title: Perfect prisms and perfectoid rings\nby Sanath Devalapurkar as p art of STAGE\n\n\nAbstract\nWe will show that the category of perfect pris ms is equivalent to the category of perfectoid rings\, and use this to pro ve some structural results about perfectoid rings.\n\nReferences: Lecture IV of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Konrad Zou (Bonn) DTSTART:20210409T170000Z DTEND:20210409T183000Z DTSTAMP:20260404T094309Z UID:STAGE/28 DESCRIPTION:Title: The prismatic site\nby Konrad Zou (Bonn) as part of STAGE\n\n\nA bstract\nWe will introduce the prismatic site and finally define the prism atic complex and the Hodge-Tate complex.\nWe define the Hodge-Tate compari son map\, which relates the Kähler differentials to the cohomology of the Hodge-Tate complex.\nFinally\, we will introduce the Čech-Alexander comp lex\, which computes the prismatic complex in the affine case.\n\nReferenc es: Lecture V of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Avi Zeff (Columbia) DTSTART:20210416T170000Z DTEND:20210416T183000Z DTSTAMP:20260404T094309Z UID:STAGE/29 DESCRIPTION:Title: The Hodge-Tate and crystalline comparison theorems\nby Avi Zeff (Columbia) as part of STAGE\n\n\nAbstract\nWe will briefly review crystall ine cohomology and its relationship to prismatic cohomology\, and sketch a proof of the crystalline comparison theorem and of the Hodge-Tate compari son theorem as a corollary.\n\nReferences: Lecture VI of Bhatt's notes< /a>. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Danielle Wang DTSTART:20210423T170000Z DTEND:20210423T183000Z DTSTAMP:20260404T094309Z UID:STAGE/30 DESCRIPTION:Title: The $q$-de Rham complex\nby Danielle Wang as part of STAGE\n\n\n Abstract\nReferences: Lecture X of Bhatt's notes. For more details \, see the Bhatt-Scholze paper< /a>.\n\nIn this talk\, we define the q-de Rham complex\, show that it is a q-deformation of the usual de Rham complex\, and state a conjecture about the coordinate independence of this construction (to be proved next lectu re using q-crystalline cohomology).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng DTSTART:20210430T170000Z DTEND:20210430T183000Z DTSTAMP:20260404T094309Z UID:STAGE/31 DESCRIPTION:Title: $q$-crystalline cohomology\nby Tony Feng as part of STAGE\n\n\nA bstract\nReferences: Lecture XI of Bhatt's notes. For more details \, see the Bhatt-Scholze paper< /a>.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Zijian Yao DTSTART:20210507T170000Z DTEND:20210507T183000Z DTSTAMP:20260404T094309Z UID:STAGE/32 DESCRIPTION:Title: Extension to the singular case via derived prismatic cohomology\ nby Zijian Yao as part of STAGE\n\n\nAbstract\nReferences: Lecture VII of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhiyu Zhang DTSTART:20210514T170000Z DTEND:20210514T183000Z DTSTAMP:20260404T094309Z UID:STAGE/33 DESCRIPTION:Title: Perfections in mixed characteristic\nby Zhiyu Zhang as part of S TAGE\n\n\nAbstract\nReferences: Lecture VIII of Bhatt's notes. For more details\, see the Bhatt-S cholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Shizhang Li (University of Michigan) DTSTART:20210521T170000Z DTEND:20210521T183000Z DTSTAMP:20260404T094309Z UID:STAGE/34 DESCRIPTION:Title: The étale comparison theorem\nby Shizhang Li (University of Mic higan) as part of STAGE\n\n\nAbstract\nReferences: Lecture IX of Bhatt' s notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Angus McAndrew (Boston University) DTSTART:20210922T150000Z DTEND:20210922T163000Z DTSTAMP:20260404T094309Z UID:STAGE/35 DESCRIPTION:Title: Intersection theory with divisors\nby Angus McAndrew (Boston Uni versity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nReferences: Appendix B of Kleiman\, The Picard scheme\, Contemp. Math.\, 200 5 and/or Appendix VI.2 in Kollár\, Rational curves on algebraic varieties .\n LOCATION:https://stable.researchseminars.org/talk/STAGE/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Chen (Harvard) DTSTART:20210929T140000Z DTEND:20210929T153000Z DTSTAMP:20260404T094309Z UID:STAGE/36 DESCRIPTION:Title: Big and nef line bundles\nby Nathan Chen (Harvard) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\ nWe will give a gentle introduction to big and nef line bundles\, with an emphasis on their properties and examples. Reference: Sections 1.4 and 2.2 of Laz arsfeld\, Positivity in algebraic geometry I\, Springer\, 2004. \n\nNon-MIT participants must click here to get a "Tim Ticket" well in advanc e\; this is required for access to the seminar.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Katia Bogdanova (Harvard) DTSTART:20211006T150000Z DTEND:20211006T163000Z DTSTAMP:20260404T094309Z UID:STAGE/37 DESCRIPTION:Title: Height machine\nby Katia Bogdanova (Harvard) as part of STAGE\n\ nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nRefer ences: Sections B1-B3 in Hindry and Silverman\, Diophantine geometry\, Spr inger\, 2000 and/or Chapter 2 of Serre\, Lectures on the Mordell-Weil theo rem\, 3rd edition\, Springer\, 1997.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20211013T150000Z DTEND:20211013T163000Z DTSTAMP:20260404T094309Z UID:STAGE/38 DESCRIPTION:Title: Comparison of Weil height and canonical height\nby Alice Lin (Ha rvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bu ilding.\n\nAbstract\nReferences: Sections B4-B5 in Hindry and Silverman\, Diop hantine geometry\, Springer\, 2000 and/or Chapter 3 of Serre\, Lecture s on the Mordell-Weil theorem\, 3rd edition\, Springer\, 1997. Al so\, Theorem A of Silverman\, Heigh ts and the specialization map for families of abelian varieties\, J. Re ine Angew. Math. 342 (1983)\, 197–211.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20211020T150000Z DTEND:20211020T163000Z DTSTAMP:20260404T094309Z UID:STAGE/39 DESCRIPTION:Title: Vojta's approach to the Mordell conjecture I\nby Niven Achenjang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bu ilding.\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's p roof.\n\nReferences: Chapter 11 of Bombieri and Gubler\, Heights in diophantine geometry\, Cambridge University Press\, 2006.\nand/or Part E of Hindry and Silverman\, Diophantine geometry\, Springer\, 2000.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20211103T150000Z DTEND:20211103T163000Z DTSTAMP:20260404T094309Z UID:STAGE/40 DESCRIPTION:Title: Line bundles on complex tori\nby Vijay Srinivasan (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbst ract\nSections I.1 and I.2 of Mumford\, Abelian varieties\, Oxford University Press\, 1970.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Weixiao Lu (MIT) DTSTART:20211110T160000Z DTEND:20211110T173000Z DTSTAMP:20260404T094309Z UID:STAGE/41 DESCRIPTION:Title: Algebraization of complex tori\nby Weixiao Lu (MIT) as part of S TAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract \nSection I.3 of Mumford\, Abelian varieties\, Oxford University Pr ess\, 1970.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Ryan Chen (MIT) DTSTART:20211117T160000Z DTEND:20211117T173000Z DTSTAMP:20260404T094309Z UID:STAGE/42 DESCRIPTION:Title: Moduli spaces of curves and abelian varieties\nby Ryan Chen (MIT ) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin g.\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Yujie Xu (Harvard) DTSTART:20211201T160000Z DTEND:20211201T173000Z DTSTAMP:20260404T094309Z UID:STAGE/43 DESCRIPTION:Title: Betti map and Betti form I\nby Yujie Xu (Harvard) as part of STA GE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: TB A\n LOCATION:https://stable.researchseminars.org/talk/STAGE/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Yujie Xu (Harvard) DTSTART:20211208T150000Z DTEND:20211208T163000Z DTSTAMP:20260404T094309Z UID:STAGE/45 DESCRIPTION:Title: Betti map and Betti form II\nby Yujie Xu (Harvard) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: T BA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20211215T150000Z DTEND:20211215T163000Z DTSTAMP:20260404T094309Z UID:STAGE/46 DESCRIPTION:Title: The height inequality and applications\nby Aashraya Jha (Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simo ns Building.\n\nAbstract\nWe shall look at section 7 of Ziyang Gao's summa ry "Recent Developments of the Uniform Mordell–Lang\nConjecture". We sha ll state the Height Inequality from the paper "Uniformity in Mordell-Lang for curves" by Dimitrov-Gao-Habegger and an application to show a statemen t similar to the New Gap Principle.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20211027T140000Z DTEND:20211027T153000Z DTSTAMP:20260404T094309Z UID:STAGE/47 DESCRIPTION:Title: Vojta's approach to the Mordell conjecture II\nby Niven Achenjan g (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons B uilding.\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's proof.\n\nReferences: Chapter 11 of Bombieri and Gubler\, Heights in diophantine geometry\ , Cambridge University Press\, 2006.\nand/or Part E of Hindry and Silverman\, Diophantine geometry\, Springer\, 2000.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng (MIT) DTSTART:20220223T150000Z DTEND:20220223T163000Z DTSTAMP:20260404T094309Z UID:STAGE/48 DESCRIPTION:Title: Uniform Mordell: review and preview 1\nby Tony Feng (MIT) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbst ract: TBA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng (MIT) DTSTART:20220302T150000Z DTEND:20220302T163000Z DTSTAMP:20260404T094309Z UID:STAGE/49 DESCRIPTION:Title: Uniform Mordell: review and preview 2\nby Tony Feng (MIT) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbst ract: TBA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Cong Wen (Boston University) DTSTART:20220316T140000Z DTEND:20220316T153000Z DTSTAMP:20260404T094309Z UID:STAGE/50 DESCRIPTION:Title: Intersection theory and height inequality 1\nby Cong Wen (Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim ons Building.\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20220330T140000Z DTEND:20220330T153000Z DTSTAMP:20260404T094309Z UID:STAGE/52 DESCRIPTION:Title: Intersection theory and height inequality 2\nby Xinyu Zhou (Bost on University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT S imons Building.\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20220406T140000Z DTEND:20220406T153000Z DTSTAMP:20260404T094309Z UID:STAGE/53 DESCRIPTION:Title: Height bounds for nondegenerate varieties\nby Alice Lin (Harvard ) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin g.\n\nAbstract\nWe will prove the Silverman-Tate theorem in Appendix 5 of [DGH]\, which upper-bounds the difference between the Neron-Tate height an d the Weil height of a point $P$ in an abelian scheme $\\pi: \\mathcal{A}\ \to S$ in terms of the height of the point $\\pi(P)$ in the base scheme. T hen\, we'll apply this result\, together with last week's Proposition 4.1 of [DGH]\, to prove Theorem 1.6 in [DGH]\, which gives a lower bound on th e Neron-Tate height of $P$ in a nondegenerate subvariety $X$ of $\\mathcal {A}\\to S$ in terms of the height of $\\pi(P)$. For this application\, we follow Section 5 of [DGH].\n LOCATION:https://stable.researchseminars.org/talk/STAGE/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20220413T140000Z DTEND:20220413T153000Z DTSTAMP:20260404T094309Z UID:STAGE/54 DESCRIPTION:Title: Proof of the new gap principle 1\nby Niven Achenjang (MIT) as pa rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nA bstract\nOver the next two talks to prove Proposition 7.1 of [DGH] which\, roughly-speaking\, bounds the number of points on a curve within a fixed distance of a given point. In this talk we prepare for the proof of this p roposition by proving a series of lemmas from section 6 of [DGH]. Specific ally\, after stating Proposition 7.1 of [DGH]\, we will prove Theorem 6.2 (which shows non-degeneracy of a certain subvariety of the universal abeli an variety) followed by Lemmas 6.3 and 6.1 (which will be used to obtain t he bound in Proposition 7.1).