The geometry of coherent sheaves: From derived categories to Higgs bundles
2023-02-06 15:00 - 17:30 (Europe/Berlin) | Frankfurt, Hilbertraum, Rober-Mayer-Str. 6-8, Raum 302
GAUS-Workshop: “Invariants and curve counting”
15:00-16:00: Luca Battistella (Frankfurt):
Logarithmic and orbifold Gromov-Witten invariants
Abstract: Logarithmic Gromov-Witten theory can be thought of as the study of curves in open manifolds, or, in other words, curves with tangency conditions to a boundary divisor. When the divisor is smooth, several techniques have been developed to compute the invariants, most notably orbifold stable maps. When the divisor is normal crossings, on the other hand, the logarithmic theory remains hardly accessible. The strategy of rank reduction, i.e. looking at the components of the boundary one at a time, is more directly applicable to other theories than the logarithmic one (as shown in Nabijou-Ranganathan and B.-Nabijou-Tseng-You) due to tropical obstructions. Inspired by one of the distinguishing features of the logarithmic theory – namely, birational invariance [Abramovich-Wise] – in joint work with Nabijou and Ranganathan we show that, when the genus is zero, tropical obstructions can be disposed of by blowing up the target sufficiently. The slogan is that the logarithmic theory is the limit orbifold theory under birational modifications along the boundary divisor. If time permits I will discuss work in progress towards understanding negative contact.
16:20-17:20: Georg Oberdieck (Stockholm):
Pandharipande-Thomas theory of elliptic threefolds and Jacobi forms
Abstract: Pandharipande-Thomas theory is the study of the intersection theory of the moduli space of stable pairs of a threefold. The intersection numbers, called Pandharipande-Thomas invariants, may be viewed as counting curves on the threefold subject to given incidence conditions. In this talk we explore the properties of the generating series of Pandharipande-Thomas invariants of elliptically fibered threefolds. There will be two main conjectures: Quasi-Jacobi Property and Holomorphic Anomaly Equations. Together these essentially determine the modular properties of the generating series. The conjectures are motivated by the case of Calabi-Yau threefolds where by mirror symmetry computations Huang-Katz-Klemm conjectured that the series of PT invariants are Jacobi forms. I discuss several examples, in particular the equivariant geometry of K3xA^1. Here the conjectures lead to explicit new formulas for the invariants. Based on joint work with Maximilian Schimpf.

Asymptotic equidistribution for partition statistics and topological invariants
2023-02-07 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Throughout mathematics, the equidistribution properties of certain objects are a central theme studied by many authors. In my talk I am going to speak about a joint project with William Craig and Joshua Males, where we provide a general framework for proving asymptotic equidistribution, convexity, and log-concavity of coefficients of generating functions on arithmetic progressions.
Giulia Cesana (University of Cologne)
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

Revisiting derived crystalline cohomology
2023-02-10 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar Arithmetic Geometry
Zhouhang Mao (Paris)
Projectively generated âˆž-categories and left derived functors turn out to be important in derived geometry. In this talk, we will present the result of the âˆž-category of surjections of animated rings being projectively generated, the notion of animated PD-pairs â€” surjections of animated rings with a â€œderivedâ€ PD-structure, and how to use these tools to study the crystalline and prismatic cohomology. In particular, we will deduce various comparison theorems without finiteness conditions.
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

tba
2023-04-14 08:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Sachi Hashimoto (MPI Leipzig)
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

Arithmetic theta series from CM cycles
2023-04-18 14:00 - 16:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 308
Seminar: Non-archimedean geometry
Lucas Gerth (Universität Frankfurt)
Abstract: We study arithmetic analogues of theta series. Given a simplectic vector space V and a Schwartz function f on V, there is a collection of cycles Z(n,f), consisting of CM points, on the Siegel modular variety. Assuming that f satisfies a strong regular semisimple condition at some prime p, we show that the generating series of the degrees of the cycles Z(n,f) is a modular form, We identify it explicitly with a classical theta series for a quaternion unitary similitude group. The proof relies on the p-adic uniformization of the supersingular locus on the Siegel modular variety.

