Abstract: I will explain how a natural amalgam of classical Hodge theory with the nc Hodge structures arising from Gromov-Witten theory gives rise to new additive invariants of smooth projective varieties called Hodge atoms. Combined with Iritani's blow-up formula, Hodge atoms provide obstructions to birational equivalence. I will discuss applications to classical rationality problems. This is joint work with L. Katzarkov, M. Kontsevich, and T. Y. Yu.

Recordings

Monday February 16

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Wednesday February 18

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Monday February 23

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Wednesday February 25

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Abstract:  In this talk I will discuss an analytic approach to de Rham cohomology using condensed techniques. The main character in this theory is the so-called "analytic de Rham stack". This stacky or geometric approach has several advantages, one of them being that its definition is uniform in archimedean and non-archimedean geometry, another advantage is that it satisfies very strong descent properties which are helpful to define "de Rham cohomology". I will discuss its construction and some of its applications.  

Abstract: Given a family f : X -> S of algebraic varieties (often smooth and projective), one wants to understand points s of S where the fibre X_s acquires extra structure. Recent work has used G-functions together with a "Hasse principle" due to Bombieri (and an improvement of Andre) to constrain such points s. We survey these developments with a focus on recent work by the author. 

Abstract: In studying the asymptotics of the Mahler measure of certain families of cyclotomic integers (so-called Gaussian periods) one bumps into a family of Laurent hypersurfaces C_k that we have dubbed `cyclovarieties’. In the talk, I will introduce and motivate these, and then focus on two aspects: the geometry of the family, and the asymptotics of their Mahler measure. 

The cyclovarieties form an incomplete linear system in the anticanonical system on very particular toric varieties. In general, there are a lot of mysteries, but we know more when k is twice a power of an odd prime number: the ambient variety is a product of smooth toric del Pezzo varieties, the Newton polytope is reflexive, and, after resolving the base locus, the families are `log Calabi-Yau’ in the sense of Gross-Hacking-Keel, with specific vanishing cohomology groups. 

The Laurent hypersurface can be thought of as describing a random planar walk (with dependencies), and the corresponding probability density function is related to a Picard-Fuchs equation, allowing one to establish a central limit theorem for the Mahler measure of C_k for k twice a prime number tending to infinity. (Joint work with David Hokken and Berend Ringeling.)

 Abstract: Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of Calabi-Yau complete intersections in some sense. Exoflops essentially create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to a given Calabi-Yau complete intersection. We will explain this technique, provide some foundational results about this and then some new applications proven recently for Calabi-Yau orbifolds. This talk contains results from a series of joint works with D. Favero (UMinn) and A. Malter (BIMSA). 

Abstract: In 1990, Witten conjectured that the generating series of intersection numbers of psi classes is a tau function of the KdV hierarchy. This was first proved by Kontsevich. In 2017, Norbury conjectured that the generating series of intersection numbers of psi classes times a negative square root of the canonical bundle is also a tau function of the KdV hierarchy. In joint work with N. Chidambaram and A. Giacchetto, we proved Norbury’s conjecture and obtained polynomial relations among kappa classes which had been recently conjectured by Kazarian--Norbury. We also introduced a new collection of cohomology classes, which correspond to negative r-th roots (previously r=2) of the canonical bundle and form a cohomological field theory (CohFT), the negative analogue of Witten’s r-spin CohFT, which turns out to be geometrically much simpler. We proved that the corresponding intersection numbers can be computed recursively using topological recursion (which I will briefly introduce) and, equivalently, W-constraints. The strategy draws inspiration from our proof, together with S. Charbonnier, of Witten’s r-spin conjecture from 1993 (Faber—Shadrin—Zvonkine’s theorem from 2010) that claims that (positive) r-spin intersection numbers satisfy the r-KdV hierarchy. We also obtain new (tautological) relations on the moduli space of curves in a (negative) analogous way to Pandharipande--Pixton--Zvonkine. The talk will be an overview of these four topics (r=2/>2; positive/negative) and their connections. 

Abstract: I'll start by giving a quick introduction to o-minimality and its connections with Diophantine geometry. I'll then discuss how various structures from algebraic geometry and Hodge theory fit in the o-minimal setting, and give some examples for the type of consequences one can derive from this o-minimal perspective. Finally I'll describe a refinement to the o-minimal framework that should conjecturally apply to these types of geometric structures. 

Abstract: Calabi-Yau varieties are objects of great interest in algebraic geometry and also play a significant role in string theory. Their importance is due, among other aspects, to the possibility of constructing pairs of families of varieties that are symmetric under mirror symmetry. When these varieties arise as hypersurfaces in toric Fano varieties, classical results by Batyrev, Berglund-Hübsch-Krawitz, among others, provide constructions of families of Calabi–Yau varieties that are mirror to each other, using the combinatorial features of the underlying varieties. In this talk, we will explain how this theory can be extended to complete intersections, yielding a more general construction that produces new examples of dual families of Calabi-Yau varieties. This is joint work with Michela Artebani (Universidad de Concepción) and Robin Guilbot (Université de Toulouse).