BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ananth Shankar (MIT) DTSTART:20200506T150000Z DTEND:20200506T160000Z DTSTAMP:20260404T111133Z UID:CMI/1 DESCRIPTION:Title: A finiteness criterion for 2-dimensional representations of surface gro ups.\nby Ananth Shankar (MIT) as part of CMI seminar series\n\n\nAbstr act\nLet C be a a complex algebraic curve of genus \\geq 1\, and let pi be its fundamental group. Let \\rho: pi\\rightarrow \\GL_2(\\C) be a semisim ple 2-dimensional representation\, such that \\rho(\\alpha) has finite ord er for every simple closed loop \\alpha. We will prove that \\rho has fini te image. If time permits\, we will mention applications of this result to the Grothendieck-Katz p-curvature conjecture. This is joint work with Ana nd Patel and Junho Peter Whang.\n LOCATION:https://stable.researchseminars.org/talk/CMI/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Chandrasekhar Raju (École Polytechnique Fédérale Laussane) DTSTART:20200508T123000Z DTEND:20200508T133000Z DTSTAMP:20260404T111133Z UID:CMI/2 DESCRIPTION:Title: Connections between the circle method\, trace formula and bounds for th e subconvexity problem.\nby Chandrasekhar Raju (École Polytechnique F édérale Laussane) as part of CMI seminar series\n\n\nAbstract\nAfter int roducing the sub-convexity problem for L-functions in a general context\, we will focus our attention to the particular case of Rankin-Selberg L-fun ctions. We will briefly trace the history of this particular problem start ing from Kowalski\, Michel\, and Vanderkam\, with a lot of authors in betw een upto the seminal work of Michel\, Venkatesh. I will then try to explai n how the circle method enters this question by sketching an argument of M unshi for what is perhaps the simplest case i.e character twists of GL(2) L-functions. I will end the talk by explaining how we can solve the Subcon vexity problem for Rankin-Selberg L-functions in the combined level aspect by a very easy version of the circle method and see how this approach is connected to earlier work on the same problem.\n LOCATION:https://stable.researchseminars.org/talk/CMI/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Akash Sengupta (Columbia University) DTSTART:20200511T150000Z DTEND:20200511T160000Z DTSTAMP:20260404T111133Z UID:CMI/3 DESCRIPTION:Title: Geometric invariants and geometric consistency of Manin's conjecture.\nby Akash Sengupta (Columbia University) as part of CMI seminar series\ n\n\nAbstract\nLet X be a Fano variety with an associated height function defined over a number field. Manin's conjecture predicts that\, after remo ving a thin set\, the growth of the number of rational points of bounded h eight on X is controlled by certain geometric invariants (e.g. the Fujita invariant of X). I will talk about how to use birational geometric methods to study the behaviour of these invariants and propose a geometric descri ption of the thin set in Manin's conjecture. Part of this is joint work wi th Brian Lehmann and Sho Tanimoto.\n LOCATION:https://stable.researchseminars.org/talk/CMI/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohan Swaminathan (Princeton University) DTSTART:20200513T150000Z DTEND:20200513T160000Z DTSTAMP:20260404T111133Z UID:CMI/4 DESCRIPTION:Title: A concrete approach to virtual classes in genus 0 Gromov--Witten theory \nby Mohan Swaminathan (Princeton University) as part of CMI seminar s eries\n\n\nAbstract\nGromov--Witten theory is concerned with counting curv es inside (smooth) projective varieties satisfying some incidence conditio ns (e.g. how many rational degree d curves pass through 3d-1 generic point s in the complex projective plane?). In general (due to complications aris ing from the fact that spaces of curves may not be smooth)\, instead of co unting curves directly\, we need to use intersection theory on the space o f curves to define certain "virtual counts". In the first half of the talk \, we will provide the necessary background (spaces of curves\, "expected dimension"\, compactness and some examples of curve counting). In the seco nd half of the talk\, we will describe a concrete differential geometric a pproach to these "virtual counts" for genus 0 curves in projective varieti es (using ideas coming from the theory of psuedo-holomorphic curves in sym plectic manifolds).\n LOCATION:https://stable.researchseminars.org/talk/CMI/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Akashdeep Dey (Princeton University) DTSTART:20200518T150000Z DTEND:20200518T160000Z DTSTAMP:20260404T111133Z UID:CMI/5 DESCRIPTION:Title: A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory.