Abstract: One can construct an A-model analogue of the Gauss-Manin connection using genus 0 Gromov-Witten invariants, which also make perfect sense over fields of positive characteristics and p-adic fields. I will explain how arithmetic gadgets including p-curvature, Fontaine-Laffaile modules, and over-convergent Frobenii (conjecturally) arise in this context, and discuss certain extensions in the q-difference setting based on quasimap K-theory. This is based on joint works with Lee, Pomerleano, and Seidel.

Abstract: A recent manuscript written by Claude under the direction of Levent Alpoge constructs a 1-parameter family of compact complex threefolds diffeomorphic to S^6. I will describe the main geometric ideas behind the construction and outline the proof. If time permits, I will remark on possible future research directions.

Abstract: McMullen constructed automorphisms of complex K3 surfaces with positive entropy—and thus chaotic dynamics on a large set—that nonetheless have a Siegel disk: an open region on which the automorphism is holomorphically conjugate to an irrational rotation. Strikingly, such an automorphism forces the K3 surface to be non-projective, so McMullen's examples cannot be written down using explicit equations. Instead they are obtained indirectly via the global Torelli theorem.

In this talk we construct a p-adic analogue: a positive-entropy automorphism with a Siegel disk on a non-projective rigid-analytic K3 surface over a p-adic field. Our construction produces the surface directly, without the Torelli theorem, which makes it considerably more flexible.

Abstract: When does an algebraic differential equation have solutions which are themselves algebraic functions? This question goes back at least to Fuchs, in 1875. Its conjectural answer, due to Grothendieck and Katz in the linear case and Ekedahl–Shepherd-Barron–Taylor and Bost in general, situates it as one of number theory. I’ll explain these conjectures and some progress towards them, joint with Josh Lam, for isomonodromy differential equations. This class of differential equations includes, for example, the Painlevé VI equation. This work is in part an excuse to study “motivic” aspects of the space of all algebraic differential equations.

Abstract: AI systems are now more-or-less autonomously solving some interesting open problems, resulting in an abundance of PDFs. This presents a problem for our profession, which was already bottlenecked by our ability to understand one another’s work even prior to AI. I’ll survey the recent history of AI and math, discuss some experiments using AI tools non-autonomously, and speculate a bit about the future.

Abstract: In non-commutative algebraic geometry, an algebraic variety is

replaced by a DG algebra A, and the role of de Rham cohomology is played

by periodic cyclic homology HP(A). It is a standard expectation that all

the additional motivic-type structure on de Rham cohomology should also

exist in non-commutative generality. There has been a lot of progress on

this in the p-adic situation, but the archimedian case is much more

problematic. The missing ingredient is complex conjugation. While a

Hodge filtration on periodic cyclic homology exists by definition, we

also expect that HP(A) for a good A over C carries a functorial

R-structure, and it is not clear at all where it might come from. I am

going to present a proposal towards this based on L^2-completed version

of HP. This is still work-in-progress, but the proposal looks viable, so

hopefully, this won't be a complete waste of audience's time.

Abstract: The homotopy groups of spheres classify continuous functions from an n-dimensional sphere to an m-dimensional sphere up to continuous deformation. They are of great interest as they control many classification problems in geometric topology, but are rich and complicated. I will explain some of what we know about them, and describe the telescopic approach to understanding them systematically. I will then explain how algebraic K-theory has refined our understanding of the telescopic approach, and can be used to obtain asymptotic lower bounds on the complexity of these homotopy groups that were previously out of reach.

Abstract:  Over the last few years I studied with Michael Groechenig, Moritz Kerz and in part with Johan de Jong properties of complex local systems on smooth quasi-projective varieties which are expected by Simpson's general conjectures to be motivic.  It is a mathematical program. I’ll try to give an overview of some of the problems which are currently on my mind.  Best would be if we could develop the talk a bit  like a discussion. Feel free to ask questions, I‘ll try to think of them and, if possible, to answer. 

Abstract: Here "derived commutative ring" refers to a nonconnective generalization of animated commutative rings,

first defined by Mathew; in other words, it's a common home for the homotopy theories of simplicial and cosimplicial

commutative rings. I first encountered the notion while studying Hochschild homology and de Rham cohomology.

In this talk, I'll describe a different presentation of the category. It's an instance of a fairly simple and very general pattern,

of which there are other interesting instances too. I plan to also discuss one more, a notion of "synthetic E-infinity ring",

and the relevance of this to the study of topological Hochschild homology and prismatic cohomology (this part will be

based on joint work with Devalapurkar, Hahn, and Senger).