On quasimodular forms associated to projective representations of symmetric groups
2023-04-18 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
We explain how one can naturally associate a quasimodular form to a representation of a symmetric group. We determine its growth and explain how this construction is applied to several problems in enumerative geometry. Finally, we discuss the difference between linear and projective representations. This is based on joint work with Adrien Sauvaget.
Jan-Willem van Ittersum (MPIM Bonn)
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

The geometry of coherent sheaves: From derived categories to Higgs bundles
2023-04-24 15:00 - 17:30 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
GAUS-Workshop: “Recent developments in GIT”
14:00-15:00: Victoria Hoskins (Nijmegen, speaking remotely): An introduction to geometric invariant theory
Abstract: The aim of this survey talk is to give an introduction to geometric invariant theory in order to prepare the audience for the subsequent talks as requested by the organisers. I will start by explaining how group actions often appear in moduli problems and we will see how constructing algebra-geometric quotients is related to 19th century invariant theory. I will explain why the theory is simplest for non-reductive group actions and, in this case, I will explain how Mumford constructs quotients (of certain open ‘semistable’ subsets) using geometric invariant theory, as well as giving combinatorial and numerical criteria for semistability. If there is time, I will briefly mention some recent developments to extend GIT to certain non-reductive group actions.
15:20-16:20: Joshua Jackson (Sheffield): Advances in Non-reductive GIT and applications

Abstract: Following from the previous talk on reductive GIT, I will survey recent developments in extending this theory to non-reductive groups, with a particular focus on applications to moduli theory. Time permitting, I will then indicate how non-reductive GIT can be used in the study of sheaves, Higgs bundles, hypersurfaces, and singular curves.
16:40-17:40: Dario Weissmann (Essen): A stacky approach to identify the semi-stable locus of vector bundles
Abstract: I report on recent joint work with Xucheng Zhang focusing on our Theorem A for vector bundles in characteristic 0: The semi-stable locus in the stack of bundles over a smooth projective curve is the maximal open locus admitting a schematic good moduli space. This gives an intrinsic motivation for semi-stability of vector bundles. Historically, semi-stability appeared in the quest for a moduli space of bundles and the classical construction of this moduli space uses a non-canonical GIT-construction. Theorem A also provides us with natural examples of good moduli spaces which are only algebraic spaces and not schemes.

Comparison of tame and log-étale cohomology
2023-04-25 14:00 - 16:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 308
Seminar: Non-archimedean geometry
Cancelled: postponed by one week
Amine Koubaa (Universität Frankfurt)
Abstract: tba

Almost holomorphic Drinfeld modular forms
2023-04-25 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Oguz Gezmis (Heidelberg University)
In his series of papers from 1970s, Shimura analyzed a non-holomorphic operator, nowadays called the Maass-Shimura operator, and later extensively studied almost holomorphic modular forms. He also discovered their role on constructing class fields as well as the connection with periods of CM elliptic curves. In this talk, our first goal is to introduce their positive characteristic counterpart, almost holomorphic Drinfeld modular forms. We further relate them to Drinfeld quasi-modular forms which leads us to generalize the work of Bosser and Pellarin to a certain extend. Moreover, we introduce the Maass-Shimura operator $delta_k$ in our setting for any nonnegative integer k and investigate the relation between the periods of CM Drinfeld modules and the values at CM points of arithmetic Drinfeld modular forms under the image of $delta_k$. If time permits, we also reveal how to construct class fields by using such values. This is a joint work with Yen-Tsung Chen.
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

Gluing sheaves along Harder-Narasimhan strata of Bun_2
2023-05-02 14:00 - 16:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 308
Seminar: Non-archimedean geometry
Jonathan Miles (Universität Frankfurt)
Abstract: We compute some examples of gluing sheaves on the moduli stack of rank 2 vector bundles on the Fargues-Fontaine curve. In the case of prime-to-p torsion coefficients, the category D_ét(Bun_G) can be thought of as an approximation of the automorphic data appearing in the geometrization of the local Langlands correspondence due to Fargues-Scholze. The stratification of Bun_G arising from the Harder-Narasimhan slope formalism on G-isocrystals yields a semi-orthogonal decomposition of D_ét(Bun_G) into the derived categories of smooth representations of inner forms of Levi subgroups of G. Between such categories there is a full six functor formalism that can be used to compute how sheaves arising on a quasi-compact open substack interact with sheaves on higher strata via nearby cycles functors, which can be interpreted as some derived analogue of Jacquet restriction functors for parabolic subgroups of G up to inner twisting. We restrict to G=GL_2 and to sufficiently nice coefficients (notably this includes an algebraic closure of F_ell and Z/ell^n Z for almost all ell prime to p), and we will explain how these computations fundamentally reduce to the étale cohomology of local Shimura varieties (more generally local shtuka spaces).