\nby Akashdeep Dey (Princeton University) as part of CMI seminar series\n \n\nAbstract\nMin-max theory for the area functional was developed by Almg ren and Pitts to construct closed minimal hypersurfaces in an arbitrary cl osed Riemannian manifold. There is an alternate approach via PDE to the co nstruction of minimal hypersurfaces. This approach is based on the study o f the limiting behaviour of solutions to the Allen-Cahn equation. In my ta lk\, I will briefly describe the Amgren-Pitts min-max theory and the Allen -Cahn min-max theory and discuss the question to what extent these two the ories agree.\n LOCATION:https://stable.researchseminars.org/talk/CMI/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alapan Mukhopadhyay (University of Michigan\, Ann Arbor) DTSTART:20200520T150000Z DTEND:20200520T160000Z DTSTAMP:20260404T111133Z UID:CMI/6 DESCRIPTION:Title: Singularities in positive characteristics.\nby Alapan Mukhopadhyay (University of Michigan\, Ann Arbor) as part of CMI seminar series\n\n\nAb stract\nIn the first half\, we shall look at the several notions of singul arities of a polynomial function at a point from both analytic and algebro -geomteric point of view. We will indicate the surprising similarities of the results coming from these two seemingly different directions. In the s econd half\, these ideas will be put into a more general context detailing more on the positive characteristic side. We shall discuss the notion of F-split\, F-regular schemes\, how these notions are related to the charact eristic zero singularities. We shall end by mentioning some open problems relating singularities in characteristic zero and positive characteristic. This talk is an exposition of the ideas developed in the last fifty years in different contexts.\n LOCATION:https://stable.researchseminars.org/talk/CMI/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Arnab Saha (Indian Institute of Technology (IIT) Gandhinagar) DTSTART:20200525T123000Z DTEND:20200525T133000Z DTSTAMP:20260404T111133Z UID:CMI/7 DESCRIPTION:Title: $p$-adic Hodge theory and delta geometry\nby Arnab Saha (Indian Ins titute of Technology (IIT) Gandhinagar) as part of CMI seminar series\n\n\ nAbstract\nWe will talk about a new $p$-adic Galois representation that co mes from $\\delta$-geometry. Here\, we construct a new filtered Isocrystal associated to an abelian scheme that is different from the usual crystall ine cohomology. In the case of elliptic curves\, depending on a modular pa rameter\, this Isocrystal is also weakly admissible which leads to a new c rystalline Galois representation attached to the elliptic curve via the Fo ntaine functor. This is joint work with Jim Borger. \n\nWe will dedicate t he first half of the talk on giving an overview of $p$-adic Hodge theory a nd $\\delta$-geometry.\n LOCATION:https://stable.researchseminars.org/talk/CMI/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Utsav Choudhury (Indian Statistical Institute (ISI)\, Kolkata) DTSTART:20200710T123000Z DTEND:20200710T140000Z DTSTAMP:20260404T111133Z UID:CMI/8 DESCRIPTION:Title: Unstable motivic homotopy theory and few commutative algebra problems\nby Utsav Choudhury (Indian Statistical Institute (ISI)\, Kolkata) as p art of CMI seminar series\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/CMI/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Siddhi Pathak (Pennsylvania State University) DTSTART:20200529T123000Z DTEND:20200529T133000Z DTSTAMP:20260404T111133Z UID:CMI/9 DESCRIPTION:Title: Arithmetic nature of special values of L-functions\nby Siddhi Patha k (Pennsylvania State University) as part of CMI seminar series\n\n\nAbstr act\nThe study of L-functions has occupied a center stage in number theory since the work of Riemann and Dirichlet. A standard example of an L-funct ion is the Riemann zeta-function\, $\\zeta(s)$\, given by the series $\\s um_{n=1}^{\\infty} n^{-s}$ when $\\Re(s)>1$. The aim of this talk will be to discuss the question of determining the arithmetic nature (that is\, ra tional/irrational and algebraic/transcendental) of values of L-functions a t positive integers. For example\, it is expected that the values $\\zeta( m)$ are transcendental for all integers $m >1$. However\, the only known c ases of this conjecture are the even zeta-values\, which Euler had explici tly evaluated in the 1730s. Among the odd zeta-values\, Apery proved that $\\zeta(3)$ is irrational\, whereas the irrationality of the remaining odd zeta-values remains a mystery. \n\nIn this talk\, we will discuss various facets of this problem. If time permits\, we will prove that for a fixed odd positive integer m\, all the values $\\zeta_K(m)$ are irrational as K varies over imaginary quadratic fields\, with at most one possible excepti on. This is joint work with Ram Murty. This talk will be accessible to a w ide audience.