BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Antoine Ducros (Sorbonne University) DTSTART:20211001T120000Z DTEND:20211001T130000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/1 DESCRIPTION:Title: Non-standard analysis and non-archimedean geometry\nby Anto ine Ducros (Sorbonne University) as part of Non-Archimedean and Tropical G eometry\n\n\nAbstract\nIn a joint work with E. Hrushovski and F. Loeser\, we show that certain one-parameter families of complex integrals have a li mit that can be expressed as an integral on a Berkovich space over the fie ld $\\mathbb C((t))$ (in the sense of the theory of integration of real di fferential forms on Berkovich spaces\, developed by Chambert-Loir and myse lf). In this talk I will present this result\, but rather focus on the gen eral method we introduced to prove it\, which we hope will be useful for a lot of other situations involving a "t-adic limit of one-parameter famili es of complex objects". It consists in introducing a huge non-standard mod el of $\\mathbb C$ also equipped with a non-archimedean absolute value\; w orking on such a model enables by design to deal at the same time with lim its of usual complex objects (through non-standard analysis) as well as wi th non-archimedean objects\, allowing for a direct comparison between thes e two worlds.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Joe Rabinoff (Duke University) DTSTART:20211001T131500Z DTEND:20211001T141500Z DTSTAMP:20260404T094533Z UID:NonArchTrop/2 DESCRIPTION:Title: Weakly smooth forms and Dolbeault cohomology of curves\nby Joe Rabinoff (Duke University) as part of Non-Archimedean and Tropical Geo metry\n\n\nAbstract\nGubler and I work out a theory of weakly smooth forms on non-Archimedean analytic spaces closely following the construction of Chambert-Loir and Ducros\, but in which harmonic functions are forced to b e smooth. We call such forms "weakly smooth". We compute the Dolbeault coh omology groups of rig-smooth\, compact non-Archimedean curves with respect to this theory\, and show that they have the expected dimensions and sati sfy Poincaré duality. We carry out this computation by giving an alternat ive characterization of weakly smooth forms on curves as pullbacks of cert ain "smooth forms" on a skeleton of the curve. This yields an isomorphism between the Dolbeault cohomology of the skeleton\, which can be computed u sing standard combinatorial methods\, and the Dolbeault cohomology of the curve.\nThis work is joint with Walter Gubler.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Yue Yu (California Institute of Technology) DTSTART:20211001T143000Z DTEND:20211001T153000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/3 DESCRIPTION:Title: Generalizing GKZ secondary fan using Berkovich geometry\nby Tony Yue Yu (California Institute of Technology) as part of Non-Archimede an and Tropical Geometry\n\n\nAbstract\nGelfand-Kapranov-Zelevinski introd uced the notion of secondary fan in the study of the Newton polytopes of d iscriminants and resultants. It also controls the geometric invariant theo ry for toric varieties. We propose a generalization of the GKZ secondary f an to general Fano varieties using ideas from Berkovich geometry and Mori theory. Furthermore\, inspired by mirror symmetry\, we propose a synthetic construction of a universal family of Kollár-Shepherd-Barron-Alexeev sta ble pairs over the toric variety associated to the generalized secondary f an. This generalizes the families of Kapranov-Sturmfels-Zelevinski and Ale xeev in the toric case. We gave a detailed construction and proved the sta bility in the case of del Pezzo surfaces. This is joint work with Hacking and Keel.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Kris Shaw (University of Oslo) DTSTART:20211015T120000Z DTEND:20211015T130000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/4 DESCRIPTION:Title: A tropical approach to the enriched count of bitangents to quar tic curves\nby Kris Shaw (University of Oslo) as part of Non-Archimede an and Tropical Geometry\n\n\nAbstract\nUsing A1 enumerative geometry Lars on and Vogt have provided an enriched count of the 28 bitangents to a quar tic curve. In this talk\, I will explain how these enriched counts can be computed combinatorially using tropical geometry. I will also introduce an arithmetic analogue of Viro’s combinatorial patchworking for real algeb raic curves which\, in some cases\, retains enough data to recover the enr iched counts. Finally\, I will comment on a possible tropical approach to the enriched count of the 27 lines on a cubic surface of Kass and Wickelgr en. This talk is based on joint work with Hannah Markwig and Sam Payne.