Mass equidistribution for Saito-Kurokawa lifts
2023-05-02 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Abhishek Saha (Queen Mary University of London)
The Quantum Unique Ergodicity (QUE) conjecture was proved in the classical case for Maass forms of full level in the eigenvalue aspect by Lindenstrauss and Soundararajan, and for holomorphic forms in the weight aspect by Holowinsky and Soundararajan. In this talk, I will discuss some joint work with Jesse Jaasaari and Steve Lester on the analogue of the QUE conjecture in the weight aspect for holomorphic Siegel cusp forms of degree 2 and full level. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for Saito–Kurokawa lifts as the weight tends to infinity. As an application, we prove the equidistribution of zero divisors of Saito-Kurokawa lifts.
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

A p-Adic 6-Functor Formalism on Rigid-Analytic Varieties
2023-05-04 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Lucas Mann (Münster)
Abstract: Using Clausen-Scholze’s theory of condensed mathematics, we construct a full 6-functor formalism for p-adic sheaves on rigid-analytic varieties. As a special case of this formalism we obtain Poincaré duality for the étale F_p-cohomology of smooth proper rigid-analytic varieties. By applying the formalism to classifying stacks of p-adic groups, we obtain new insights into the p-adic Langlands program.

p-adic Gross--Zagier and rational points on modular curves
2023-05-05 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Faltings’ theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings’ theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross–Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross–Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross–Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points.

The SYZ conjecture for families of hypersurfaces
2023-05-05 14:00 - 15:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2023)
Léonard Pille-Schneider (ENS, Paris)
Abstract: Let X -> D* be a polarized family of complex Calabi-Yau manifolds, whose complex structure degenerates in the worst possible way. The SYZ conjecture predicts that the fibers X_t, as t ->0, degenerate to a tropical object; and in particular the program of Kontsevich and Soibelman relates it to the Berkovich analytification of X, viewed as a variety over the non-archimedean field of complex Laurent series.
I will explain the ideas of this program and some recent progress in the case of hypersurfaces.

Tropical spin Hurwitz numbers
2023-05-05 15:30 - 16:30 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2023)
Lou-Jean Cobigo (Universität Tübingen)
Abstract: Classical Hurwitz numbers count the number of branched covers of a fixed target curve that exhibit a certain ramification behaviour. It is an enumerative problem deeply rooted in mathematical history.
A modern twist: Spin Hurwitz numbers were introduced by Eskin-Okounkov-Pandharipande for certain computations in the moduli space of differentials on a Riemann surface.
Similarly to Hurwitz numbers they are defined as a weighted count of branched coverings of a smooth algebraic curve with fixed degree and branching profile. In addition, they include information about the lift of a theta characteristic of fixed parity on the base curve.
In this talk we investigate them from a tropical point of view. We start by defining tropical spin Hurwitz numbers as result of an algebraic degeneration procedure, but soon notice that these have a natural place in the tropical world as tropical covers with tropical theta characteristics on source and target curve.
Our main results are two correspondence theorems stating the equality of the tropical spin Hurwitz number with the classical one.

Tropical functions on skeletons: a finiteness result
2023-05-05 16:45 - 17:45 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2023)
Antoine Ducros (Sorbonne Université, Paris)
Abstract: Skeletons are subsets of non-archimedean spaces (in the sense of Berkovich) that inherit from the ambiant space a natural PL (piecewise-linear) structure, and if S is such a skeleton, for every invertible holomorphic function f defined in a neighborhood of S, the restriction of log |f| to S is PL.
In this talk, I will present a joint work with E.Hrushovski F. Loeser and J. Ye in which we consider an irreducible algebraic variety X over an algebraically closed, non-trivially valued and complete non-archimedean field k, and a skeleton S of the analytification of X defined using only algebraic functions, and consisting of Zariski-generic points. If f is a non-zero rational function on X then log |f| induces a PL function on S, and if we denote by E the group of all PL functions on S that are of this form, we prove the following finiteness result on the group E: it is stable under min and max, and there exist finitely many non-zero rational functions f_1,…f_m on X such that E is generated, as a group equipped with min and max operators, by the log |f_i| and the constants |a| for a in k^*. Our proof makes a crucial use of Hrushovski-Loeser’s model-theoretic approach of Berkovich spaces.

