The seminar webpage for the 2022-2023 Caltech Algebra/Geometry seminar is here
If a seminar is on Zoom, we meet in room 86724275712. The password is the smallest prime greater than 100, repeated twice.
Organizers: Robert Cass and Abhishek Oswal
Spring 2022
April 18, 2022 (4:00pm PT on Zoom)
Galois action on the pro-algebraic fundamental group
Alexander Petrov (Harvard University)
Given a smooth variety X over a number field, the action of the Galois group on the geometric etale fundamental group of X makes the ring of functions on the pro-algebraic completion of this fundamental group into a (usually infinite-dimensional) Galois representation. This Galois representation turns out to satisfy the following two properties:
1) Every finite-dimensional subrepresentation of it satisfies the assumptions of the Fontaine-Mazur conjecture: it is de Rham and almost everywhere unramified.
2) If X is the projective line with three punctures, the semi-simplification of every Galois representation of geometric origin is a subquotient of the ring of regular functions on the pro-algebraic completion of the etale fundamental group of X.
I will also discuss a conjectural characterization of local systems of geometric origin on complex algebraic varieties, arising from property 1) above.
May 2, 2022 (Two talks, in-person)
Categories phantoms and applications (3:00pm, Linde 187)
Ludmil Katzarkov (University of Miami)
In this talk we will introduce the notion of phantom categories. Connection to other structures will be discussed.
Quantitative theory of the trace character (4:00pm, Linde 387)
Anna Szumowicz (Caltech)
Let G be a p-adic reductive group. J.-L. Kim, S. W. Shin and N. Templier conjectured that for every regular element \gamma of G, the trace character \theta_\pi (\gamma) grows much faster than deg(\pi) when deg(\pi) tends to infinity and \pi runs over irreducible supercuspidal representations of G. J.-L. Kim, S.W Shin and N. Templier proved the conjecture under some additional conditions. We discuss the further progress on the problem.
May 9, 2022 (4:00pm, Linde 387)
Normalization in the integral models of Shimura varieties of Hodge (abelian) type
Yujie Xu (Harvard University)
Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, Rapoport-Zink etc.) have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of abelian type constructed by Kisin (resp. Kisin-Pappas). I will talk about my recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models. I will also mention an application to toroidal compactifications of such integral models.
May 16, 2022 (4:00pm, Linde 387)
Perverse mod p sheaves on the affine Grassmannian
Robert Cass (Caltech)
Perverse sheaves have important applications in representation theory and number theory. In this talk we will consider the case of mod p étale sheaves on the affine Grassmannian over a field of characteristic p. Despite the pathological behavior of such sheaves, they encode the structure of mod p Hecke algebras. We will focus on a mod p version of the geometric Satake equivalence. Along the way we will encounter connections between Frobenius splittings and the general theory of perverse mod p étale sheaves.
May 23, 2022 (Two talks, in-person)
Robba cohomology for dagger spaces in positive characteristic (4:00pm, Linde 387)
Koji Shimizu (UC Berkeley)
We will discuss a p-adic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic p. After explaining the motivation, we will define a site (Robba site) and discuss its basic properties.
A tensor 2-product of 2-representations of $\mathfrak{sl}_2$ (5:15pm, Linde 387)
Matthew McMillan (UCLA)
The categorification program for TQFTs has long sought a braided monoidal structure for the 2-category of 2-representations of Kac-Moody 2-algebras. Such structure requires a general construction for the tensor 2-product of 2-reps. Webster has given a diagrammatic categorification for products of simple 2-reps, and with Losev defined some axioms that determine these uniquely. Rouquier has formulated a general construction, but it returns $\mathcal{A}_\infty$-categories as the products of 2-reps given by (dg)-algebras. Manion-Rouquier applied this construction in the case of $\mathfrak{gl}(1|1)$ where homotopical complications disappear. We present a general construction preserving the 2-category of algebras for the case of $\mathfrak{sl}_2$, specifically the product of the fundamental rep $\mathcal{L}(1)$ and an arbitrary rep. We study the output for $\mathcal{L}(1)$ times $\mathcal{L}(n)$ and compare with the known categorification in this case.