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20220420T140000Z DTEND:20220420T153000Z DTSTAMP:20260404T094309Z UID:STAGE/55 DESCRIPTION:Title: Proof of the new gap principle 2\nby Aashraya Jha (Boston Univer sity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bui lding.\n\nAbstract\nIn this talk\, we will prove Proposition 7.1 of [DGH]\ , the so called "New Gap Principle". We will first prove a couple of lemma s (Lemma 6.3 and Lemma 6.4 of [DGH]) using techniques from enumerative geo metry which bounds the number of points on a given curve lying in proper s ubsets of a certain product of varieties . We then use height bounds of po ints on non degenerate varieties (Theorem 1.6 and Theorem 6.2 of [DGH]) al ong with lemmas proven to use an inductive argument to prove the New Gap P rinciple.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Fei Hu (Harvard) DTSTART:20220427T140000Z DTEND:20220427T153000Z DTSTAMP:20260404T094309Z UID:STAGE/56 DESCRIPTION:Title: Uniformity for rational points\nby Fei Hu (Harvard) as part of S TAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract \nWe discuss the proof of Proposition 8.1 in [DGH]\, which gives a uniform bound for the intersection of rational points $C(\\overline\\mathbb{Q})$ of a curve $C$ of large modular height in an abelian variety $A$ and a fin ite rank subgroup $\\Gamma\\subseteq A(\\overline\\mathbb{Q})$.\nThe numbe r of large points can be handled by a standard application of the Vojta an d\nMumford inequalities.\nThe key of [DGH] is to bound the number of those small points using the so-called New Gap Principle.\n\nWe then deduce the uniform boundedness of rational/torsion points of curves in [DGH]\, i.e.\ , their Theorems 1.1\, 1.2\, and 1.4\, from the above Proposition 8.1 (for curves of large modular height) and some other classical results (taking care of curves of small modular height).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Anlong Chua DTSTART:20220504T140000Z DTEND:20220504T153000Z DTSTAMP:20260404T094309Z UID:STAGE/57 DESCRIPTION:Title: Unlikely intersection theory and the Ax-Schanuel theorem\nby Anl ong Chua as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nCounting dimensions heuristically tells us whether geometric objects are "likely" or "unlikely" to intersect. For instance\, Bezout's theorem tells us that two curves in $\\mathbb{P}^2$ always inters ect. On the other hand\, two curves in $\\mathbb{P}^3$ are unlikely to int ersect. In number theory\, one is often concerned with unlikely intersecti on problems — for example\, when does a subvariety of an abelian variety contain many torsion points?\n\nIn this talk\, I will try to explain the connections between functional transcendence\, unlikely intersections\, an d number theory. Time permitting\, I will discuss the answer to the questi on posed above and more. On our journey\, we will pass through the fascina ting world of o-minimality\, which I hope to describe in broad strokes.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond van Bommel (Massachusetts Institute of Technology) DTSTART:20220511T140000Z DTEND:20220511T153000Z DTSTAMP:20260404T094309Z UID:STAGE/58 DESCRIPTION:Title: Proof of the amplification principle 1\nby Raymond van Bommel (M assachusetts Institute of Technology) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe will recall the def initions of the Betti map and Betti rank\, and look at the degeneration lo cus of abelian schemes. We will see how these notions are related to each other\, and the bi-algebraic structure that we saw in the previous talk.\n \nAll participants should abide by MIT's COVID policies https://now.mit.ed u/policies/events/\n LOCATION:https://stable.researchseminars.org/talk/STAGE/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Hyuk Jun Kweon (Massachusetts Institute of Technology) DTSTART:20220518T140000Z DTEND:20220518T153000Z DTSTAMP:20260404T094309Z UID:STAGE/59 DESCRIPTION:Title: Proof of the amplification principle 2\nby Hyuk Jun Kweon (Massa chusetts Institute of Technology) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nIn the previous talk\, we proved several results on the Betti rank. In this talk\, we will prove mor e generalized versions of these results. Then we will prove that the rank of Betti become maximal if we take enough iterated fibered products\, unde r some mild conditions.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Schiavone (MIT) DTSTART:20220913T150000Z DTEND:20220913T163000Z DTSTAMP:20260404T094309Z UID:STAGE/60 DESCRIPTION:Title: Brauer groups of fields\nby Sam Schiavone (MIT) as part of STAGE \n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTo pics: Definition of Brauer group in terms of central simple algebras (also known as Azumaya algebras over a field)\; definition of Brauer group in t erms of Galois cohomology\; cyclic algebras\; Brauer groups of finite fiel ds\, local fields\, and global fields (without proofs).\n\nReferences: Poonen\, Rationa l \npoints on varieties\, Section 1.5. See also Gille and Szamuel y\, Central simple algebras and Galois cohomology\, Sections 2.4-2.6\, for some of the topics. Also see Milne\, Class field theory\, Chapter IV and Theo rem VIII.4.2.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20220920T150000Z DTEND:20220920T163000Z DTSTAMP:20260404T094309Z UID:STAGE/61 DESCRIPTION:Title: Review of étale cohomology\nby Kenta Suzuki (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\ nTopic: A crash course on étale cohomology covering étale morphisms\, si tes and cohomology\, and the étale site.\n\nReferences: Poonen\, Rational \npoints on varieties\, Sections 3.5 (just enough to define étale morphism) a nd 6.1-6.4\; or Milne\, Lectures on é\;tale cohomology.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Peng (MIT) DTSTART:20220927T150000Z DTEND:20220927T163000Z DTSTAMP:20260404T094309Z UID:STAGE/62 DESCRIPTION:Title: Brauer groups of schemes\nby Hao Peng (MIT) as part of STAGE\n\n Lecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics : Étale cohomology of $\\mathbb{G}_m$\; definition of cohomological Braue r group of a scheme\; Azumaya algebras\; definition of Azumaya Brauer grou p\; comparison (without proof).\n\nReference: Poonen\, Rational \npoints on varieties\, Section 6.6. See also Colliot-Thé\;lè\;ne and Skorobogatov\ , The Brauer-Grothendieck group\, Sections 3.1-3.3 and Chapter 4.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Haoshuo Fu (MIT) DTSTART:20221004T150000Z DTEND:20221004T163000Z DTSTAMP:20260404T094309Z UID:STAGE/63 DESCRIPTION:Title: The Hochschild-Serre spectral sequence\nby Haoshuo Fu (MIT) as p art of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\n Abstract\nTopics: Spectral sequences\; spectral sequence from a compositio n of functors\; the Hochschild-Serre spectral sequence in group cohomology and étale cohomology.\n\nReference: Poonen\, Rational \npoints on varieties\, Section 6.7.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Weixiao Lu (MIT) DTSTART:20221011T150000Z DTEND:20221011T163000Z DTSTAMP:20260404T094309Z UID:STAGE/64 DESCRIPTION:Title: Residue homomorphisms and examples of Brauer groups\nby Weixiao Lu (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Residue homomorphisms\; purity\; examples o f Brauer groups of schemes.\n\nReferences: Poonen\, Rational \npoints on varieties< /a>\, Sections 6.8-6.9\; Colliot-Thé\;lè\;ne and Skorobogatov\, Th e Brauer-Grothendieck group\, Section 3.7.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20221018T150000Z DTEND:20221018T163000Z DTSTAMP:20260404T094309Z UID:STAGE/65 DESCRIPTION:Title: The Brauer-Manin obstruction\nby Vijay Srinivasan (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbst ract\nTopics: Brauer evaluation\; Brauer set\; Brauer-Manin obstruction to the local-global principle or to weak approximation\; Brauer evaluation i s locally constant.\n\nReference: Poonen\, Rational \npoints on varieties\, Sec tions 8.2.1-8.2.4.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20221025T150000Z DTEND:20221025T163000Z DTSTAMP:20260404T094309Z UID:STAGE/66 DESCRIPTION:Title: The Brauer-Manin obstruction for conic bundles\nby Aashraya Jha (Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Iskovskikh's example\; Brauer gr oups of conic bundles.\n\nReference: Poonen\, Rational \npoints on varieties\, Section 8.2.5\; and Skorobogatov\, Torsors and rational points\, Se ction 7.1.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Anne Larsen (MIT) DTSTART:20221101T150000Z DTEND:20221101T163000Z DTSTAMP:20260404T094309Z UID:STAGE/67 DESCRIPTION:Title: Torsors of algebraic groups over a field\nby Anne Larsen (MIT) a s part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\ n\nAbstract\nTopics: Torsors of groups\; torsors of algebraic groups over a field\; examples\; classification by $H^1$\; operations on torsors.\n\nR eference: Poonen \, Rational \npoints on varieties\, Sections 5.12.1-5.12.5.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Rüd (MIT) DTSTART:20221108T160000Z DTEND:20221108T173000Z DTSTAMP:20260404T094309Z UID:STAGE/68 DESCRIPTION:Title: Torsors over finite fields\, local fields\, and global fields\nb y Thomas Rüd (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Torsors over fields of dimension $\\le 1$\; torsors over local fields\; local-global principle for torsors over global fields.\n\nReference: Poonen\, Rational \npoints on varieties\, Se ctions 5.12.6-5.12.8.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20221115T160000Z DTEND:20221115T173000Z DTSTAMP:20260404T094309Z UID:STAGE/69 DESCRIPTION:Title: Torsors over a scheme\nby Alice Lin (Harvard) as part of STAGE\n \nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopi cs: Torsors over a scheme\; torsor sheaves\; torsors and $H^1$\; geometric operations on torsors.\n\nReference: Poonen\, Rational \npoints on varieties\, Sections 6.5.1-6.5.6.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Daishi Kiyohara (MIT) DTSTART:20221122T160000Z DTEND:20221122T173000Z DTSTAMP:20260404T094309Z UID:STAGE/70 DESCRIPTION:Title: Unramified torsors\nby Daishi Kiyohara (MIT) as part of STAGE\n\ nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopic : Unramified torsors.\n\nReference: Poonen\, Rational \npoints on varieties\, S ection 6.5.7.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Kush Singhal (Harvard) DTSTART:20221129T160000Z DTEND:20221129T173000Z DTSTAMP:20260404T094309Z UID:STAGE/71 DESCRIPTION:Title: An example of descent\nby Kush Singhal (Harvard) as part of STAG E\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nT opics: Example of descent on a genus 2 curve\; explanation in terms of twi sts of a Galois covering.\n\nReference: Poonen\, Rational \npoints on varieties \, Section 8.3.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20221206T160000Z DTEND:20221206T173000Z DTSTAMP:20260404T094309Z UID:STAGE/72 DESCRIPTION:Title: The descent obstruction\nby Niven Achenjang (MIT) as part of STA GE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\n Topics: Evaluation of torsors\; Selmer set\; weak Mordell-Weil theorem\; d escent obstruction.\n\nReference: Poonen\, Rational \npoints on varieties\, Sec tions 8.4.1-8.4.5 and 8.4.7.