Hyperbolic Localization and Second Adjointness
Following an idea of Drinfeld, we present an abstract proof of Braden's
result on hyperbolic localization, relating sheaves on a space with
$\mathbb G_m$-action to their restriction to the attractor, repeller and
fixed point locus. Our argument is purely formal and works in an
abstract setting, and it roughly reduces the general statement to the
base case of $\mathbb G_m$ acting on $\mathbb A1$. This allows us to
extend the result to stacks and to arbitrary 6-functor formalisms.
Applying the result to the classifying stack of a reductive group, we
obtain new results on second adjointness for smooth representations in
natural characteristic, and we expect a similar version for locally
analytic representations as well as other applications to various
geometric incarnations of second adjointness in the Langlands program.
The main technical difficulty lies in handling the higher category
theory involved in the argument, and we have found efficient ways to
deal with "enriched (infty,2)-categories" and lax functors between them.
This is joint work with Claudius Heyer.
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A sketch of the proof of modularity of abelian surfaces
Venue: Endenicher Allee 60, Room 1.007, Math. Center, University of Bonn
Quantum groups and recovering Jones and HOMFLY
Habiro Cohomology
(33èmes Journées Arithmétiques) Wild Betti sheaves
Plenary talk at the conference "33èmes Journées Arithmétiques", Luxembourg:
https://www.uni.lu/fstm-en/conferences/ja25/
This talk will be streamed from the MPIM lecture hall, Bonn.
(Seminar SAG) On the exceptional locus of O’Grady’s nonsymplectic resolutions
In this talk, we focus on some singular moduli spaces of sheaves on a K3 surface. More precisely, for any integer n > 1, we consider the moduli space M(n) associated with the Mukai vector 2(1,0,1-n). Looking for new deformation classes of hyper-Kähler manifolds, O’Grady constructed an explicit resolution of every M(n). O’Grady’s resolution is crepant and does give a hyper-Kähler manifold only if n=2. If n>2, it turns out that no crepant resolution exists for M(n), but one may still look for a categorical crepant resolution.
We will report on the preliminary step in this direction, which consists in a geometric analysis of O’Grady’s resolution and of its exceptional locus.
(LDT seminar) Non-flexible loops of loose Legendrians in 3 dimensions
In recent years, significant progress has been made in understanding families of Legendrians in 3-dimensional contact topology. For instance, the homotopy type of the component of the max-tb Legendrian unknot (or any algebraic link) in the standard contact 3-sphere is now completely understood. However, the existence of non-flexible families of Legendrians has remained an open question. In this talk, we will explore this question in the context of loops of Legendrians. Specifically, we will show that in every closed overtwisted contact 3-manifold, there exists a non-contractible yet formally contractible loop of loose Legendrians. The proof relies on h-principles and parametrized Legendrian surgery, drawing inspiration from Dave Gay’s work on diffeomorphisms of the 4-sphere. This talk is based on joint work in progress with Fabio Gironella.
Reshetikhin-Turaev invariants
Q&A on Lagrangians, Quilts, and Moduli Spaces
We will collect questions and then decide on one or two topics to discuss in more depth.
A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory
Quinn's Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension n of space, depending on the choice of a homotopy finite space B. For instance, B can be the classifying space of a finite group or a finite 2-group.
In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Quinn's Finite Total Homotopy TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space B. I will shown how to compute these once-extended TQFTs when B is the classifying space of a homotopy 2-type, represented by a crossed module of groups.
Reference: Faria Martins J, Porter T: "A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids." arXiv:2301.02491 [math.CT]
Local-global compatibility at l = p over CM fields (Oberseminar Arithm. Geometry and Representation Theory)
In joint work in progress with Hevesi, Thorne and Whitmore we prove that the global Galois representations associated with cuspidal, cohomological automorphic representations are compatible with the (semisimplified) local Langlands correspondence. In this talk I will explain the context and meaning of this statement. Moreover, I will sketch the proof of the main new technical ingredient, which is a uniform bound on the torsion in the cohomology of certain unitary Shimura varieties.
Vorlesung: Habiro Cohomology
(ARGOS seminar) The Cousin maps and relation with the Sen operator
Venue: Endenicher Allee 60, Room 1.007, Math. Center, University of Bonn
Global formality for embeddings of varieties
I will discuss a joint work in progress with Damien Calaque aimed at globalizing a local formality result by Calaque-Felder-Ferrario-Rossi. Concretely, for an embedding of smooth varieties $X \hookrightarrow Y$, I will explain the construction of a global $L_\infty$-quasi-isomorphism between the polyvector field-valued differential-graded Lie algebra controlling the Kodaira-Spencer deformations of $X$ inside of $Y$ and the Hochschild cochain complex of an $A_\infty$-category describing the $A_\infty$-bimodule deformations of the structure sheaf of $X$ associated with the relative Kodaira-Spencer class. I will report on how this formality map is used to derive certain general isomorphisms between Ext-algebras of embeddings of varieties.
Lectures on The interrelation between Hilbert’s Tenth Problem and arithmetic geometry
Lectures 3 & 4 by Carlo
Thursday 26 June, 10:30-12:00 & 13:00-14:30
These lectures focus on the use of elliptic curves in proving H10/R for any finitely generated ring R; using many existing reductions, one arrives at a problem of controlling Selmer groups of an analogue of the Manin elliptic curve, which is done by new tools using additive combinatorics.
Local to Global results in Floer theory from neck-stretching
Deformations of log-symplectic forms
By a log-symplectic form I mean a meromorphic closed non-degenerate 2-form omega on a complex manifold X that has only simple poles along a divisor D. I will discuss how the de Rham class of omega in H^2(X\D) determines the deformations of the triple (X,D,omega). Some open problems about topology of the divisor deforming D will be discussed.
Double ∞-categories
I will give a leisurely introduction to double ∞-categories, provide a range of examples, and explain why these structures are useful. In particular, I will sketch how they play a key role in an abstract framework that systematically develops different generalizations of ∞-category theories (like equivariant and enriched versions). This was part of my PhD thesis.
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