Forthcoming events in this series
The monoidal structure on strict polynomial functors and adjoints of the Schur functor
Abstract
Firstly, we will discuss how the category of strict polynomial functors can be endowed with a monoidal structure, including concrete calculations. It is well-known that the above category is equivalent to the category of modules over the Schur algebra. The so-called Schur functor in turn relates the category of modules over the Schur algebra to the category of representations of the symmetric group which posseses a monoidal structure given by the Kronecker product. We show that the Schur functor is monoidal with respect to these structures.
Finally, we consider the right and left adjoints of the Schur functor. We explain how these can be expressed in terms of one another using Kuhn duality and the central role the monoidal structure on strict polynomial functors plays in this context.
Classifying $A_{\mathfrak{q}}(\lambda)$ modules by their Dirac cohomology
Abstract
We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules of real reductive groups, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.
Center of quiver Hecke algebras and cohomology of quiver varieties
Abstract
I will explain how to relate the center of a cyclotomic quiver Hecke algebras to the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E. Vasserot.
Analytic and Arithmetic Geometry Workshop: Quasi-abelian categories in analytic geometry
Abstract
I will describe a categorical approach to analytic geometry using the theory of quasi-abelian closed symmetric monoidal categories which works both for Archimedean and non-Archimdedean base fields. In particular I will show how the weak G-topologies of (dagger) affinoid subdomains can be characterized by homological method. I will end by briefly saying how to generalize these results for characterizing open embeddings of Stein spaces. This project is a collaboration with Oren Ben-Bassat and Kobi Kremnizer.
Analytic and Arithmetic Geometry Workshop: Overconvergent global analytic geometry
Abstract
We will discuss our approach to global analytic geometry, based on overconvergent power series and functors of functions. We will explain how slight modifications of it allow us to develop a derived version of global analytic geometry. We will finish by discussing applications to the cohomological study of arithmetic varieties.
Analytic and Arithmetic Geometry Workshop: On the arithmetic deformation theory of Shinichi Mochizuki in 80 minutes
Abstract
I will talk in down to earth terms about several main features of this theory.
Analytic and Arithmetic Geometry Workshop: Variations on quadratic Chabauty
Abstract
We describe how p-adic height pairings allow us to find integral points on hyperelliptic curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss how to carry out this ``quadratic Chabauty'' method over quadratic number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).
Hamiltonian reduction, matrix factorization and categorical braid group actions.
Geometric Satake Equivalence
Abstract
Both sides of the geometric Langlands correspondence have natural Hecke
symmetries. I will explain an identification between the Hecke
symmetries on both sides via the geometric Satake equivalence. On the
abelian level it relates the topology of a variety associated to a group
and the representation category of its Langlands dual group.
Transversal slices to conjugacy classes in algebraic groups and Lustig's partition.
Abstract
I shall show that for every conjugacy class O in a connected semisimple algebraic group G over an algebraically closed field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.
Towards the compatibility of Geometric Langlands with the extended Whittaker model
Abstract
Let $G$ be a connected reductive group and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $Bun_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called $coeff_{G}$, from the DG category of D-modules on $Bun_G$ to a certain DG category $Wh(G, ext)$, called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor $$\mathbb{L}_G : \operatorname{IndCoh}_{N}(LocSys_{\check{G}} ) \to D(Bun_G)$$ with the Whittaker model. For $G = GL_n$ and $G = PGL_n$, we prove that $coeff_G$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.
Introduction to Factorization
Abstract
Factorization is a property of global objects that can be built up from local data. In the first half, we introduce the concept of factorization spaces, focusing on two examples relevant for the Geometric Langlands programme: the affine Grassmannian and jet spaces.
In the second half, factorization algebras will be defined including a discussion of how factorization spaces and commutative algebras give rise to examples. Finally, chiral homology is defined as a way to give global invariants of such objects.
What is the [Categorical] Weil Representation?
Abstract
Motivic L-functions
Abstract
This talk will be a brief introduction to some standard conjectures surrounding motivic L-functions, which might be viewed as the arithmetic motivation for Langlands reciprocity.
The Ran space and contractibility of the space of rational maps
Abstract
We will define the Ran space as well as Ran space versions of some of the prestacks we've already seen, and explain what is meant by the homology of a prestack. Following Gaitsgory and possibly Drinfeld, we'll show how the Ran space machinery can be used to prove that the space of rational maps is homologically contractible.
Generic maps
Abstract
D-modules on prestacks
Abstract
This talk will be an introduction to the notion of D-modules on
prestacks. We will begin by discussing Grothendieck's definition of
crystals of quasi-coherent sheaves on a smooth scheme X, and briefly
indicate how the category of such objects is equivalent to that of
modules over the sheaf of differential operators on X. We will then
explain what we mean by a prestack and define the category of
quasi-coherent sheaves on them. Finally, we consider how the
crystalline approach may be used to give a suitable generalization
of D-modules to this derived setting.
An introduction to infinity categories.
Abstract
Infinity categories simultaneously generalize topological spaces and categories. As a result, their study benefits from a combination of techniques from homotopy theory and category theory. While the theory of ordinary categories provides a suitable context to analyze objects up to isomorphism (e.g. abelian groups), the theory of infinity categories provides a reasonable framework to study objects up to a weaker concept of identification (e.g. complexes of abelian groups). In the talk, we will introduce infinity categories from scratch, mention some of the fundamental results, and try to illustrate some features in concrete examples.
The geometric Langlands conjecture
Abstract
Higher differential operators and genera of algebraic varieties
Abstract
We will describe a generalization of the algebra of differential operators, which gives a
geometric approach to quantization of cotangent field theories. This construction is compatible
with "integration" thus giving a local-to-global construction of volume forms on derived mapping
spaces using a version of non-abelian duality. These volume forms give interesting invariants of
varieties such as the Todd genus, the Witten genus and the B-model operations on Hodge
cohomology.
Pointed Hopf Algebras with triangular decomposition.
Abstract
In this talk, two concepts are brought together: Algebras with triangular decomposition (as studied by Bazlov & Berenstein) and pointed Hopf algebra. The latter are Hopf algebras for which all simple comodules are one-dimensional (there has been recent progress on classifying all finite-dimensional examples of these by Andruskiewitsch & Schneider and others). Quantum groups share both of these features, and we can obtain possibly new classes of deformations as well as a characterization of them.
Cluster combinatorics and geometrical models (part II)
Abstract
In this talk I will introduce cluster categories and report on some new results on cluster categories of type E_6.