Comparison of tame and log-étale cohomology
2023-05-09 14:00 - 16:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 308
Seminar: Non-archimedean geometry
Amine Koubaa (Universität Frankfurt)
Abstract:
Given a regular scheme $X$ and a normal crossing divisor $D$ one may concider two different cohomology groups.
The first one is the log étale cohomology developed by Illusie, K. Kato and many others: We associate a logarithmic structure $M$ to $X$ and define the log étale site over $(X,M)$.The second one is the tame cohomology developed by Hübner and Schmidt. Here we consider the tame site over the discretely ringed adic space $Spa(Xbackslash D,X)$. Tame morphisms are those which are étale and induce at most tamely ramified extension on the valuations.We construct a comparison morphism between these cohomology groups and prove that they are equal for schemes over $mathbb{F}_p$ and locally constant finite sheaves once we assume resolution of singularities.“

p-adic Gross-Zagier and rational points on modular curves
2023-05-09 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Sachi Hashimoto (MPI Leipzig)
Faltings’ theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings’ theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross-Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross-Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross-Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points.
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

Tropical perspectives in enumerative geometry
2023-05-10 16:45 - 18:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Frankfurter Seminar – Kolloquium des Instituts für Mathematik
Renzo Cavalieri (Colorado State University, Fort Collins)
Abstract: Enumerative geometry is an ancient branch of mathematics that aims to count the number of geometric objects that satisfy some constrains: the primordial enumerative geometric statement is that there is a unique straight line that passes through two distinct points in a plane. While enumerative geometric questions are often easy to state, the attempts to answer them have both employed and spurred the development of several mathematical techniques.
This talk will be a broad and hopefully friendly survey of how tropical geometry has become an important actor for several enumerative problems especially related to counting curves. I will use Hurwitz theory as the running example, and show how tropical geometry provides us not only with an interesting approach to classical Hurwitz theory, but also allows us to define „new“ enumerative problems of Hurwitz type. Much of the work presented has been collaborative work with Paul Johnson, Hannah Markwig, Dhruv Ranganathan and Johannes Schmitt.

Algebraicity and p-adic interpolation of critical Hecke L-values
2023-05-11 14:00 - 16:00 | Mainz, Hilbertraum (05-432)
Johannes Sprang (Essen)
Abstract: Euler’s beautiful formula on the values of the Riemann zeta function at the positive even integers can be seen as the starting point of the investigation of special values of L-functions. In particular, Euler’s result shows that all critical zeta values are rational up to multiplication with a particular period, here the period is a power of 2πi. Conjecturally this is expected to hold for all critical L-values of motives. In this talk, I will explain a joint result with Guido Kings on the algebraicity of critical Hecke L-values up to explicit periods for totally imaginary fields. If time permits, I will discuss the construction of p-adic L-functions for such fields as an application.

A quadratically refined tropical Bézout theorem
2023-05-12 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Sabrina Pauli (Düsseldorf)
Abstract: Results from motivic homotopy theory allow to study questions in enumerative geometry over an arbitrary field k. In this case the answer to these questions is not a number but a quadratic form carrying arithmetic information about the count. Using tropical geometry one can translate questions from enumerative geometry to questions in combinatorics which are often easier to solve. In my talk I will present one of the first examples of how to use tropical geometry for questions in enumerative geometry over an arbitrary field k, namely a proof of Bézout’s theorem for tropical curves. This is joint work with Andrés Jaramillo Puentes.

Endoscopy for GSp(4) and rational points on elliptic curves
2023-05-16 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
I report on a joint project with M. Bertolini , M.A. Seveso and R. Venerucci, aimed at studying the equivariant BSD conjecture for rational elliptic curves twisted by certain self-dual 4-dimensional Artin representations in situations of odd analytic rank. We use the endoscopy for GSp(4) to construct Selmer classes related to the relevant (complex and p-adic) L-values via explicit reciprocity laws.
Fabrizio Andreatta (University of Milan)
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

Prismatic F-crystals associated with strongly divisible modules
2023-05-23 14:00 - 16:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 308
Seminar: Non-archimedean geometry
Matti Würthen (Universität Frankfurt)
Abstract: The talk will be about the relationship between two different categories associated with the category of lattices in crystalline representations with small Hodge-Tate weights. In particular, I will explain how to attach a prismatic Frobenius crystal to a (crystalline) strongly divisible module.
Time permitting, I will also sketch how this can be extended to higher dimensions.