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/72/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20221213T160000Z DTEND:20221213T173000Z DTSTAMP:20260404T094309Z UID:STAGE/73 DESCRIPTION:Title: The étale-Brauer obstruction and insufficiency of the obstructions< /a>\nby Xinyu Zhou (Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: The étale-Br auer set\; comparison with the descent set\; insufficiency of the obstruct ions for a quadric bundle over a curve.\n\nReference: Poonen\, Rational \npoints on var ieties\, Sections 8.5.2-8.5.3 and 8.6.2.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20230213T210000Z DTEND:20230213T223000Z DTSTAMP:20260404T094309Z UID:STAGE/74 DESCRIPTION:Title: Complex tori and abelian varieties\nby Kenta Suzuki (MIT) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb stract\nWe will define abelian varieties and discuss polarization. We then discuss Riemann's criterion for when a period matrix gives rise to an abe lian variety\, and if we have time\, will see how the Siegel upper-half sp ace parametrizes abelian varieties.\n\nReference: Section 1.1 (and maybe 1 .2) of Genestier a nd Ngo\, Lectures on Shimura varieties.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/74/ END:VEVENT BEGIN:VEVENT SUMMARY:Meng Yan (Brandeis University) DTSTART:20230227T210000Z DTEND:20230227T223000Z DTSTAMP:20260404T094309Z UID:STAGE/75 DESCRIPTION:Title: Quotients of the Siegel upper half space\nby Meng Yan (Brandeis University) as part of STAGE\n\nLecture held in Room 2-135 in the MIT Simo ns Building.\n\nAbstract\nWe will first talk about Riemann's theorem of po larization of complex tori and then give canonical bijections between pola rized abelian varieties and Siegel upper half-spaces. If time permits\, we will also define principal level structures on abelian varieties to build isomorphisms to smooth complex analytic spaces.\n\nReference: The end of Section 1.1\, and Section 1.2 of Genestier and Ngo\, Lectures on Shimura varieties.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/75/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Peng (MIT) DTSTART:20230306T213000Z DTEND:20230306T230000Z DTSTAMP:20260404T094309Z UID:STAGE/76 DESCRIPTION:Title: Moduli space of abelian varieties I\nby Hao Peng (MIT) as part o f STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstr act\nSections 2.1-2.3 of Genestier and Ngo\, Lectures on Shimura varieties. We will firs t finish the part on classifying isomorphism of polarized Abelian varietie s over \\mathbb C\, then introduce dual Abelian schemes\, calculate cohomo logy of line bundles on Abelian varieties\, and verify the representabilit y of the moduli problem \\mathcal A classifying Abelian varieties with pol arizations and Level strictures.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/76/ END:VEVENT BEGIN:VEVENT SUMMARY:Zifan Wang (MIT) DTSTART:20230313T203000Z DTEND:20230313T220000Z DTSTAMP:20260404T094309Z UID:STAGE/77 DESCRIPTION:Title: Moduli space of abelian varieties II\nby Zifan Wang (MIT) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb stract\nSections 2.4.-2.6 of Genestier and Ngo\, Lectures on Shimura varieties. We will finish the proof that the functor $\\mathcal{A}$ is represented by a smoot h quasiprojective scheme. In particular\, to show the smoothness of $\\mat hcal{A}$\, we review Grothendieck and Messing's theorem on deformations of abelian schemes. Finally\, if we have time\, we will give an adelic descr iption of $\\mathcal{A}$ and define Hecke operators.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/77/ END:VEVENT BEGIN:VEVENT SUMMARY:Kush Singhal (Harvard University) DTSTART:20230320T203000Z DTEND:20230320T220000Z DTSTAMP:20260404T094309Z UID:STAGE/78 DESCRIPTION:Title: Review of reductive algebraic groups\nby Kush Singhal (Harvard U niversity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simon s Building.\n\nAbstract\nThis will be a crash course on the theory of redu ctive algebraic groups. We will run through basic definitions and results on affine algebraic groups\, reductive groups\, and tori. This will be fol lowed by a discussion on the Lie algebra and the adjoint representation of a reductive group. Finally\, if time allows\, we will briefly discuss Bor el and parabolic subgroups and their relation to (generalized) flag variet ies. No proofs will be given due to time constraints. We will mostly follo w parts of Milne's book on Algebraic Groups (available at https://math.ucr .edu/home/baez/qg-fall2016/Milne_iAG.pdf) specifically various subsections of chapters 1-4\, 8\, 9\, 12\, 14\, 18\, & 19. I will thus be covering th e prerequisites for Milne's notes on Shimura Varieties (https://www.jmilne .org/math/xnotes/svi.pdf)\, as well as the beginning few subsections of Ch apters 2 and 5 of these notes.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/78/ END:VEVENT BEGIN:VEVENT SUMMARY:Gefei Dang (MIT) DTSTART:20230403T203000Z DTEND:20230403T220000Z DTSTAMP:20260404T094309Z UID:STAGE/79 DESCRIPTION:Title: Hodge structures and variations\nby Gefei Dang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstrac t\nWe will first introduce Hodge structures\, give some examples\, and rep hrase them as representations of the Deligne torus $\\mathbb{S}$. Then we will talk about Hodge tensors\, polarizations\, and variations of Hodge st ructures. Finally\, we will briefly introduce hermitian symmetric domains and realize them as parameter spaces for variations of Hodge structures.\n \nReference: Milne\, Introduction to Shimura varieties\, Chapter 2.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/79/ END:VEVENT BEGIN:VEVENT SUMMARY:Anne Larsen (MIT) DTSTART:20230410T203000Z DTEND:20230410T220000Z DTSTAMP:20260404T094309Z UID:STAGE/80 DESCRIPTION:Title: Hermitian symmetric domains and locally symmetric varieties\nby Anne Larsen (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MI T Simons Building.\n\nAbstract\nThe goal of this week's talk is to give th e necessary background for the definition of a Shimura variety\, to be giv en next week. In the first part of the talk\, we will discuss hermitian sy mmetric domains and their groups of automorphisms (including the homomorph ism from U_1 associated with each point and Cartan involutions on the asso ciated real adjoint group). In the second part\, we will define arithmetic groups and state some of the main theorems about the algebraic variety st ructure and group of automorphisms of the quotients of hermitian symmetric domains by torsion-free arithmetic groups.\n\nReference: Milne\, Introduction to Shimura varie ties\, Chapter 1 and 3.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/80/ END:VEVENT BEGIN:VEVENT SUMMARY:Eunsu Hur (MIT) DTSTART:20230424T203000Z DTEND:20230424T220000Z DTSTAMP:20260404T094309Z UID:STAGE/81 DESCRIPTION:Title: Shimura data and Shimura varieties\nby Eunsu Hur (MIT) as part o f STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstr act\nPrimarily cover Chapter 4-5 of Milne.\n\nDefine congruence subgroup a nd relate to compact open subgroups of $G(\\mathbb{A}_f)$\, no proofs nece ssary. Define connected Shimura datum\, equivalence via Prop. 4.8. Proposi tion 4.9. Define connected Shimura variety. Cover Example 4.14 on Hilbert modular varieties. Give the adelic description in Prop 4.18 and Prop 4.19 .\n\nRemind us of $G^{\\mathrm{der}}$ and $G^{\\mathrm{ad}}$. Define Shimu ra datum\, compare to connected Shimura datum. Give Ex 5.6. Cover Prop 5.7 \, Cor 5.8\, Prop 5.9. Define Shimura varieties. Define a morphism of Shim ura varieties.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/81/ END:VEVENT BEGIN:VEVENT SUMMARY:Dylan Pentland (Harvard University) DTSTART:20230501T203000Z DTEND:20230501T220000Z DTSTAMP:20260404T094309Z UID:STAGE/82 DESCRIPTION:Title: Classification of Shimura varieties\nby Dylan Pentland (Harvard University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simo ns Building.\n\nAbstract\nPrimarily cover Chapters 6-8 of Milne.\n\nRemind us of the definition of a Shimura datum\, and maybe give SV2*-SV6 on p.63 . Sketch the construction of the Siegel modular variety in Chapter 6 and w hy it satisfies SV1-SV6. Show that the Siegel modular variety parametrizes polarized abelian varieties over $\\mathbb{C}$ with symplectic level stru cture.\n\nSummarize Hodge type Shimura varieties as in Chapter 7.\n\nIf yo u have time\, sketch what changes to go from Siegel modular varieties to P EL Shimura varieties (Chapter 8). It would be great to cover some idea of Shimura varieties of abelian type (Chapter 9).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/82/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20230508T203000Z DTEND:20230508T220000Z DTSTAMP:20260404T094309Z UID:STAGE/83 DESCRIPTION:Title: Complex Multiplication\, Shimura-Taniyama formula\nby Niven Ache njang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simo ns Building.\n\nAbstract\nMotivated by the desire to construct canonical m odels of Shimura curves (in the final talk)\, we introduce the theory of c omplex multiplication (CM) of abelian varieties. After briefly discussing the connection between CM and canonical models\, we will cover the basic p roperties of CM abelian varieties\, state the Shimura-Taniyama formula (wi thout proof)\, and then give the main theorem of complex multiplication. O ur main reference for all of this will be chapters 10 and 11 of Milne.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/83/ END:VEVENT BEGIN:VEVENT SUMMARY:Aaron Landesman (MIT) DTSTART:20230515T203000Z DTEND:20230515T220000Z DTSTAMP:20260404T094309Z UID:STAGE/84 DESCRIPTION:Title: Canonical models of Shimura varieties\nby Aaron Landesman (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building. \n\nAbstract\nDefinition of canonical model in Chapter 12\, uniqueness in Chapter 13\, existence in Chapter 14.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/84/ END:VEVENT BEGIN:VEVENT SUMMARY:Daishi Kiyohara (Harvard) DTSTART:20231130T210000Z DTEND:20231130T223000Z DTSTAMP:20260404T094309Z UID:STAGE/85 DESCRIPTION:Title: The Eisenstein quotient\nby Daishi Kiyohara (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstra ct\nDefine the Eisenstein quotient and show it has the properties necessar y to deduce the nonexistence of rational $p$-torsion in elliptic curves ov er $\\mathbb{Q}$ for $p \\ge 11\, p\\ne 13$.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/85/ END:VEVENT BEGIN:VEVENT SUMMARY:Barinder Banwait (Boston University) DTSTART:20231207T210000Z DTEND:20231207T223000Z DTSTAMP:20260404T094309Z UID:STAGE/86 DESCRIPTION:Title: Points of order 13\nby Barinder Banwait (Boston University) as p art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n Abstract\nProve that an elliptic curve over $\\mathbb{Q}$ cannot have a ra tional point of order $13$\, following the paper of Mazur and Tate.\n\nRef erences: [MT]\n LOCATION:https://stable.researchseminars.org/talk/STAGE/86/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20230907T200000Z DTEND:20230907T213000Z DTSTAMP:20260404T094309Z UID:STAGE/87 DESCRIPTION:Title: Overview of the proof\nby Vijay Srinivasan (MIT) as part of STAG E\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nS tate the main theorem and give a summary of the ingredients of the proof.