\n LOCATION:https://stable.researchseminars.org/talk/CMI/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Ronno Das (University of Chicago) DTSTART:20200708T123000Z DTEND:20200708T133000Z DTSTAMP:20260404T111133Z UID:CMI/10 DESCRIPTION:Title: Noncollinear points in the projective plane\nby Ronno Das (Univers ity of Chicago) as part of CMI seminar series\n\n\nAbstract\nWe will look at the space of n points in the projective plane such that no three lie on the same line. There is an action of the symmetric group by permuting the points and we can compute the action on cohomology (for n < 7) by countin g the numbers of such n tuples over the finite field F_q\, with a 'twist'. Unfortunately for n > 6 such a computation is still hard and we expect th e answer for large n to be arbitrarily complicated (in the sense of Mnëv' s universality). This talk will be based on joint work with Ben O'Connor.\ n LOCATION:https://stable.researchseminars.org/talk/CMI/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Arul Shankar (University of Toronto) DTSTART:20200603T150000Z DTEND:20200603T160000Z DTSTAMP:20260404T111133Z UID:CMI/11 DESCRIPTION:Title: Squarefree sieves in arithmetic statistics\nby Arul Shankar (Unive rsity of Toronto) as part of CMI seminar series\n\n\nAbstract\nA classical question in analytic number theory is: given a polynomial with integer co efficients\, how often does it take squarefree values? In arithmetic stati stics\, we are particularly interested in the case of discriminant polynom ials. In this talk\, I will present several different cases of this questi on. First\, we will consider a classical result of Davenport--Heilbronn wh ich considers the case of discriminants of binary cubic forms. Then\, I wi ll discuss joint work with Bhargava in which we consider the case of discr iminants of ternary cubic forms.\n\nThird\, I will describe joint and ongo ing work with Bhargava and Wang\, in which we consider different families of degree-n polynomials in one variable\, and determine the proportion of those having squarefree discriminant. Finally\, I will describe various ap plications of these results\n LOCATION:https://stable.researchseminars.org/talk/CMI/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Abhishek Oswal (University of Toronto) DTSTART:20200608T150000Z DTEND:20200608T160000Z DTSTAMP:20260404T111133Z UID:CMI/12 DESCRIPTION:Title: A non-archimedean definable Chow theorem.\nby Abhishek Oswal (Univ ersity of Toronto) as part of CMI seminar series\n\n\nAbstract\nO-minimali ty has had some striking applications to number theory.\nThe utility of o- minimal structures originates from the remarkably\ntame topological proper ties satisfied by sets definable in such\nstructures. Despite the rigidity that it imposes\, the theory is\nsufficiently flexible to allow for a ran ge of analytic constructions.\nAn illustration of this `tame' property is the following surprising\ngeneralization of Chow's theorem proved by Peter zil and Starchenko -\nA closed analytic subset of a complex algebraic vari ety that is also\ndefinable in an o-minimal structure\, is in fact algebra ic. While the\no-minimal machinery aims to capture the archimedean order t opology of the\nreal line\, it is natural to wonder if such a machinery ca n be set up over\nnon-archimedean fields. In this talk\, we explore a non- archimedean\nanalogue of an o-minimal structure and describe a version of the definable\nChow theorem in this context.\n LOCATION:https://stable.researchseminars.org/talk/CMI/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Baskar Balasubramanyam (Indian Institute of Science Education and Research (IISER)\, Pune) DTSTART:20200611T123000Z DTEND:20200611T133000Z DTSTAMP:20260404T111133Z UID:CMI/13 DESCRIPTION:Title: Construction of p-adic L-functions\nby Baskar Balasubramanyam (Ind ian Institute of Science Education and Research (IISER)\, Pune) as part of CMI seminar series\n\n\nAbstract\nIn this talk\, I will discuss some of t he constructions of p-adic L-functions that interpolates families of class ical special values. Finally\, I will talk about the construction of the p -adic adjoint L-functions using overconvergent cohomology. I will try to k eep this talk accessible to as wide an audience as possible.