June 6, 2022 (4:00pm, Linde 387)
On the local structure of arc spaces
Tommaso de Fernex (University of Utah)
The arc space of a variety is an infinite dimensional scheme whose geometric structure captures, in a way that is not yet fully understood, certain features of the singularities of the variety. Focusing on its local rings and invariants of these rings such as embedding dimension and codimension, we explore the local structure of arc spaces. Our main tools rely on a formula for the sheaf of differentials on arc spaces and some recent finiteness results on the fibers of the map induced at the level of arc spaces from an arbitrary morphism of schemes over a field. The talk is based on joint work with Christopher Chiu and Roi Docampo.
Winter 2022
January 18, 2022 (Tuesday 10:00am PT on Zoom)
Weyl symmetry for curve counting invariants via spherical twists
Tim-Henrik Buelles (ETH Zurich)
The curve counting invariants of Calabi—Yau 3-folds exhibit symmetries induced by the action of derived autoequivalences. I will give a brief overview of the subject and explain some recent development. In joint work with M. Moreira we obtain the Weyl symmetry along a ruled divisor, a new rationality result and functional equation for the generating function of stable pairs invariants. The underlying derived autoequivalence involves spherical twists.
January 24, 2022 (in-person)
Stratified étale homotopy theory
Peter Haine (UC Berkeley)
Étale homotopy theory was invented by Artin and Mazur in the 1960s as a way to associate to a scheme S, a homotopy type with fundamental group the étale fundamental group of S and whose cohomology captures the étale cohomology of S with locally constant constructible coefficients. In this talk we’ll explain how to construct a stratified refinement of the étale homotopy type that classifies constructible étale sheaves of spaces. We’ll also explain how this refinement gives rise to a new, concrete definition of the étale homotopy type. We’ll then explain how to use condensed math to upgrade this result from discrete rings to rings with a topology such as ℤℓ , ℚℓ , or 𝔽q⟦t⟧. This is joint work with Clark Barwick and Saul Glasman.
January 31, 2022 (4:00pm PT on Zoom)
Hamiltonian flows in Calabi-Yau categories
Nick Rozenblyum (University of Chicago)
A classical result of Goldman states that the character variety of an oriented surface is a symplectic algebraic variety, and that the Goldman Lie algebra of free loops on the surface acts by Hamiltonian vector fields on the character variety. I will describe a vast generalization of this result in the setting of noncommutative Calabi-Yau geometry. One incarnation of this result is a higher-dimensional version of Goldman's theorem: the Chas-Sullivan string Lie algebra acts on by Hamiltonian vector fields on the (derived) character stack of a closed oriented manifold. Other incarnations include Hitchin's integrable system and the action of the necklace Lie algebra on Nakajima quiver stacks. This is joint work with Christopher Brav.
February 7, 2022 (4:00pm PT on Zoom)
Abelian Varieties not Isogenous to Jacobians over global fields
Ananth Shankar (University of Wisconsin, Madison)
We prove that over an arbitrary global field, for every g > 3 there exists an abelian variety of dimension g, which is not isogenous to a Jacobian. We will also discuss this question over finite fields. This is joint work with Jacob Tsimerman.
February 14, 2022 (4:00pm PT on Zoom)
Varieties of general type with doubly exponential asymptotics.
Chengxi Wang (UCLA)
We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior. (Joint work with Louis Esser and Burt Totaro .)
February 28, 2022 (in-person)
Sectorial descent for wrapped Fukaya categories and applications to mirror symmetry
Sheel Ganatra (University of Southern California)
I will describe a package of structural results (joint with Pardon-Shende) for computing partially wrapped Fukaya categories of Weinstein manifolds and sectors, culminating in a descent formula for such categories with respect to so-called "(Weinstein) sectorial" coverings. I will also discuss what is known about finding such coverings from different types of input data, with an emphasis on examples. A particularly interesting case (based on works in progress with Auroux-Hanlon-Jeffs and Hanlon-Hicks-Ward) is when the relevant input data comes from a geometric mirror construction, in which case the associated coverings and descent formulae can be used to give a local-to-global understanding of mirror symmetry.