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Melody Chan (Brown University) DTSTART:20211015T131500Z DTEND:20211015T141500Z DTSTAMP:20260404T094533Z UID:NonArchTrop/5 DESCRIPTION:Title: Homology representations of compactified configurations on grap hs applied to $M_{2\,n}$\nby Melody Chan (Brown University) as part of Non-Archimedean and Tropical Geometry\n\n\nAbstract\nThe homology of a co mpactified configuration space of a graph is\nequipped with commuting acti ons of a symmetric group and the outer\nautomorphism group of a free group . We construct an efficient free\nresolution for these homology representa tions. Using the Peter-Weyl\nTheorem for symmetric groups\, we consider ir reducible representations\nindividually\, vastly simplifying the calculati on of these homology\nrepresentations from the free resolution.\nAs our ma in application\, we obtain computer calculations of the top\nweight ration al cohomology of the moduli spaces M_{2\,n}\, equivalently\nthe rational h omology of the tropical moduli spaces Δ_{2\,n}\, as a\nrepresentation of S_n acting by permuting point labels for all n≤10.\nWe further give new multiplicity calculations for specific irreducible\nrepresentations of S_n appearing in cohomology for n≤17. Our approach\nproduces information ab out these homology groups in a range well\nbeyond what was feasible with p revious techniques. Joint work with\nChristin Bibby\, Nir Gadish\, and Cla udia He Yun.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniele Turchetti (University of Warwick) DTSTART:20211015T143000Z DTEND:20211015T153000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/6 DESCRIPTION:Title: Schottky spaces and moduli of curves over $\\mathbb{Z}$\nby Daniele Turchetti (University of Warwick) as part of Non-Archimedean and Tropical Geometry\n\n\nAbstract\nSchottky uniformization is the descriptio n of an analytic curve as the quotient of an open dense subset of the proj ective line by the action of a Schottky group.\nAll Riemann surfaces can b e uniformized in this way\, as well as some non-archimedean curves\, calle d Mumford curves.\nIn this talk\, I will present a construction of universal Mumford curves: analytic spaces that parametrize both arc himedean and non-archimedean uniformizable curves of a fixed genus.\nThis result relies on the existence of suitable moduli spaces for marked Schott ky groups\, that can be built using the theory of Berkovich spaces over ri ngs of integers of number fields developed by Poineau.\n\nAfter introducin g Berkovich analytic geometry from the beginning\, I will describe univers al Mumford curves and explain how these can be related to combinatorial st ructures arising from the theories of tropical moduli and geometric group theory.\nThis is based on joint work with Jérôme Poineau.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Elise Goujard (Bordeaux University) DTSTART:20211029T120000Z DTEND:20211029T130000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/7 DESCRIPTION:Title: Counting (tropical) covers: quasimodularity of generating funct ions\nby Elise Goujard (Bordeaux University) as part of Non-Archimedea n and Tropical Geometry\n\n\nAbstract\nCounting (weighted) ramified coveri ngs of the sphere or the torus lead \nto several applications\, one of the m is the (weighted) count of \ninteger points in some moduli spaces of fla t surfaces\, leading to the \nevaluation of the Masur-Veech volumes or the Siegel-Veech constants of \nthese moduli spaces. Both these quantities ar e relevant to the study \nof dynamics in polygonal billiards for instance\ , but also to other \ndynamical problems on flat surfaces (such as the cou nt of closed \ngeodesics).\nWith that motivation in mind\, I will explain a joint work with Martin \nMöller on the generating series of these count s : they are quasimodular and this property holds "graph by graph". Some o f our results can be stated in terms of tropical covers and I will detail the relation of our results with a previous work of Böhm-Bringmann-Buchho lz-Markwig.\nI will also give several questions and conjectures about the \n"completed cycles" that appear naturally in this setting.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Walter Gubler (University of Regensburg) DTSTART:20211029T131500Z DTEND:20211029T141500Z DTSTAMP:20260404T094533Z UID:NonArchTrop/8 DESCRIPTION:Title: Forms on Berkovich spaces based on harmonic tropicalizations\nby Walter Gubler (University of Regensburg) as part of Non-Archimedean and Tropical Geometry\n\n\nAbstract\nChambert-Loir and Ducros introduced s mooth forms and currents on Berkovich spaces using tropicalization maps in duced by morphisms to tori. In joint work with Philipp Jell und Joe Rabino ff\, we allow more generally harmonic tropicalization maps to define a lar ger class of weakly smooth forms which has essentially the same properties as the smooth forms\, but have a better cohomological behavior.