## The Beauville-Voisin conjecture for Hilb(K3) and the Virasoro algebra

We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow groups, and using them to solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik.

## DAHA approach to algebraic knots and links

DAHA generally provide refined invariants of colored iterated links, which generalize the

WRT-invariants and HOMFLY-PT polynomials. In the uncolored case and for iterated knots,

they are conjectured to coincide with the stable reduced Khovanov-Rozansky polynomials (the most

powerful numerical invariants we have). The "intrinsic" DAHA conjectures are mostly verified

at the moment; these properties are generally difficult to check topologically. The DAHA super-

duality is an important example (a theorem for DAHA, but far from obvious in topology).

Its conjectural coincidence with the functional equation in the motivic approach (my next talk),

can be a fundamental development. We will focus on the DAHA construction in this talk, with

some explicit calculations (for trefoil and beyond).

## Vertex Operator Algebras and Modular Forms

Time: Tuesdays, 4.30 - 6 pm

Place: MPIM Lecture Hall, Vivatsgasse 7

First lecture: on April 2, 2019, end on July 2

## Recent developments in Quantum Topology

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## The singular (co)chains and rational homotopy theory, II

The central theme of this talk is the question: How much of the rational homotopy type of a space can we deduce from the (co)chains on the space? More precisely, in this talk we explain what the relationship is between the singular (co)chains and various approaches to the rational homotopy theory.

## The singular (co)chains and rational homotopy theory, I

The central theme of this talk is the question: How much of the rational homotopy type of a space can we deduce from the (co)chains on the space? More precisely, in this talk we explain what the relationship is between the singular (co)chains and various approaches to the rational homotopy theory.

## On Graphs of Hecke operators and Hall algebras

In Bombay 1979, Don Zagier observes that if the kernel of certain

operators on automorphic forms turns out to be an unitarizable

representation, over the field of rational numbers $Q$, a formula of Hecke implies the Riemann hypothesis.

Zagier calls the elements of this kernel toroidal automorphic forms.

Moreover, Zagier asks what happens if $Q$ is replaced by a global function field and remarks that the space of unramified toroidal automorphic forms can be expected to be finite dimensional.

Motivated by these questions, Oliver Lorscheid introduces, in 2012,

the ***graphs of Hecke operators*** for global function fields.

This theory allowed him to prove, among other things, that the space

of unramified toroidal automorphic forms for a global function field

is indeed, finite dimensional. The graphs of Hecke operators

introduced by Lorscheid encode the action of Hecke operators on

automorphic forms.

On the other hand, Ringel(1990), Kapranov (1997), Burban and

Schiffmann (2012) et al. have been developing the theory of *Hall

algebras * of coherent sheaves over a smooth geometric irreducible projective

curve over a finite field (in general for a finitary category).

For this talk we discuss the connection between graphs of Hecke

operators and Hall algebras.

In the elliptic case, Atiyah's work on vector bundles (1957) allow us

to describe (explicitly) these graphs.

## Deformations of VB-groupoids

VB-groupoids can be understood as vector bundles in the category of Lie groupoids. They encompass several classical objects, such as Lie group representations, 2-vector spaces, Lie group actions on vector bundles; moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids, in particular the adjoint representation. In this talk, I will attach to every VB-groupoid a cochain complex controlling its deformations and discuss some features, such as Morita invariance, as well as some examples and applications. I will also compare it to its infinitesimal counterpart, the linear deformation complex of a VB-algebroid, via a Van Est map. This is joint work with Luca Vitagliano.

## Hochschild cohomology and deformation quantization of affine toric varieties

For an affine toric variety we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Moreover, we show that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.

## Semistability and Approximate Hermitian-Einstein Structures

## tba

## Frobenius lifts and correspondences on GL_n

Let GL_n be the general linear group scheme over the integers.We show that the p-adic completion of GL_n possessescertain remarkable Frobenius lifts attached to integral symmetric matrices; these Frobenius liftscan be viewed as arithmetic analogues, for Spec Z, of the Levi-Civita connection attached to a metric on a manifold. We then show how these Frobeniuslifts admit algebraizations by correspondences on GL_n; the commutatorsof these correspondences can be viewed as an arithmetic analogue, for Spec Z, of Riemannian curvature.

## tba

## Vertex Operator Algebras and Modular Forms

Time: Tuesdays, 4.30 - 6 pm

Place: MPIM Lecture Hall, Vivatsgasse 7

First lecture: on April 2, 2019, end on July 2

## Recent developments in Quantum Topology

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## Recent developments in Quantum Topology

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## Vertex Operator Algebras and Modular Forms

Time: Tuesdays, 4.30 - 6 pm

Place: MPIM Lecture Hall, Vivatsgasse 7

First lecture: on April 2, 2019, end on July 2

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