\n LOCATION:https://stable.researchseminars.org/talk/CMI/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Padmavathi Srinivasan (University of Georgia) DTSTART:20200615T150000Z DTEND:20200615T160000Z DTSTAMP:20260404T111133Z UID:CMI/14 DESCRIPTION:Title: Topological invariants in arithmetic geometry\nby Padmavathi Srini vasan (University of Georgia) as part of CMI seminar series\n\n\nAbstract\ nThis will be a gentle introduction to two independent themes.\n\nThe firs t half of the talk will focus on conductors and discriminants\, two invari ants that measure degeneration in a family of a hyperelliptic curves. We w ill show how a combinatorial refinement helps us compare these two invaria nts.\n\nThe second half of the talk will be be an introduction to A^1 enum erative geometry\, i.e.\, how we may use quadratic forms to encode additio nal arithmetic information in enumerative problems in algebraic geometry.\ n LOCATION:https://stable.researchseminars.org/talk/CMI/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Ved Datar (Indian Institute of Science (IISc) Bangalore) DTSTART:20200617T123000Z DTEND:20200617T133000Z DTSTAMP:20260404T111133Z UID:CMI/15 DESCRIPTION:Title: (Inverse)-Hessian type equations and positivity in complex algebraic g eometry\nby Ved Datar (Indian Institute of Science (IISc) Bangalore) a s part of CMI seminar series\n\n\nAbstract\nIn the early 2000's Demailly a nd Paun proved that a (1\,1) cohomology class on a K\\"ahler manifold is p ositive if and only if certain intersection numbers are positive. This is a generalization of the classical Nakai criteria for ampleness of line bun dles on projective manifolds. The proof\, somewhat surprisingly\, relies o n Yau's work on the complex Monge-Ampere equation\, and his solution to th e Calabi conjecture. In 2019\, Gao Chen extended the method of Demailly-Pa un to prove that another important PDE in Kahler geometry\, namely the J-e quation\, is equivalent to the positivity of certain (twisted) intersectio n numbers\, thereby settling a conjecture of Lejmi and Szekelyhidi. In my talk\, I will describe this circle of ideas\, concluding with a recent res ult obtained in collaboration with Vamsi Pingali extending the work of Gao Chen to more general inverse Hessian type equations\, thereby settling a conjecture of Szekelyhidi for projective manifolds. In the process we obta in an equivariant version of Gao Chen's result\, and in particular recover some results of Collins and Szekelyhidi on the J-equation on toric manifo lds.\n LOCATION:https://stable.researchseminars.org/talk/CMI/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Swarnava Mukhopadhyay (Tata Institute of Fundamental Research (TIF R) Mumbai) DTSTART:20200622T123000Z DTEND:20200622T133000Z DTSTAMP:20260404T111133Z UID:CMI/16 DESCRIPTION:Title: Graph potentials and Moduli spaces of vector bundles of curves\nby Swarnava Mukhopadhyay (Tata Institute of Fundamental Research (TIFR) Mumb ai) as part of CMI seminar series\n\n\nAbstract\nWe construct and study gr aph potentials\, a collection of Laurent polynomials associated with color ed graphs of small valency. The potentials we construct are related to the moduli spaces of vector bundles of rank two with fixed determinant on alg ebraic curves. We will discuss relations between these graph potentials an d Gromov-Witten invariants of the moduli spaces. This is joint work with P . Belmans and S. Galkin.\n LOCATION:https://stable.researchseminars.org/talk/CMI/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Yajnaseni Dutta (University of Bonn) DTSTART:20200624T123000Z DTEND:20200624T133000Z DTSTAMP:20260404T111133Z UID:CMI/17 DESCRIPTION:Title: Positivity of direct image sheaves\nby Yajnaseni Dutta (University of Bonn) as part of CMI seminar series\n\n\nAbstract\nPositivity properti es of direct image sheaves have deep implications in\nthe geometry of fami lies of varieties. For instance\, the existence of\nenough global sections of pushforwards of higher tensors of the relative\ncanonical bundle of a family\, puts certain restrictions on the kinds of\nvarieties that can app ear on the fibres etc. I will discuss some these\npositivity properties\, especially the ones that come as a generalisation\nof the Fujita conjectur e and its application to the Iitaka conjecture. This is\npartially a joint work with Takumi Murayama.