Periods, Power Series, and Integrated Algebraic Numbers
2023-05-24 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Tobias Kaiser (Universität Passau)
Abstract:
Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the polynomial ring over the rationals and evaluate it at a rational number. We follow this path and close these algebraic power series under taking iterated antiderivatives and nearby algebraic and geometric operations. We obtain a system of rings of power series whose coefficients form a countable real closed field. Using techniques from o-minimality we are able to show that every period belongs to this field. In the setting of o-minimality we define exponential integrated algebraic numbers and show that exponential periods and the Euler constant are exponential integrated algebraic number. Hence they are a good candiate for a natural number system extending the period ring and containing important mathematical constants.

An equivariant local epsilon constant conjecture
2023-05-26 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Alessandro Cobbe (Universität Heidelberg)
Abstract: The local epsilon constant conjecture in the formulation by Breuning of 2004 fits into the general framework of the equivariant Tamagawa number conjecture (ETNC) and should be interpreted as a consequence of the expected compatibility of the ETNC with the functional equation of Artin-L-functions. It relates local epsilon constants, which are associated to L-functions, to some terms which originate from local Galois cohomology groups of Z_p(1). We will also look at more general versions of the conjecture, obtained by twisting Z_p(1) with unramified representations. This is joint work with Werner Bley.

Discontinuity property of a certain Habiro series at roots of unity
2023-05-30 09:00 - 10:00 | Zoom
International Seminar on Automorphic Forms
Toshiki Matsusaka (Kyushu University)
The object of this talk is a family of q-series originating from Habiro’s work on the Witten-Reshetikhin-Turaev invariants. The q-series usually make sense only when q is a root of unity, but for some instances, it also determines a holomorphic function on the open unit disc. Such an example is Habiro’s unified WRT invariant H(q) for the Poincaré homology sphere. In 2007, Hikami observed its discontinuity at roots of unity. More precisely, the value of H(ζ) at a root of unity is 1/2 times the limit value of H(q) as q tends towards ζ radially within the unit disc. In this talk, we give an explanation of the appearance of the 1/2 factor and generalize Hikami’s observations by using Bailey’s lemma and the theory of mock/false theta functions.
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

On Emerton's factorization of completed cohomology
2023-05-30 14:00 - 16:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 308
Seminar: Non-archimedean geometry
Pierre Colmez (CNRS, Sorbonne Université, Paris)
Abstract: Emerton has given a factorization of the completed cohomology of the tower of modular curves, separating the contributions of all the groups that act (i.e., the absolute Galois group of ${mathbb Q}$ and the ${mathrm GL}_2({mathbb Q}_ell)$ for all primes $ell$).
I will explain how one can use p-adic Hodge theory to construct a Kirillov model for the completed cohomology and obtain a more direct construction of this factorization.

Hidden structures on de Rham cohomology of p-adic analytic varieties
2023-05-31 16:15 - 17:15 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Wieslawa Niziol (CNRS, Sorbonne Université, Paris)
Abstract: I will survey what we know about extra structures (Hodge filtration, Frobenius, monodromy) appearing on de Rham cohomology of analytic varieties over local fields of mixed characteristic.

Motivic homotopy theory beyond A^1-invariance
2023-06-01 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Ryomei Iwasa (Paris)
Abstract: The basic question I’d like to address in this talk is the following: How to do homotopy theory in algebraic geometry while keeping the affine line A^1 non-contractible? I will explain that tensor invertibility of the pointed projective line P^1 supplies homotopies between projective bundle sections in a non-trivial but canonical way. This dramatically expands the scope of motivic homotopy theory, and non-A^1-invariant theories such as syntomic cohomology, prismatic cohomology, algebraic K-theory, and topological cyclic homology can be studied from this perspective. In particular, I’ll explain that algebraic and Selmer K-theory are described by Snaith-type formulas. Based on joint work with Toni Annala and Marc Hoyois.

Intersections of components of Emerton-Gee stack for GL2
2023-06-02 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Kalyani Kansal (Johns Hopkins University)
Abstract: The Emerton-Gee stack for GL2 is a stack of (phi, Gamma)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and it can be viewed as a moduli stack of mod p representations of a p-adic Galois group. We compute criteria for codimension one intersections of the irreducible components of X. We interpret these criteria in terms of representation theory of GL2, motivated by conjectural categorical p-adic and mod p Langlands correspondence. We also give a representation-theoretic cohomological criterion for the number of top-dimensional components in a codimension one intersection.