\ n LOCATION:https://stable.researchseminars.org/talk/STAGE/87/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhao Yu Ma (MIT) DTSTART:20230914T200000Z DTEND:20230914T213000Z DTSTAMP:20260404T094309Z UID:STAGE/88 DESCRIPTION:Title: Abelian varieties\nby Zhao Yu Ma (MIT) as part of STAGE\n\nLectu re held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nBasic defin itions regarding abelian varieties and abelian schemes (isogenies\, dual a belian variety\, polarizations)\, Poincaré reducibility theorem\, weak Mo rdell-Weil theorem\n LOCATION:https://stable.researchseminars.org/talk/STAGE/88/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20230921T200000Z DTEND:20230921T213000Z DTSTAMP:20260404T094309Z UID:STAGE/89 DESCRIPTION:Title: Group schemes\nby Alice Lin (Harvard) as part of STAGE\n\nLectur e held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nPreliminarie s on the theory of group schemes with emphasis on finite flat group scheme s (connected-étale sequence\, Cartier duality\, Frobenius/Verschiebung\, Raynaud's theorem)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/89/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Schiavone (MIT) DTSTART:20230928T200000Z DTEND:20230928T213000Z DTSTAMP:20260404T094309Z UID:STAGE/90 DESCRIPTION:Title: Elliptic curves over a local field\nby Sam Schiavone (MIT) as pa rt of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nA bstract\nBasic theory of Weierstrass equations over a DVR (including semis table reduction theorem\, Néron-Ogg-Shafarevich criterion)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/90/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Lu (Harvard) DTSTART:20231005T200000Z DTEND:20231005T213000Z DTSTAMP:20260404T094309Z UID:STAGE/91 DESCRIPTION:Title: Néron models\nby Frank Lu (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nNéron models of elliptic curves\, Néron models of abelian varieties (omitting proof o f existence)\, reduction types of Néron models\n LOCATION:https://stable.researchseminars.org/talk/STAGE/91/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Hu (Harvard) DTSTART:20231012T200000Z DTEND:20231012T213000Z DTSTAMP:20260404T094309Z UID:STAGE/92 DESCRIPTION:Title: Relative Picard functor\nby Daniel Hu (Harvard) as part of STAGE \n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDe fine the relative Picard functor and discuss representability\, discuss th e case of curves and abelian schemes.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/92/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (MIT) DTSTART:20231019T200000Z DTEND:20231019T213000Z DTSTAMP:20260404T094309Z UID:STAGE/93 DESCRIPTION:Title: Jacobians\nby Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLectur e held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDiscuss Jaco bians of smooth curves\, reduced proper curves\, and families of semistabl e curves.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/93/ END:VEVENT BEGIN:VEVENT SUMMARY:Eunsu Hur (MIT) DTSTART:20231026T200000Z DTEND:20231026T213000Z DTSTAMP:20260404T094309Z UID:STAGE/94 DESCRIPTION:Title: Modular forms and Hecke operators\nby Eunsu Hur (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstra ct\nTheory of modular forms and Hecke operators over $\\mathbb{C}$\, modul ar curves over $\\mathbb{C}$ and geometric interpretation of modular forms \, the divisor $[0]-[\\infty]$ on $X_0(p)$\n LOCATION:https://stable.researchseminars.org/talk/STAGE/94/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20231102T200000Z DTEND:20231102T213000Z DTSTAMP:20260404T094309Z UID:STAGE/95 DESCRIPTION:Title: Integral models of modular curves\nby Niven Achenjang (MIT) as p art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n Abstract\nSmooth models of $X_0(N)\, X_1(N)$ over $\\mathbb{Z}[1/N]$ and m odels over $\\mathbb{Z}$ à la Deligne-Rapoport and Katz-Mazur\, consequen ces for the structure of the Néron model of $J_0(p)$\n LOCATION:https://stable.researchseminars.org/talk/STAGE/95/ END:VEVENT BEGIN:VEVENT SUMMARY:Hari Iyer (Harvard) DTSTART:20231109T210000Z DTEND:20231109T223000Z DTSTAMP:20260404T094309Z UID:STAGE/96 DESCRIPTION:Title: Galois representations and modular forms\nby Hari Iyer (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building .\n\nAbstract\nEichler-Shimura relation on the special fiber of $J_0(N)$\, associating Galois representations and abelian varieties to weight 2 cusp forms.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/96/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20231116T210000Z DTEND:20231116T223000Z DTSTAMP:20260404T094309Z UID:STAGE/97 DESCRIPTION:Title: Criterion for the nonexistence of rational $p$-torsion\nby Kenta Suzuki (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Si mons Building.\n\nAbstract\nReduce the proof of the main theorem to showin g that there exists a rank-$0$ quotient $A$ of $J_0(p)$ such that $[0]\\ne [\\infty]$ in $A$.\n\nReference: [Ma1] Section III.5\n LOCATION:https://stable.researchseminars.org/talk/STAGE/97/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond van Bommel (Massachusetts Institute of Technology) DTSTART:20240208T210000Z DTEND:20240208T223000Z DTSTAMP:20260404T094309Z UID:STAGE/98 DESCRIPTION:Title: Height functions\nby Raymond van Bommel (Massachusetts Institute of Technology) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nThis talk will be a survey of the theory of heights. We will consider heights for projective varieties over number fie lds and function fields. We will not only consider finiteness of points of bounded degree and height\, but also the number of such points. If time a llows\, we will consider how the heights associated to different line bund les on a projective variety are related.\n\nThe contents of this talk are based on Chapter 2 of the book. Another good source is Lang's book.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/98/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou DTSTART:20240222T210000Z DTEND:20240222T223000Z DTSTAMP:20260404T094309Z UID:STAGE/99 DESCRIPTION:Title: Normalized heights\nby Xinyu Zhou as part of STAGE\n\nLecture he ld in Room 2-131 in the MIT Simons Building.\n\nAbstract\nChapter 3 of Ser re\, Lectures on the Mordell-Weil theorem\n LOCATION:https://stable.researchseminars.org/talk/STAGE/99/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven T. Achenjang (MIT) DTSTART:20240229T210000Z DTEND:20240229T223000Z DTSTAMP:20260404T094309Z UID:STAGE/100 DESCRIPTION:Title: The Mordell-Weil theorem and Chabauty's theorem\nby Niven T. Ac henjang (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Si mons Building.\n\nAbstract\nChapter 4 and Section 5.1 of Serre\, Lectures on the Mordell-Weil theorem.\n\nThis talk will be split into two parts. In the first part\, we will discuss the Mordell-Weil Theorem\, which states that the abelian group of rational points on an abelian variety $A$ define d over a global field $K$ is finitely generated. We will show that this th eorem follows from some classical finiteness results in algebraic number t heory along with the theory of heights built up in previous talks. Time pe rmitting\, we will conclude the first part by proving a theorem of Neron w hich gives an asymptotic count for the number of points of bounded height on an abelian variety of rank $\\rho$. In the second part\, we will turn o ur attention towards curves of genus $g\\ge2$. For such curves $C/K$\, we will prove Chabauty's Theorem that $C(K)$ is finite if $\\operatorname{ran k}\\operatorname{Jac}(C)(K) < g$ (finiteness of $C(K)$ is now known even w hen $C$'s Jacobian has large rank).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/100/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Lu (Harvard) DTSTART:20240307T210000Z DTEND:20240307T223000Z DTSTAMP:20260404T094309Z UID:STAGE/101 DESCRIPTION:Title: The Demyanenko-Manin method and Mumford's inequality\nby Frank Lu (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Sim ons Building.\n\nAbstract\nIn this talk\, we will discuss two theorems reg arding the number of rational points on curves of genus $g \\geq 2:$ the D emyanenko-Manin theorem and Mumford's inequality. We will begin with the D emyanenko-Manin theorem\, which tells us how the existence of enough funct ions $f_i: C \\rightarrow A\,$ for some abelian variety $A\,$ allows us to show the number of rational points on $C$ is finite. After outlining the proof of this theorem and discussing an application to modular curves\, we will then sketch a proof of Mumford's inequality\, which gives an asympto tic bound on the number of points of bounded height without knowing Faltin g's theorem.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/101/ END:VEVENT BEGIN:VEVENT SUMMARY:Bjorn Poonen (MIT) DTSTART:20240321T200000Z DTEND:20240321T213000Z DTSTAMP:20260404T094309Z UID:STAGE/103 DESCRIPTION:Title: Siegel's method\nby Bjorn Poonen (MIT) as part of STAGE\n\nLect ure held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nWe will sh ow how theorems about diophantine approximation (e.g.\, Roth's theorem tha t irrational algebraic numbers cannot be approximated too well by rational numbers) can be used to prove one of the most famous theorems of 20th cen tury arithmetic geometry\, Siegel's theorem that a hyperbolic affine curve can have only finitely many integral points. The proof is ineffective\, however: 95 years later it is still not known if there is an algorithm tha t takes as input the equation of a curve and returns the list of its integ ral points.\n\nReference: Chapter 7 of Serre\, Lectures on the Mordell-Wei l theorem.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/103/ END:VEVENT BEGIN:VEVENT SUMMARY:Shiva Chidambaram (MIT) DTSTART:20240404T200000Z DTEND:20240404T213000Z DTSTAMP:20260404T094309Z UID:STAGE/104 DESCRIPTION:Title: Baker's method\nby Shiva Chidambaram (MIT) as part of STAGE\n\n Lecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nIn thi s talk\, we will discuss Baker's theorem on lower bounds for linear forms in logarithms\, and how it gives effective bounds for quasi-integral point s on $\\mathbb{P}^1 \\setminus \\{0\,1\,\\infty\\}$. Using coverings\, thi s further yields effective bounds for quasi-integral points on elliptic\, superelliptic and certain hyperelliptic affine curves. We will also discus s an application towards finding elliptic curves with good reduction outsi de a given finite set of places.\n\nReference: Chapter 8 of Serre\, Lectur es on the Mordell-Weil theorem.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/104/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Hu (Harvard) DTSTART:20240411T200000Z DTEND:20240411T213000Z DTSTAMP:20260404T094309Z UID:STAGE/105 DESCRIPTION:Title: The Hilbert irreducibility theorem\nby Daniel Hu (Harvard) as p art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n Abstract\nI will introduce the notion of thin sets and discuss some applic ations to Galois groups of polynomials. Then\, I will state and prove Hilb ert's irreducibility theorem following Serre's account of Lang's proof.\n\ nReference: Chapter 9 of Serre\, Lectures on the Mordell-Weil theorem.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/105/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan DTSTART:20240418T200000Z DTEND:20240418T213000Z DTSTAMP:20260404T094309Z UID:STAGE/106 DESCRIPTION:Title: Construction of Galois extensions\nby Vijay Srinivasan as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbst ract\nThe inverse Galois problem asks whether every finite group can be re alized as the Galois group of a finite extension of $\\mathbb{Q}$. In this talk\, we will discuss the cases of $S_n$ and $\\text{PGL}_2(\\mathbb{F}_ p)$. These methods can also be adapted to apply to the simple groups $A_n$ and $\\text{PSL}_2(\\mathbb{F}_p)$ (for many $p$).