\n LOCATION:https://stable.researchseminars.org/talk/CMI/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Nivedita Bhaskar (University of Southern California) DTSTART:20200629T150000Z DTEND:20200629T160000Z DTSTAMP:20260404T111133Z UID:CMI/18 DESCRIPTION:Title: Brauer p-dimensions of complete discretely valued fields\nby Nived ita Bhaskar (University of Southern California) as part of CMI seminar ser ies\n\n\nAbstract\n(This is joint work with Bastian Haase) To every centra l simple algebra A over a field F\, one can associate two numerical Brauer class invariants called the index(A) and the period(A). It is well known from that index(A) divides high powers of per(A). The Brauer dimension of a field F is defined to be the least number n such that index(A) divides p eriod(A)^n for every central simple algebra A defined over any finite exte nsion of F. Similarly there exist analogous notions of Brauer-p-dimensions of fields. The 'period-index' questions revolve around bounding the Braue r (p) dimensions of arbitrary fields.\n\nIn this talk\, we will look at th e period-index question over complete discretely valued fields in the so-c alled 'bad characteristic' case (i.e when the residue field has characteri stic p). We will give a flavour of the known results for this question and discuss progress for the cases when the residue fields have small 'p-rank s'. Finally\, we will propose a (still open!) conjecture which very precis ely relates the Brauer p-dimensions of the complete discretely valued fiel ds to the p-ranks of the residue fields\, along with some evidence via a f amily of examples. The key idea involves working with Kato's filtrations a nd bounding the symbol length of the second Milnor K group modulo p in a c oncrete manner\, which further relies on the machinery of differentials in characteristic p as developed by Cartier.\n LOCATION:https://stable.researchseminars.org/talk/CMI/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Chiranjib Mukherjee (University of Münster) DTSTART:20200701T123000Z DTEND:20200701T133000Z DTSTAMP:20260404T111133Z UID:CMI/19 DESCRIPTION:Title: The Kardar-Parisi-Zhang equation in $d\\geq 3$ and the Gaussian free f ield\nby Chiranjib Mukherjee (University of Münster) as part of CMI s eminar series\n\n\nAbstract\nThe Kardar-Parisi-Zhang (KPZ) equation is a s ingular stochastic partial differential equation (SPDE) and belongs to a l arge class of models known as the KPZ universality class\, which is believ ed to exhibit very different behavior than Gaussian universality class and describe \nthe long-time of a wide class of systems including some noisy SPDEs\, driven lattice gases\, randomly growing interfaces and directed po lymers in random media. In spatial dimension one\, recently this class has been studied extensively based on approximations by exactly solvable mode ls. which no longer exist if perturbations appear in the approximating m odels\, or when higher dimensional models are investigated. When the spati al dimension is at least three\, it was conjectured that two disjoint univ ersality classes co-exist when the long-time/large-scale behavior of the s olutions are studied. We will report some recent progress along these dire ctions.\n LOCATION:https://stable.researchseminars.org/talk/CMI/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Agnid Banerjee (TIFR Centre For Applicable Mathematics\, Bangalore ) DTSTART:20200706T123000Z DTEND:20200706T133000Z DTSTAMP:20260404T111133Z UID:CMI/20 DESCRIPTION:Title: The structure of the regular and the singular set of the free boundary in the obstacle problem for fractional heat equation.\nby Agnid Baner jee (TIFR Centre For Applicable Mathematics\, Bangalore) as part of CMI se minar series\n\n\nAbstract\nIn this talk\, I will discuss the structure of the free boundary in the obstacle problem for fractional powers of the he at operator. Our results are derived from the study of a lower dimensional obstacle problem for a class of local\, but degenerate\, parabolic equati ons. The analysis will be based on new Almgren\, Weiss and Monneau type mo notonicity formulas and the associated blow-up analysis. This is a joint w ork with D. Danielli\, N. Garofalo and A. Petrosyan.\n LOCATION:https://stable.researchseminars.org/talk/CMI/20/ END:VEVENT END:VCALENDAR