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Navid Nabijou (University of Cambridge) DTSTART:20211029T143000Z DTEND:20211029T153000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/9 DESCRIPTION:Title: Tropical expansions and rubber torus actions\nby Navid Nabi jou (University of Cambridge) as part of Non-Archimedean and Tropical Geom etry\n\n\nAbstract\nGiven a normal crossings pair (X\,D)\, its tropicaliza tion can be defined as the cone over the dual intersection complex of D. A polyhedral subdivision of the tropicalization induces a degeneration of X \, called a tropical expansion. To first approximation\, this is obtained by attaching additional "bubble" irreducible components to X\, along strat a in D. Tropical expansions form a natural class of degenerations which ad mit nice combinatorial descriptions. They have been studied and exploited by many authors\, in many contexts.\n\nWe investigate automorphisms of tro pical expansions covering the identity on X. We discover a purely tropical description: the so-called rubber torus is the torus associated to the mo duli space of tropical edge lengths in the polyhedral subdivision\, and it s action on each component of the expansion is encoded in a linear "tropic al position map." Our main application is to logarithmic enumerative geome try\, which I will motivate\, but the talk will not focus on this. This is joint work with Francesca Carocci.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Jan Steffen Müller (University of Groningen) DTSTART:20211112T130000Z DTEND:20211112T140000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/10 DESCRIPTION:Title: $p$-adic Arakelov theory on abelian varieties and quadratic Ch abauty\nby Jan Steffen Müller (University of Groningen) as part of No n-Archimedean and Tropical Geometry\n\n\nAbstract\nI will discuss a new co nstruction of p-adic height functions on abelian varieties over number fie lds using Besser's p-adic Arakelov theory. In analogy with Zhang's constru ction of Néron-Tate heights via adelic metric\, these heights are given i n terms of canonical p-adic adelic metrics on line bundles. As an applicat ion\, I will describe a new and simplified approach to the quadratic Chaba uty method for the computation of rational points on certain curves. This is joint work in progress with Amnon Besser and Padmavathi Srinivasan.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Klaus Künnemann (University of Regensburg) DTSTART:20211112T141500Z DTEND:20211112T151500Z DTSTAMP:20260404T094533Z UID:NonArchTrop/11 DESCRIPTION:Title: Pluripotential theory for tropical toric varieties and non-arc himedean Monge-Ampère equations\nby Klaus Künnemann (University of R egensburg) as part of Non-Archimedean and Tropical Geometry\n\n\nAbstract\ nTropical toric varieties are partial compactifications of finite dimensio nal real vector spaces associated with rational polyhedral fans. We introd uce pluripotential theory on tropical toric varieties. This theory provide s a canonical correspondence between complex and non-archimedean pluripote ntial theories of invariant plurisubharmonic functions on toric varieties. We apply this correspondence to solve invariant non-archimedean Monge-Amp ère equations on toric and abelian varieties over arbitrary non-trivially valued non-archimedean fields. This is joint work with José Ignacio Burg os Gil\, Walter Gubler and Philipp Jell.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Stefan Wewers (Ulm University) DTSTART:20211112T153000Z DTEND:20211112T163000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/12 DESCRIPTION:Title: Explicit models of curves via nonarchimedian geometry\nby Stefan Wewers (Ulm University) as part of Non-Archimedean and Tropical Geo metry\n\n\nAbstract\nI will report on my long term efford to make the comp utation of the semistable reduction of curves over p-adic fields effective and practical. I will focus on some particular cases\, and on the use of methods from nonarchimedian analytic geometry to achieve this goal.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Vlerë Mehmeti (University Paris-Saclay) DTSTART:20211126T130000Z DTEND:20211126T140000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/13 DESCRIPTION:Title: A Hasse Principle on Berkovich Analytic Curves\nby Vlerë Mehmeti (University Paris-Saclay) as part of Non-Archimedean and Tropical Geometry\n\nLecture held in Seminar Room of the CMLS\, École Polytechniqu e (Palaiseau\, France).\n\nAbstract\nPatching techniques\, under various f orms and inspired from results in complex analysis\, have in the past been used as an approach to the inverse Galois problem. \nRecently\, these tec hniques have become a very important tool in the study of local-global pri nciples. I will explain how patching can be adapted to Berkovich \nanalyti c curves. Working in this setting\, one can then obtain several local-glob al principles\, all of which are applicable to quadratic forms.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Erwan Brugallé (Nantes University) DTSTART:20211126T140000Z DTEND:20211126T150000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/14 DESCRIPTION:Title: Euler characteristic and signature of real semi-stable degener ations\nby Erwan Brugallé (Nantes University) as part of Non-Archimed ean and Tropical Geometry\n\nLecture held in Seminar Room of the CMLS\, É cole Polytechnique (Palaiseau\, France).