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/106/ END:VEVENT BEGIN:VEVENT SUMMARY:Jane Shi DTSTART:20240502T200000Z DTEND:20240502T213000Z DTSTAMP:20260404T094309Z UID:STAGE/107 DESCRIPTION:Title: Construction of elliptic curves of large rank\nby Jane Shi as p art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n Abstract\nIn this talk\, I will first prove Néron's theorem and explain h ow we can use it as a basis to construct elliptic curves of large rank. Th en\, I will discuss two methods for constructing elliptic curves of rank a t least 9 and one method for constructing elliptic curves of rank at least 10. If there is more time\, I will discuss approaches for generating elli ptic curves of rank at least $ 11$.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/107/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Larsen (Massachusetts Institute of Technology) DTSTART:20240425T200000Z DTEND:20240425T213000Z DTSTAMP:20260404T094309Z UID:STAGE/108 DESCRIPTION:Title: The large sieve\nby Daniel Larsen (Massachusetts Institute of T echnology) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simon s Building.\n\nAbstract\nIn this talk\, we will prove a version of the lar ge sieve inequality\, a result from analytic number theory that will event ually be used to give bounds on thin sets. Along the way\, we will prove t he Davenport-Halberstam theorem and generally try to understand how the su pport of a function's Fourier transform influences the function's behavior .\n\nReference: Chapter 12 of Serre\, Lectures on the Mordell-Weil theorem .\nWearing a mask is welcomed\, but optional.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/108/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Peng (MIT) DTSTART:20240509T200000Z DTEND:20240509T213000Z DTSTAMP:20260404T094309Z UID:STAGE/109 DESCRIPTION:Title: Applications of the large sieve to thin sets\nby Hao Peng (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building .\n\nAbstract\nWe review the proof of Cohen-Serre bound on the number of r ational points on projective and affine varieties using the large sieve me thod and Lang-Weil bound on rational points on varieties over finite field s. Notice that stronger bounds are known now by work of Browning\, Heath-B rown and Salberger.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/109/ END:VEVENT BEGIN:VEVENT SUMMARY:Elia Gorokhovsky (Harvard) DTSTART:20240909T143000Z DTEND:20240909T160000Z DTSTAMP:20260404T094309Z UID:STAGE/110 DESCRIPTION:Title: Complex analytic spaces\, vector bundles\, and GAGA\nby Elia Go rokhovsky (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDefinition of the category of complex an alytic spaces\, and statements of GAGA.\n\nReferences: \n\n LOCATION:https://stable.researchseminars.org/talk/STAGE/110/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20240916T143000Z DTEND:20240916T160000Z DTSTAMP:20260404T094309Z UID:STAGE/111 DESCRIPTION:Title: Local systems\, fundamental groupoid\, definition of connection \nby Xinyu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-449 i n the MIT Simons Building.\n\nAbstract\nThe definition of local systems vs . vector bundles. Definition of the fundamental group and fundamental gro upoid of a nice topological space. Statement of equivalence between the c ategory of local systems and the category of finite-dimensional representa tions of the fundamental group. Definition of connection on a vector bund le.\n\nReference: Deligne\, up to Section 2.9.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/111/ END:VEVENT BEGIN:VEVENT SUMMARY:Atticus Wang (MIT) DTSTART:20240923T143000Z DTEND:20240923T160000Z DTSTAMP:20260404T094309Z UID:STAGE/112 DESCRIPTION:Title: Integrable connections\nby Atticus Wang (MIT) as part of STAGE\ n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDef inition of curvature and integrable connection. Statement of the equivale nce between the category of local systems and the category of vector bundl es with integrable connection. Variants: schemes\, relative setting.\n\nR eference: Deligne\, 2.10 to the end of Section 2\; Conrad\, Classical moti vation for the Riemann-Hilbert correspondence. Notes from the talk are att ached under "slides".\n LOCATION:https://stable.researchseminars.org/talk/STAGE/112/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20240930T143000Z DTEND:20240930T160000Z DTSTAMP:20260404T094309Z UID:STAGE/113 DESCRIPTION:Title: Relationship to PDEs and $n$th order differential equations\nby Kenta Suzuki (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nVia local trivializations of vector bund les and connections\, we translate the conditions of horizontal sections a nd flat connections in terms of classical differential equations. We then associate vector bundles with connections to higher order differential equ ations and finally prove an equivalence between the categories of these ob jects (with additional data).\n\nReference: Deligne\, Sections 3 and 4.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/113/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenz Kallal (Princeton University) DTSTART:20241007T143000Z DTEND:20241007T160000Z DTSTAMP:20260404T094309Z UID:STAGE/114 DESCRIPTION:Title: Second order differential equations and projective connections\ nby Kenz Kallal (Princeton University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nReference: Deligne\, Section 5.\n\nIn the previous section\, Deligne sets up an equivalence of categories between order-n differential equations on line bundles on curve s and rank-n vector bundles with connection plus the extra data of a certa in cyclic morphism. \n\nIn section 5\, Deligne reinterprets the special ca se n = 2 in terms of a connection on a certain bundle and another uniformi zation datum called a projective connection. I will prove this alternative equivalence of categories\, focusing on the different ways of viewing and computing with projective connections.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/114/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Barz (Princeton University) DTSTART:20241209T153000Z DTEND:20241209T170000Z DTSTAMP:20260404T094309Z UID:STAGE/115 DESCRIPTION:Title: Irregular connections and the Stokes phenomena\nby Michael Barz (Princeton University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDeligne's book focuses mostly on con nections with regular singularities -- in the 1970s\, Deligne found connections with irregular singularities to be pathological (see his arti cle "Pourquoi un géomètre algébriste s'intéresse-t-il aux connexions i rrégulières?"). But since then\, Deligne\, Malgrange\, Sibuya\, and many others have noticed that irregular connections are home to many interesti ng phenomena which seem to mirror things occurring for ell-adic sheaves.\n \nRegular connections are the simplest to understand since\, by Riemann-Hi lbert\, they are completely determined by the monodromy of their solutions . Unfortunately\, this fails for irregular connections -- there are nontri vial irregular connections whose solutions have no monodromy. In this talk we describe the Stokes data which one can use to help understand irregula r connections.\n\nReference: Malgrange\, Équations Différentielles à Coefficients Polynomiaux\, chapters 3 and 4\nBabbitt and Varadarajan\ , Local moduli for meromorphic differential equations\nDeligne\, Ma lgrange\, and Ramis\, Singularités Irrégulières: Correspondance et d ocuments\, particularly the 19.4.78 letter from Deligne to Malgrange.\ n LOCATION:https://stable.researchseminars.org/talk/STAGE/115/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20241021T143000Z DTEND:20241021T160000Z DTSTAMP:20260404T094309Z UID:STAGE/116 DESCRIPTION:Title: Multivalued functions\, and abstract derivations and connections\nby Xinyu Zhou (Boston University) as part of STAGE\n\nLecture held in R oom 2-449 in the MIT Simons Building.\n\nAbstract\nOur next goal is to stu dy extensions of vector bundles and connections on an open (non-compact) s urfaces. In this talk\, we introduce relevant concepts and their basic pro perties. We first generalize the idea of multivalued functions to sheaves and show its relations to monodromy representations. We also introduce and study the valuations of a function under the action of a connection.\n\nD eligne\, Sections I.6 and the beginning of II.1.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/116/ END:VEVENT BEGIN:VEVENT SUMMARY:Julia Meng (MIT) DTSTART:20241028T143000Z DTEND:20241028T160000Z DTSTAMP:20260404T094309Z UID:STAGE/117 DESCRIPTION:Title: Regular connections in dimension 1\nby Julia Meng (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbs tract\nDefinition of meromorphic vector bundles with regular connections i n dimension one. Meromorphic vector bundles on the punctured disc and mono dromy transformations. Statement that two meromorphic vector bundles on th e punctured disc with regular connections are isomorphic if and only if th ey have the same monodromy.\n\nReference: Deligne\, section II.1.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/117/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (MIT) DTSTART:20241104T153000Z DTEND:20241104T170000Z DTSTAMP:20260404T094309Z UID:STAGE/118 DESCRIPTION:Title: Connections with regular singularities in higher dimension\nby Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nFollowing chapter 2 of Deligne\, we will introduce aspects of regularity for connections on smooth varieties. This includes moderate growth conditions and\, in the normal crossing divi sor case\, connections with regular singularities. Time-permitting\, we wi ll discuss the relations between the various characterizations of regular singularities.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/118/ END:VEVENT BEGIN:VEVENT SUMMARY:Oakley Edens (Harvard) DTSTART:20241118T153000Z DTEND:20241118T170000Z DTSTAMP:20260404T094309Z UID:STAGE/119 DESCRIPTION:Title: Regular connections in dimension $n$\nby Oakley Edens (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building .\n\nAbstract\nFollowing Deligne chapter $2$\, we discuss the relationship between several characterizations of regular connections in higher dimens ions. In particular\, we show that the regularity is "local in codimension $1$ at $\\infty$". Time permitting\, we will prove the Riemann-Hilbert co rrespondence for regular connections.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/119/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20241125T153000Z DTEND:20241125T170000Z DTSTAMP:20260404T094309Z UID:STAGE/120 DESCRIPTION:Title: Proof of the Riemann-Hilbert Correspondence\nby Xinyu Fang (Har vard) as part of STAGE\n\nLecture held in Room 2-255 in the MIT Simons Bui lding.\n\nAbstract\nAfter a quick review of the key concepts we learned be fore\, I will present the statement and a sketch of the proof of the Riema nn-Hilbert correspondence\; namely\, the equivalence of categories between algebraic vector bundles with regular integrable connections and holomorp hic vector bundles with an integrable connection on a complex algebraic va riety (and its analytification\, respectively). After that\, I will also p resent a simple example to illustrate why we should have the regularity co ndition imposed on the algebraic side. We will follow Chapter II Section 5 of Deligne.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/120/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20241202T153000Z DTEND:20241202T170000Z DTSTAMP:20260404T094309Z UID:STAGE/121 DESCRIPTION:Title: de Rham cohomology and Gauss-Manin connections\nby Aashraya Jha (Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe state the equivalence of algebraic a nd analytic de Rham cohomologies for vector bundles with regular integrabl e connections and discuss a relative version. We then discuss the Gauss-Ma nin connection\, which is obtained on the derived pushforward sheaves of a n integrable connection and is a prominent example of connections consider ed in practice. We show that the Gauss-Manin connection is regular if the integrable connection we start with is regular. We will try to provide exa mples along the way to elucidate the theory.