Abstract: Recall that Grothendieck introduced a technology which associates with any non-singular algebraic variety X over a  field k the category of etale covers of X. The limit \tilde{X} of such a category can serve as a formal  cover space for the lattice of varieties that are etale covers of X. However, classical analytic cover spaces  such as the complex plane C or the upper half plane H serve as a cover for a much broader family of varieties, which dont have to form a lattice.  I will explain how to use Hrushovski's fusion construction to construct a formal cover space for non-singular curves of genus at most 1 over k and state its model-theoretic properties such as quasi-minimality and QE. The hypothetical analogue of H will also be mentioned. 

Abstract: Mirror P=W conjecture, introduced by Harder-Katzarkov-Przyjalkowski, is a refined Hodge number symmetry between mirror log Calabi-Yau manifolds U and Y. In case that U has a good compactification (X,D) where X is Fano and D is an anti-canonical divisor, one can understand this conjecture from mirror symmetry for the Fano pair (X,D). Its mirror candidate is a hybrid Landau-Ginzburg (LG) model (Y,h:Y->C^N), a multi-potential analogue of an ordinary LG model. We will review this story for the first part of the talk. In the second part, we will discuss one of the applications, a topological mirror construction of d-semistable Calabi-Yau varieties. 

Abstract: the aim of the seminar consists of providing an overview of the theory of two-dimensional cohomological Hall algebras (COHAs) and their connections to, from one side, the study of the topology of certain moduli spaces (Nakajima quiver varieties, moduli spaces of framed sheaves on the projective plane, moduli spaces of semistable sheaves on smooth projective surfaces), and, on the other hand, quantum groups and vertex algebras. The following two examples will be discussed in detail: the COHA of the affine plane and the COHA of the minimal resolution of an ADE singularity.

Abstract: There are many situations in which the derived category of coherent sheaves on a smooth projective variety admits a semiorthogonal decomposition that reflects something interesting about its geometry. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. Then, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin's conjecture. 

Abstract: This talk is based on joint work with Kris Shaw and Arthur Renaudineau. Our work is an attempt to generalize (to higher codimension) “Viro's combinatorial patchworking”, which is a powerful technique for studying the possible topological types of real algebraic hypersurfaces. In our generalization, given a real family over the punctured disc of algebraic varieties in a toric variety, the main goal is to give a combinatorial description  of the topology of a real fibre. This description works under the condition that the tropical limit of the family is a smooth tropical variety and is based on real phase structures for tropical varieties. In my talk, I will try to explain all these notions and possibly mention a few other connections, for example, to oriented matroids. 

Abstract: The absolute Galois group of a number field is a central object of interest in algebraic number theory. Algebraic geometry provides us with a large class of p-adic representations of such a group: conjecturally, these “geometric” representations are characterized by some local properties, i.e. properties of their restrictions to decomposition groups at rational primes. By the Langlands principle of functoriality, the property of being geometric should remain true after composition with an L-homomorphism. We investigate what this means on the level of the local conditions that characterize geometricity, and how the picture changes when one tries to deform it p-adically. 

Abstract: Scattering amplitudes predict the probability of interaction of elementary particles. Without assuming any previous knowledge in string theory, we will review the state of the art in the study of (perturbative) string theory scattering amplitudes, which can be expressed in terms of interesting periods. We will focus on the relations between amplitudes for open strings (corresponding to gauge theories) and closed strings (corresponding to gravity), and on how such relations are proven or conjectured to originate from the theory of single-valued periods, recently developed by Francis Brown.

Abstract: Siegel introduced the notion of E-function in a landmark 1929 paper with the goal of generalising the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers. E-functions are power series with algebraic coefficients that are solutions of a linear differential equation and satisfy some growth conditions of arithmetic nature. Besides the exponential function, examples include Bessel functions and a rich family of hypergeometric series. Siegel asked whether all E-functions are polynomial expressions in these hypergeometric series. I explain why the answer to Siegel's question is negative, and then try to amend it by describing how E-functions arise from geometry in the form of "exponential period functions" and why it might seem reasonable, in the light of other conjectures, to expect that all E-functions are of this kind.

Abstract: The Hitchin fibration may be viewed as a birational map on the cotangent bundle of Bun_G, and the generic Hitchin fibers thus provide spectral data for Higgs sheaves over Bun_G. These will be logarithmic with singularities over the 'wobbly locus' of bundles that are not very stable. Ron Donagi and Tony Pantev have proposed to endow the resulting logarithmic Higgs sheaves with parabolic data, yielding flat connections on open subsets of Bun_G. They propose that these should be the Hecke eigensheaves of the geometric Langlands correspondence. They have proven this for rank 2 bundles on P^1 minus 5 points, and we are currently working on the case of rank 2 bundles on curves of genus 2. 

Abstract: In joint work with Peter Scholze on "condensed mathematics", we propose that for many purposes, it's beneficial to replace the usual base category of sets with the richer base category of condensed sets.  We will discuss what this replacement means for algebraic geometry.  In particular, it gives more kinds of closed and open subsets, and a new,  more powerful notion of quasicoherent sheaf. 

Abstract: Associated to decorated trivalent graphs I will describe a family of Laurent polynomials called graph potentials. These polynomials satisfy interesting symmetry and compatibility properties for different choices of graphs, leading to the construction of a topological quantum field theory which efficiently computes the classical periods as the partition function.

Under mirror symmetry graph potentials are related to moduli spaces of rank 2 bundles (with fixed determinant of odd degree) on a curve of genus $g\geq 2$, which is a class of Fano varieties of dimension $3g-3$. I will discuss how enumerative mirror symmetry relates classical periods to quantum periods in this setting. Time permitting I will touch upon aspects of homological mirror symmetry for these Fano varieties and their mirror partners.

This is joint work with Sergey Galkin and Swarnava Mukhopadhyay.