\n\nAbstract\nIt is interesting to compare the Euler characteristic of the real part of a real algebraic var iety to the signature of its complex part. For example\, a theorem by Iten berg and Bertrand states that both quantities are equal for "primitive T-h ypersurfaces". After defining these latter\, I will give a motivic proof o f this theorem via the motivic nearby fiber of a real semi-stable degenera tion. This proof extends in particular the original statement by Itenberg and Bertrand to non-singular tropical varieties.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Jérôme Poineau (University of Caen Normandy) DTSTART:20211126T153000Z DTEND:20211126T163000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/15 DESCRIPTION:Title: Analytic dynamics over $\\mathbf{Z}$ and torsion points of ell iptic curves\nby Jérôme Poineau (University of Caen Normandy) as par t of Non-Archimedean and Tropical Geometry\n\nLecture held in Seminar Room of the CMLS\, École Polytechnique (Palaiseau\, France).\n\nAbstract\nLet $Y$ be a Berkovich space over $\\mathbf{Z}$. Recall that such a space nat urally contains non-Archimedean parts (such as usual $p$-adic Berkovich sp aces) and Archimedean parts (such as complex analytic spaces). Denote by $ X$ the relative projective line over $Y$. For each point $y$ in $Y$\, let $\\mu_y$ be a measure defined on the fiber $X_y$ (which is an analytic pr ojective line over the complete residue field associated to $y$). Inspire d by the work of Favre on endomorphisms on hybrid Berkovich spaces\, we pr ove general continuity results for families of measures of the form $(\\mu _y)_{y\\in Y}$ coming from dynamical systems on $X$. Following a strategy by DeMarco-Krieger-Ye\, we then deduce new cases of a conjecture of Bogomo lov-Fu-Tschinkel on uniform bounds on the number of common images on $P^1$ of torsion points of two elliptic curves.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Claudia Fevola (Max Planck Institute Leipzig) DTSTART:20211210T130000Z DTEND:20211210T140000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/16 DESCRIPTION:Title: Kp Solitons from Tropical Limits\nby Claudia Fevola (Max P lanck Institute Leipzig) as part of Non-Archimedean and Tropical Geometry\ n\n\nAbstract\nIn this talk\, we study solutions to the Kadomtsev-Petviash vili equation whose underlying algebraic curves undergo tropical degenerat ions. Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which pa rametrizes all tau functions arising from such a sum. After introducing so litons solutions\, we compute tau functions from points on the Sato Grassm annian that represent Riemann-Roch spaces.\nThis is joint work with Daniel e Agostini\, Yelena Mandelshtam and Bernd Sturmfels.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/16/ END:VEVENT BEGIN:VEVENT SUMMARY:David Holmes (Leiden University) DTSTART:20211210T141500Z DTEND:20211210T151500Z DTSTAMP:20260404T094533Z UID:NonArchTrop/17 DESCRIPTION:Title: Piecewise polynomials and intersection theory\nby David Ho lmes (Leiden University) as part of Non-Archimedean and Tropical Geometry\ n\n\nAbstract\nRather than studying the Chow ring of a variety X\, it is b ecoming\nincreasing popular to study some kind of limit of Chow rings over \nblowups of X\; for example this arises in enumerative geometry and in th e\nstudy of singular hermitian metrics. Philosophically this limit might b e\nthought of as the Chow ring of the Riemann-Zariski space of X\, or\nper haps of the valuativisation of X\; but we will not dwell on such\nquestion s. Rather\, we will explain how piecewise-polynomial functions on\nthe tro picalisation of X give an efficient way to write down certain\n'tautologic al' elements of this 'limit Chow ring'\, and describe\napplications to enu merative problems\, and a Sage implementation for the\nmoduli space of sta ble curves.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Dimitri Wyss (École Polytechnique Fédérale de Lausanne) DTSTART:20211210T153000Z DTEND:20211210T163000Z DTSTAMP:20260404T094533Z UID:NonArchTrop/18 DESCRIPTION:Title: DT-invariants from non-archimedean integrals\nby Dimitri W yss (École Polytechnique Fédérale de Lausanne) as part of Non-Archimede an and Tropical Geometry\n\n\nAbstract\nLet $M(\\beta\,\\chi)$ be the modu li space of one-dimensional semi-stable sheaves on a del Pezzo surface $S$ \, supported on an ample curve class $\\beta$ and with Euler-characteristi c $\\chi$.\n\nWorking over a non-archimedean local field $F$\, we define a natural measure on the $F$-points of $M(\\beta\,\\chi)$. We prove that th e integral of a certain gerbe on $M(\\beta\,\\chi)$ with respect to this m easure is independent of $\\chi$ if $S$ is toric. A recent result of Mauli k-Shen then implies that these integrals compute the Donaldson-Thomas inva riants of $M(\\beta\,\\chi)$. A similar result holds for suitably twisted Higgs bundles. This is joint work with Francesca Carocci and Giulio Orecch ia.\n LOCATION:https://stable.researchseminars.org/talk/NonArchTrop/18/ END:VEVENT END:VCALENDAR