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/121/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20250206T220000Z DTEND:20250206T233000Z DTSTAMP:20260404T094309Z UID:STAGE/122 DESCRIPTION:Title: Overview of the Lawrence-Venkatesh proof\nby Niven Achenjang (M IT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Build ing.\n\nAbstract\nThe Mordell Conjecture states that a curve of genus $g\\ ge2$ over a number field can only have finitely many rational points. This was first proved by Faltings in his famous 1983 paper\, but more recently \, a new proof was given by Brian Lawrence and Akshay Venkatesh using $p$- adic methods. In this talk\, after briefly setting up the context of Morde ll's conjecture\, we will discuss\, in broad strokes\, the various ideas a nd results which go into the Lawrence-Venkatesh proof.\n\nReferences: \n\n $\\bullet$ Poonen\, A $p$-adic approach t o rational points on curves\n\n$\\bullet$ Poonen\, $p$-adic approaches to rati onal and integral points on curves\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\n LOCATION:https://stable.researchseminars.org/talk/STAGE/122/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard University) DTSTART:20250213T220000Z DTEND:20250213T233000Z DTSTAMP:20260404T094309Z UID:STAGE/123 DESCRIPTION:Title: A tour of the Betti\, étale\, de Rham\, and crystalline cohomology \nby Xinyu Fang (Harvard University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nThis talk is a cras h course on the properties of various cohomology theories that will be use d in the Lawrence-Venkatesh proof. These include the Betti cohomology\, é tale cohomology\, de Rham cohomology and crystalline cohomology. We review relevant structures on each of these cohomology groups of a smooth proper variety\, state some comparison theorems\, and explain how they come toge ther to form the "big diagram" in the Lawrence-Venkatesh argument.\n\n[Upd ate] I uploaded my outline for the talk to (slides) below. It is only a sk eleton of the talk instead of a complete write-up\, so if you want to revi ew the material\, I recommend reading the first reference (it's only 2 pag es)\, and if you would like more detail\, look into the other references.\ n\nReference: \n\n$\\bullet$ Poonen\, $p$-adic approaches to rational and integral points on curves\, Sections 5 and 6.\n\nFor more details:\n\n$\\bulle t$ Deligne\, Hodge cycles on abelian varieties\, Section 1 (for Betti an d de Rham cohomology).\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine pro blems and $p$-adic period mappings\, Section 3 (to see how cohomology theories are going to be used).\n\n$\\bullet$ Brinon and Conrad\, CMI summer school not es on $p$-adic Hodge theory\, Section 9.1 (for details on the ring $B_ {cris}$ and the functor $D_{cris}$).\n\n$\\bullet$ Nicole\, Cris is for Crystalline (notes for a seminar talk on crystalline cohomolog y) (for the definition and motivations for crystalline cohomology).\n\ n$\\bullet$ Voisin\, H odge theory and complex algebraic geometry I\, Chapter II (for de Rham cohomology and the Hodge decomposition).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/123/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohit Hulse (MIT) DTSTART:20250220T220000Z DTEND:20250220T233000Z DTSTAMP:20260404T094309Z UID:STAGE/124 DESCRIPTION:Title: Period maps and the Gauss-Manin connection\nby Mohit Hulse (MIT ) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin g.\n\nAbstract\nFor a family of smooth projective varieties over a number field\, we have a complex period map and a $p$-adic period map\, and they are both governed by the (algebraic) Gauss-Manin connection. After some pr eliminaries\, we introduce these objects and prove some bounds on the dime nsions of their images.\n\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\, Section 3.\n\nFor more details:\n\n$\\bullet$ Deligne\, Hodge cycles on abelian varieties\ , Section 2 (for Gauss-Manin connection).\n\n$\\bullet$ Katz and Oda\, On the differentiation of De Rham cohomology classes with respect to parameters\n\n$\\bullet$ Voisin\, Hodge Theory and Complex Algebraic Geometry I\, Part III.\n\n$\\bullet$ Hotta\, Takeuchi and Tanisaki\, D-Modules\, Perverse Sheaves\, and Representation Theory ( for more on Riemann-Hilbert).\n LOCATION:https://stable.researchseminars.org/talk/STAGE/124/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20250227T220000Z DTEND:20250227T233000Z DTSTAMP:20260404T094309Z UID:STAGE/125 DESCRIPTION:Title: Families of varieties with good reduction\nby Xinyu Zhou (Bosto n University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Si mons Building.\n\nAbstract\nWe review some constructions on crystalline re presentations and cohomology. Then we present a result in Lawrence-Venkate sh that shows the points in a residue disk that define semisimple represen tations are contained in a proper analytic subset. The proof illustrates t he basic strategy in Lawrence-Venkatesh: to show the finiteness of a set o f points\, one only need to show its image in the period domain is contain ed in a Zariski-closed subset with dimension smaller than that of the orbi t of a point under the complex monodromy group.\n\nReference:\n\n$\\bullet $ L awrence and Venkatesh\, Diophantine problems and $p$-adic period mappings< /a>\, Section 3.\n\n$\\bullet$ Faltings\, Crystalline cohomology and p-adic Galoi s-representations. Algebraic analysis\, geometry\, and number theory ( Baltimore\, MD\, 1988)\, 25–80.\n\n$\\bullet$ Illusie\, Crystalline cohomology. Section 3.Mot ives (Seattle\, WA\, 1991)\, 43–70.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/125/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20250306T220000Z DTEND:20250306T233000Z DTSTAMP:20260404T094309Z UID:STAGE/126 DESCRIPTION:Title: The $S$-unit equation\nby Kenta Suzuki (MIT) as part of STAGE\n \nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe p rove that the $S$-unit equation\, $x+y=1$ where $x\,y\\in\\mathcal O_S^\\t imes$\, has finitely many solutions by using $p$-adic period mappings. To do so we analyze the monodromy and period mapping for (a small modificatio n of) the Legendre family. Although not logically necessary for the proof of Falting's theorem\, many of the key ideas are already present in this s pecial case.\n\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\, Section 4.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/126/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20250403T210000Z DTEND:20250403T223000Z DTSTAMP:20260404T094309Z UID:STAGE/128 DESCRIPTION:Title: Outline of the argument for Mordell's conjecture\nby Xinyu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simon s Building.\n\nAbstract\nI will present an outline of the argument for the proof of Mordell's conjecture\, following Section 5 of Lawrence-Venkatesh . Specifically\, I will give an overview of two key inputs: the existence of a good abelian-by-finite family (the Kodaira-Parshin family) and the fi niteness of rational points whose fiber along the finite map has large Gal ois orbits (proven using p-adic Hodge theory). Then\, I will explain how t o reduce Mordell's conjecture to these key inputs.\n\nReference:\n\n$\\bul let$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappin gs\, Section 5.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/128/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Lu (Harvard) DTSTART:20250410T210000Z DTEND:20250410T223000Z DTSTAMP:20260404T094309Z UID:STAGE/129 DESCRIPTION:Title: Abelian-by-finite families I\nby Frank Lu (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstrac t\nIn this talk\, we will begin the proof that there are only finitely man y rational points whose pre-images along the finite map have large Galois orbits\, introduced in the previous talk. The proof of this statement requ ires two lemmas: a generic simplicity statement\, and a finiteness stateme nt if we consider only the rational points corresponding to a given simple Galois representation. We will begin by presenting the proof\, assuming t hese two lemmas\, before proving the finiteness lemma. \n\nWe will follow part of Lawrence and Venkatesh\, Diophantine problems and $p$-adic period map pings\, Section 6.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/129/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20250417T210000Z DTEND:20250417T223000Z DTSTAMP:20260404T094309Z UID:STAGE/130 DESCRIPTION:Title: Abelian-by-finite families II\nby Vijay Srinivasan (MIT) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb stract\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problem s and $p$-adic period mappings\, second half of Section 6.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/130/ END:VEVENT BEGIN:VEVENT SUMMARY:Elia Gorokhovsky (Harvard) DTSTART:20250424T210000Z DTEND:20250424T223000Z DTSTAMP:20260404T094309Z UID:STAGE/131 DESCRIPTION:Title: The Kodaira--Parshin family\nby Elia Gorokhovsky (Harvard) as p art of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\n Abstract\nIn this talk\, we describe the construction of an abelian-by-fin ite family parametrized by a prime $q$ which is amenable to direct monodro my computations. This family\, the Kodaira-Parshin family\, serves as inpu t to the arguments in Section 6 which use an abelian-by-finite family with full monodromy to prove finiteness of rational points. The bulk of the ta lk focuses on the ``finite'' part of abelian-by-finite: we will describe t he construction of an \\'etale cover of a curve $Y$ parametrizing $G$-cove rs of $Y$ branched at a single point\, together with a universal curve.\n\ nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $ p$-adic period mappings\, Section 7.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/131/ END:VEVENT BEGIN:VEVENT SUMMARY:Joye Chen (MIT) DTSTART:20250501T214500Z DTEND:20250501T231500Z DTSTAMP:20260404T094309Z UID:STAGE/132 DESCRIPTION:Title: Mapping class groups and Dehn twists\nby Joye Chen (MIT) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb stract\nReference:\n\n$\\bullet$ Farb and Margalit\, A Primer on Mapping Class Groups.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/132/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20250508T193000Z DTEND:20250508T210000Z DTSTAMP:20260404T094309Z UID:STAGE/133 DESCRIPTION:Title: Monodromy of the Kodaira--Parshin family\nby Kenta Suzuki (MIT) as part of STAGE\n\nLecture held in Room 4-149\, in the Maclaurin Buildin gs.\n\nAbstract\nWe complete the final step of the proof\, proving that th e monodromy of the Kodaira-Parshin family is large. Let $Y$ be a compact o rientable surface or genus $g\\ge 2$ and let $Y^0$ be the puncture at a po int. Let $(Z_1\,\\pi_1)\,\\dots\,(Z_N\,\\pi_N)$ be the $\\mathrm{Aff}(q)$- covers of $Y^0$\, and let $\\mathrm{MCG}(Y^0)_0$ be those mapping classes of $Y^0$ who induce trivial mapping classes on every $Z_i$. Then $\\mathrm {MCG}(Y^0)_0$ acts on the primitive homomology $H_1^{\\mathrm{Pr}}(Z_i\,Y^ 0)$ of $Z_i$\, i.e.\, the orthogonal complement of $\\pi_i^*H_1(Z_i)\\subs et H_1(Y^0)$. We prove that $\\mathrm{MCG}(Y^0)_0\\to\\prod_{i=1}^N\\mathr m{Sp}(H_1^{\\mathrm{Pr}}(Z_i\,Y^0))$ has Zariski dense image.\n\nBy Goursa t's lemma it suffices to prove each $\\mathrm{MCG}(Y^0)_0\\to\\mathrm{Sp}( H_1^{\\mathrm{Pr}}(Z_i\,Y^0))$ has Zariski dense image\, which we prove by producing many Dehn twists on $Y^0$ inducing trivial mapping classes on $ Z_i$.\n\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine proble ms and $p$-adic period mappings\, Section 8.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/133/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Krishna + Sophie Zhu (Harvard) DTSTART:20250911T203000Z DTEND:20250911T220000Z DTSTAMP:20260404T094309Z UID:STAGE/134 DESCRIPTION:Title: Statements of the Weil conjectures\, proof for curves via the Hodge index theorem\nby Ari Krishna + Sophie Zhu (Harvard) as part of STAGE \n\nLecture held in Room 4-163.\n\nAbstract\nState the Weil conjectures fo r smooth proper varieties over finite fields. Explain the proof for curves via intersection theory on surfaces\, in particular the Hodge index theor em.\n\nReferences: Poonen\, Rational points on varieties\, Chapter 7 up to Section 7.5 .1\; Milne\, The Riem ann Hypothesis over Finite Fields: from Weil to the present day\, page s 8-10.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/134/ END:VEVENT BEGIN:VEVENT SUMMARY:Yutong Chen (MIT) DTSTART:20250918T203000Z DTEND:20250918T220000Z DTSTAMP:20260404T094309Z UID:STAGE/135 DESCRIPTION:Title: The étale site\nby Yutong Chen (MIT) as part of STAGE\n\nLectu re held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nDefine éta le morphisms\, Grothendieck topologies\, the étale site\, and sheaves the reon.\n\nReference: Poonen\, Rational points on varieties\, Section 3.5\, 6.2\, 6.3.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/135/ END:VEVENT BEGIN:VEVENT SUMMARY:Nir Elber (MIT) DTSTART:20250925T203000Z DTEND:20250925T220000Z DTSTAMP:20260404T094309Z UID:STAGE/136 DESCRIPTION:Title: Étale cohomology\nby Nir Elber (MIT) as part of STAGE\n\nLectu re held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nIn the firs t half of the talk\, we will define and explain some properties of $\\ell$ -adic cohomology\, more or less following SGA $4\\frac12$. In the second h alf of the talk\, we will explain what a Weil cohomology theory is and sta te that $\\ell$-adic cohomology is an example of a Weil cohomology theory. If we have any time remaining\, we may gesture towards other Weil cohomol ogy theories.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/136/ END:VEVENT BEGIN:VEVENT SUMMARY:Arav Karighattam (MIT) DTSTART:20251002T203000Z DTEND:20251002T220000Z DTSTAMP:20260404T094309Z UID:STAGE/137 DESCRIPTION:Title: Lefschetz trace formula and proof of rationality and functional equ ation\nby Arav Karighattam (MIT) as part of STAGE\n\nLecture held in R oom 2-105 in the MIT Simons Building.\n\nAbstract\nState the Lefschetz tra ce formula for étale cohomology\, and explain how to apply it to the Frob enius morphism to deduce the Weil conjectures\, excluding the Riemann hypo thesis.\n\nReference: Milne\, Lectures on étale cohomology\, Section 25 and 27.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/137/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (MIT) DTSTART:20251009T203000Z DTEND:20251009T220000Z DTSTAMP:20260404T094309Z UID:STAGE/138 DESCRIPTION:Title: Constructible sheaves\, L-functions and base change theorems\nb y Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLecture held in Room 2-105 i n the MIT Simons Building.\n\nAbstract\nIn this talk\, we will give an ove rview of various foundational tools used for computations in etale cohomol ogy. Time permitting\, this may include constructible sheaves\, their L-fu nctions and the Grothendieck-Lefschetz trace formula\, and the proper base change theorem.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/138/ END:VEVENT BEGIN:VEVENT SUMMARY:Jane Shi (MIT) DTSTART:20251016T203000Z DTEND:20251016T220000Z DTSTAMP:20260404T094309Z UID:STAGE/139 DESCRIPTION:Title: Katz's proof of the Riemann hypothesis for curves\nby Jane Shi (MIT) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Bui lding.\n\nAbstract\nIn this talk\, we'll study a proof of the Riemann Hypo thesis for (projective\, smooth and geometrically connected) curves over finite fields by Katz. We'll study Deligne's version of Rankin's method an d the "connect by curves" lemma\, and how they reduce a proof of RH on gen us $g$ curves to a proof of RH on Fermat curves. Finally\, we'll introduce the "persistence of purity theorem"\, which will be useful for the next t alk.\n\n(Reference: section 1-4 of A Note on Riemann Hypothesis for Curves and Hypersurfaces Over Fini te Fields)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/139/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonid Gorodetskii (MIT) DTSTART:20251023T203000Z DTEND:20251023T220000Z DTSTAMP:20260404T094309Z UID:STAGE/140 DESCRIPTION:Title: Katz's proof of the Riemann hypothesis for hypersurfaces\nby Le onid Gorodetskii (MIT) as part of STAGE\n\nLecture held in Room 2-105 in t he MIT Simons Building.\n\nAbstract\nIn this talk\, we will discuss Katz ’s proof of the Riemann Hypothesis for hypersurfaces in projective space . Building on techniques developed last time\, we will see how the persist ence of purity theorem reduces the problem to explicit cases --- the Ferma t and Gabber hypersurfaces --- and we will complete the verification using Gauss sums.\n\nReference: Katz\, A Note on Riemann Hypothesis for Curves and Hypersurfaces Over Finit e Fields\, Sections 5-8.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/140/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohit Hulse (MIT) DTSTART:20251030T203000Z DTEND:20251030T220000Z DTSTAMP:20260404T094309Z UID:STAGE/141 DESCRIPTION:Title: Deligne's proof in Weil I (Main lemma)\nby Mohit Hulse (MIT) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n \nAbstract\nAfter a quick review of $\\ell$-adic local systems and the ét ale fundamental group\, I will state and prove Deligne's "main lemma." \nI will then derive some consequences to be used in the next few talks\, and if time permits\, explain a key fact about $\\operatorname{Sp}$-invariant s used in the proof.\n\nReferences:
\n$\\bullet$ Milne\, Lectures on Étale Cohomolo gy\, Section 30.
\n$\\bullet$ Deligne\, La Conjecture de W eil. I \, Sections 1-3.
\n$\\bullet$ Fulton and Harris\, Representation Theory Appe ndix F.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/141/ END:VEVENT BEGIN:VEVENT SUMMARY:Jack Miller (Harvard) DTSTART:20251106T213000Z DTEND:20251106T230000Z DTSTAMP:20260404T094309Z UID:STAGE/142 DESCRIPTION:Title: Deligne's proof in Weil I (Lefschetz pencil)\nby Jack Miller (H arvard) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons B uilding.\n\nAbstract\nThe main player in this talk is the notion of a Lefschetz pencil\, a special kind of 1-parameter family of varieties with nice degeneration properties. Because we have discussed how the Riema nn Hypothesis for varieties over finite fields reduces to studying the mid dle cohomology of an even dimensional variety\, we will produce a Lefschet z fibration with odd dimensional fibers whose middle cohomology contains a n "interesting piece\," which we will show has big symplectic monodromy.\n \nReference: Milne\, Lectures on Étale Cohomology\, Section 31-32.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/142/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20251113T213000Z DTEND:20251113T230000Z DTSTAMP:20260404T094309Z UID:STAGE/143 DESCRIPTION:Title: Deligne's proof in Weil I (Completing the proof)\nby Xinyu Zhou (Boston University) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nIn this talk\, I will finish Deligne's proof of the Riemann Hypothesis by applying all the tools developed in the previous talks. Crucially\, the theory of Lefschetz pencils reduces the p roblem to a study of certain higher direct images on P^1. We then use the Lefschetz-Picard formula and the Main Lemma to prove an estimate of the Fr obenius-eigenvalues on the cohomology of the higher direct images\, which is sufficient to deduce the Riemann Hypothesis.\n\nReference: Milne\, Lectures on Étal e Cohomology\, Section 28 and 33.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/143/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (Princeton University) DTSTART:20251120T213000Z DTEND:20251120T230000Z DTSTAMP:20260404T094309Z UID:STAGE/144 DESCRIPTION:Title: Weights and the statement of Weil II\nby Kenta Suzuki (Princeto n University) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Si mons Building.\n\nAbstract\nThe Weil conjecture states that given a smooth projective variety over a finite field\, the Frobenius eigenvalues on the étale cohomology have specific absolute values. As is usual in algebraic geometry\, we may ask for a relative analog: what happens when there is a morphism of schemes? We will introduce weights for étale sheaves on sche mes and formulate Weil II\, which gives a relation between the weights of a sheaf to its pushforward. We will see how this recovers the Weil conject ure\, and record other consequences such as semisimplicity.\n\nReference:\ n\n1. Szamuely\, Section 7.1-7.2.\n\n2. Kiehl-Weissauer\, Weil Conjectures\, P erverse Sheaves and $l$-adic Fourier Transform\, Section I.2\, I.7.\n\ n3. Deligne\, Weil II.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/144/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Hu (Harvard) DTSTART:20251204T213000Z DTEND:20251204T230000Z DTSTAMP:20260404T094309Z UID:STAGE/145 DESCRIPTION:Title: Local Monodromy\nby Daniel Hu (Harvard) as part of STAGE\n\nLec ture held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nWe'll con tinue our discussion of Deligne's generalization of the Weil conjectures t o the relative setting. A theorem of Grothendieck allows us to define the monodromy operator on an ell-adic Galois representation that comes from co homology of smooth proper varieties over local fields. This allows us to d efine a new filtration on H^i\, called the monodromy filtration. The monod romy-weight conjecture states that it coincides with the weight filtration \, up to a shift. We'll apply the case of function fields of curves to ded uce the statement of Weil II.\n\nReference:\n\n1. Szamuely\, A course on the Weil conjectures \, Section 7.6.\n\n2. Kiehl-Weissauer\, Weil Conjectures\, Perverse Sheave s and $l$-adic Fourier Transform\, Section I.3\, I.9.\n\n3. Deligne\, La conjecture de Weil. II\, 1.3\, 1.7\, 1.8?\n LOCATION:https://stable.researchseminars.org/talk/STAGE/145/ END:VEVENT BEGIN:VEVENT SUMMARY:Elia Gorokhovsky (Harvard) DTSTART:20251211T213000Z DTEND:20251211T230000Z DTSTAMP:20260404T094309Z UID:STAGE/146 DESCRIPTION:Title: Applications of Weil II\nby Elia Gorokhovsky (Harvard) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbst ract\nWe state two applications of Deligne's theory of weights. The first is the semisimplicity theorem\, which states that a lisse\, pure $\\overli ne{\\Q_\\ell}$-sheaf on a normal base over $\\mathbb F_q$ decomposes as a direct sum of irreducible subsheaves over $\\overline{\\mathbb{F}_q}$. The second is a very general theorem about equidistribution of Frobenius elem ents in the monodromy group\, which enables proofs of several important re sults in arithmetic statistics\, such as the Sato-Tate conjecture over fun ction fields and a version of the Cohen-Lenstra heuristics.\n\nReference:\ n\n1. Szamuely. A Course on the Weil Conjectures\, Section 7.2.\n\n2. Katz . Gauss Sums\, Kloosterman Sums\, and Monodromy Groups\, Chapter 3.\n\n3. Deligne. Weil II\, Sections 3.4\, 3.5\n\nSee also:\n\n4. Katz\, Sarnak. Ra ndom Matrices\, Frobenius Eigenvalues\, and Monodromy.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/146/ END:VEVENT BEGIN:VEVENT SUMMARY:Aloysius Ng (MIT) DTSTART:20260205T220000Z DTEND:20260205T233000Z DTSTAMP:20260404T094309Z UID:STAGE/147 DESCRIPTION:Title: Torsors of algebraic groups over fields\nby Aloysius Ng (MIT) a s part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\ n\nAbstract\nTo build towards the descent obstruction\, we begin by defini ng torsors. In this talk\, we introduce $G$-torsors under smooth algebraic group $G/k$ as a generalization of simply transitive $G$-sets of group $G $. We discuss its classification and some results that arise from it\, lay ing the foundation for understanding them over arbitrary bases.\n\nReferen ce: Poonen\, Rational points on varieties\, Section 5.12.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/147/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Zhu (Harvard) DTSTART:20260212T220000Z DTEND:20260212T233000Z DTSTAMP:20260404T094309Z UID:STAGE/148 DESCRIPTION:Title: Torsors over an arbitrary base\nby Sophie Zhu (Harvard) as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbs tract\nReference: Poonen\, Rational points on varieties\, Section 6.5 (up to 6. 5.5 or 6.5.6) and review of fppf cohomology.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/148/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20260219T220000Z DTEND:20260219T233000Z DTSTAMP:20260404T094309Z UID:STAGE/149 DESCRIPTION:Title: Descent obstruction and the local-global principle for torsors\ nby Xinyu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nWe start with a quick overview of t he descent obstruction for the Hasse principle\, indicating how torsors an d $H^1(k\,G)$ show up in this context. \nNext\, we introduce contracted pr oducts and twisted torsors\, which will be important for us later. Finally \, we discuss finiteness results for torsors over local fields and the loc al-global principle for torsors. These will be useful when we discuss unra mified torsors and the descent obstruction in more detail later.\n\nRefere nce: \n\n1) Poonen\, Rational points on varieties\, Sections 8.4.7\, 5.12.5-5.1 2.8\, and 6.5.6.\n\n2) Skorobogatov\, Torsors and rational points\, Sectio n 2.2.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/149/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Krishna DTSTART:20260226T220000Z DTEND:20260226T233000Z DTSTAMP:20260404T094309Z UID:STAGE/150 DESCRIPTION:Title: Unramified torsors\nby Ari Krishna as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nWe introduce u nramified torsors as a tool for further studying local-global questions. A class $\\tau \\in H^1(k\,G)$ is unramified at a place v if it extends ove r $\\mathcal O_{k\,v}$\, i.e. if it lies in the image of $H^1(\\mathcal{O} _{k\,v}\,\\mathcal G)\\to H^1(k\,G).$ For a finite set of places $S$ and a n exact sequence $$1\\to \\mathcal G^0 \\to \\mathcal G \\to \\mathcal F \ \to 1$$ with $\\mathcal{G}^0$ having connected fibers and $\\mathcal{F}$ f inite étale\, we prove that the maps $$H^1_S(k\,\\mathcal G) \\to H^1_S( k\,\\mathcal F) \\to \\prod_{v\\in S} H^1(k_v\,F)$$ have finite fibers\, a nd that $H^{1}_S(k\,\\mathcal{G})$ is finite when $k$ is a number field. T hese finiteness results are key to proving the finiteness of Selmer sets\, which\, e.g.\, offers one route to weak Mordell-Weil in the case of abeli an varieties. Along the way\, we analyze torsors over finite fields using Lang’s theorem.\n\n\nReference: Poonen\, Rational points on varieties\, 6.5.7 .\n LOCATION:https://stable.researchseminars.org/talk/STAGE/150/ END:VEVENT BEGIN:VEVENT SUMMARY:Pitchayut (Mark) Saengrungkongka (MIT) DTSTART:20260305T220000Z DTEND:20260305T233000Z DTSTAMP:20260404T094309Z UID:STAGE/151 DESCRIPTION:Title: Examples of descent\nby Pitchayut (Mark) Saengrungkongka (MIT) as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building. \n\nAbstract\nThe main theorem of descent states that if $X$ is a smooth p roper variety over global field $k$\, $G$ is a smooth affine algebraic gro up over $k$\, and $f : Z\\to X$ is a $G$-torsor over $X$\, then the $k$-ra tional points of $X$ correspond to a union of $k$-rational points of finit ely many twists of $Z$. This reduces the problem of finding rational point s of $X$ to finding rational points of finitely many twists of $Z$. We ill ustrate this theorem through several examples\, including the Weak Mordell -Weil Theorem.\n\nReference: Poonen\, Rational points on varieties\, Section 8. 3.\, 8.4.1-2\, 8.4.4-5.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/151/ END:VEVENT BEGIN:VEVENT SUMMARY:Yutong Chen (MIT) DTSTART:20260312T210000Z DTEND:20260312T223000Z DTSTAMP:20260404T094309Z UID:STAGE/152 DESCRIPTION:Title: The descent obstruction\nby Yutong Chen (MIT) as part of STAGE\ n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nIn this talk\, we will briefly review the notion of Selmer sets and how they provide an obstruction to the existence of rational points. We will then p rove a theorem showing that\, the Selmer set is finite in interested cases . Next\, we will define the descent obstruction to the local-global princi ple and compare it with the Brauer-Manin obstruction. Finally\, we will co nstruct explicit torsors over Iskovskikh's surface to demonstrate that it exhibits a descent obstruction and\, in particular\, possesses no rational points. We may also apply descent obstruction to see the failure of stron g approximation if time permits.\n\nReference: Poonen\, Rational points on varieties\, Section 8.1\,8.4\, 8.5.1.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/152/ END:VEVENT BEGIN:VEVENT SUMMARY:Arav Karighattam (MIT) DTSTART:20260319T200000Z DTEND:20260319T213000Z DTSTAMP:20260404T094309Z UID:STAGE/153 DESCRIPTION:Title: $x^2+y^3=z^7$\nby Arav Karighattam (MIT) as part of STAGE\n\nLe cture held in Room 2-143 in the MIT Simons Building.\n\nAbstract\nThe gene ralized Fermat equation $ax^p + by^q = cz^r$ admits a fascinating trichoto my in the theory of its primitive integer solutions: if the invariant $\\c hi=1/p+1/q+1/r-1$ is positive\, there are either zero or infinitely many s olutions (Beukers)\, if $\\chi=0$ it reduces to solving certain elliptic c urves\, and if $\\chi<0$ there are only finitely many solutions (Darmon-Gr anville). The striking similarity between this result and the finiteness of rational points on algebraic curves over $\\Q$ of genus $g\\ge2$ (Falti ngs' theorem) is not a coincidence. In fact\, primitive integer solutions to the generalized Fermat equations correspond to rational points on a "s tacky curve" whose Euler characteristic is $\\chi$. The Riemann existence theorem guarantees us a finite étale covering of this stacky curve by an ordinary curve (which is in our case a branched covering of $\\mathbb{P}^ 1$ with prescribed ramification). In this talk\, I will explain this gene ral theory in the case $\\chi<0$ and focus on the explicit computations du e to Poonen-Schaefer-Stoll\, using twists of the triply branched covering $\\pi\\colon X(7)\\to\\mathbb{P}^1$ and their pullbacks to the punctured a ffine surface $S=\\text{Spec }\\Z[x\,y\,z]/(x^2+y^3-z^7)\\setminus0$ to de termine the primitive integer solutions to $x^2+y^3=z^7$. This is an exam ple of the descent obstruction applied to $G$-torsors over $S$\, where $G= \\text{Aut }\\pi=\\text{PSL}_2(\\mathbb{F}_7)$!\n\n*Note different time and location.\n\nReference: Poonen-Schaefer­-Stoll\, Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$\; Darmon\, Faltings plus eps ilon\, Wiles plus epsilon\, and the Generalized Fermat Equation\; Darmon-Granville\, On the equations $z^m=F(x\,y)$ and $Ax^p+By^ q=Cz^r$.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/153/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonid Gorodetskii (MIT) DTSTART:20260402T210000Z DTEND:20260402T223000Z DTSTAMP:20260404T094309Z UID:STAGE/154 DESCRIPTION:Title: Universal torsors\nby Leonid Gorodetskii (MIT) as part of STAGE \n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nIn this talk\, we focus on torsors under algebraic tori. For an algebraic va riety $X$ with discrete $\\operatorname{Pic}(X_{\\bar k})$ (for instance\, when $X_{\\bar k}$ is rationally connected)\, the dual group $T = \\opera torname{Hom}(\\operatorname{Pic}(X_{\\bar k})\, \\mathbb{G}_m)$ is a natur al algebraic torus associated to $X$. Universal torsors are a class of tor sors over $X$ under $T$ which\, on the one hand\, can capture the set of a ll rational points on $X$ and\, on the other\, often admit an explicit des cription. After developing the theory\, we will see how universal torsors can be used to prove the existence of rational points.\n\nReferences:\n
\n- Witten berg\, Rational points and zero-cycles on rationally connected varieti es over number fields\, Section 3.3. \n
\n- Skorobogatov\, Torsors and Rational Points\, S ection 2.3\, p.25 and the references there.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/154/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Krishna DTSTART:20260409T210000Z DTEND:20260409T223000Z DTSTAMP:20260404T094309Z UID:STAGE/155 DESCRIPTION:Title: Torsors over a diagonal cubic surface\nby Ari Krishna as part o f STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstr act\nReference: Colliot-Thélène\, Kanevsky\, Sansuc\, Arithmétique des surfaces cu biques diagonales\, Section 10(a)(b).\n(English tra nslation)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/155/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20260416T210000Z DTEND:20260416T223000Z DTSTAMP:20260404T094309Z UID:STAGE/156 DESCRIPTION:Title: Descent varieties and Brauer-Manin obstruction on diagonal cubic su rfaces\nby Xinyu Fang (Harvard) as part of STAGE\n\nLecture held in Ro om 2-139 in the MIT Simons Building.\n\nAbstract\nReference: Colliot-Thél ène\, Kanevsky\, Sansuc\, Arithmétique des surfaces cubiques diagonales\, Secti on 10(c) + Proposition 10 from (d).\n(English translati on)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/156/ END:VEVENT BEGIN:VEVENT SUMMARY:Cedric Xiao (MIT) DTSTART:20260423T210000Z DTEND:20260423T223000Z DTSTAMP:20260404T094309Z UID:STAGE/157 DESCRIPTION:Title: Finite descent on curves I\nby Cedric Xiao (MIT) as part of STA GE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\n Reference: Stol l\, Finite Descent Obstructions and Rational Points on Curves\, Sectio n 5-6 (note the erratum)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/157/ END:VEVENT BEGIN:VEVENT SUMMARY:TBA DTSTART:20260430T210000Z DTEND:20260430T223000Z DTSTAMP:20260404T094309Z UID:STAGE/158 DESCRIPTION:Title: Finite descent on curves II\nby TBA as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nReference: Stoll\, Finit e descent obstructions and rational points on curves\, Sections 7-9 (note the erratum)\n LOCATION:https://stable.researchseminars.org/talk/STAGE/158/ END:VEVENT BEGIN:VEVENT SUMMARY:TBA DTSTART:20260507T210000Z DTEND:20260507T223000Z DTSTAMP:20260404T094309Z UID:STAGE/159 DESCRIPTION:Title: Finite descent obstruction on curves and modularity\nby TBA as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\ nAbstract\nReference: Helm\, Voloch\, Finite descent obstruction on curves and modularity.\n LOCATION:https://stable.researchseminars.org/talk/STAGE/159/ END:VEVENT BEGIN:VEVENT SUMMARY:TBA DTSTART:20260514T210000Z DTEND:20260514T223000Z DTSTAMP:20260404T094309Z UID:STAGE/160 DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-139 in the M IT Simons Building.\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/STAGE/160/ END:VEVENT END:VCALENDAR