March 7, 2022 (4:00pm PT on Zoom)
Decidable problems on integral SL2-characters
Junho Peter Whang (Seoul National University)
Classical topics in the arithmetic study of quadratic forms include their reduction theory and representation problem. In this talk, we discuss their nonlinear analogues for SL2-characters of surface groups. First, we prove that the set of integral SL2-characters of a surface group with prescribed invariants can be effectively determined and finitely generated, under mapping class group action and related dynamics. Second, we prove that the set of values of an integral SL2-character of a finitely generated group is a decidable subset of the integers.
March 14, 2022 (4:00pm PT on Zoom)
Punctured log Gromov-Witten invariants
Mark Gross (University of Cambridge)
I will discuss a generalization of stable log maps developed with Abramovich, Chen and Siebert. Punctured log maps are a slight modification of ordinary stable log maps which allow for negative contact orders with divisors. This is crucial if one would like to develop a gluing formalism for stable log maps, as a splitting of a stable log map obtained by partially normalizing a domain curve is not a stable log map, but rather a punctured log map. I will sketch the definitions, and highlight some intriguing aspects of the theory.
March 21, 2022 (in-person)
Canonical Scattering Diagrams
Mark Gross (University of Cambridge)
I will discuss joint work with Bernd Siebert which gives a construction of mirrors to log Calabi-Yau manifolds by building a "scattering diagram" which describes how to assemble the mirror out of standard charts. These scattering diagrams are constructed by using punctured log Gromov-Witten invariants discussed in the previous lecture.
Fall 2021
October 18, 2021
A Categorical Künneth Formula for Weil Sheaves
Tamir Hemo (Caltech)
The failure of the product formula for fundamental groups of schemes in characteristic p > 0 was explained by Drinfeld who proved a product formula holds by taking into account the extra Frobenius symmetry. In this work, we upgrade Drinfeld’s equivalence to a Künneth-type equivalence of derived categories for appropriately defined categories of lisse and constructible Weil sheaves. This is joint work with Timo Richarz and Jakob Scholbach.
October 25, 2021
p-adic analogues of Drinfeld's lemma
Kiran Kedlaya (UC San Diego)
Drinfeld's lemma for an F_p-scheme X asserts a close relationship between X and the formal categorical quotient of the product of X with an algebraically closed field divided by the "partial Frobenius" action on the second factor. In particular, Drinfeld shows that these two have the same lisse and constructible etale sheaves.
We describe two different p-adic analogues of this result, both of which have implications for the Langlands correspondence. One is for lisse sheaves on perfectoid spaces and is related to the Fargues-Scholze construction for mixed-characteristic local Langlands; it also gives rise to a multivariate analogue of (phi, Gamma)-module theory for representations of powers of a local Galois group (the latter being joint work with Annie Carter and Gergely Zabradi). The other is for F-isocrystals and D-modules (joint work with Daxin Xu), and is needed for a crystalline analogue of V. Lafforgue's approach to geometric Langlands for reductive groups (parallel to Abe's crystalline analogue of L. Lafforgue's theorem for GL(n)).
November 1, 2021
Reflection in algebra and topology
Aaron Mazel-Gee (Caltech)
In this talk, I will discuss a new duality that was recently discovered in joint work with David Ayala and Nick Rozenblyum, which we refer to as reflection.
In essence, reflection amounts to two dual methods for reconstructing objects, based on a stratification of the category that they live in. As a classical example, an abelian group can be reconstructed on the one hand in terms of its p-completions and its rationalization, or on the other (reflected) hand in terms of its p-torsion components and its corationalization; and these both come from a certain "closed-open decomposition" of the category of abelian groups.
Examples and applications of reflection are abundant, because stratifications are abundant. In algebra, reflection recovers the derived equivalences of quivers coming from BGP reflection functors (hence the terminology "reflection"). In topology, reflection is closely related to Verdier duality, a generalization of Poincaré duality that applies to singular spaces. Moreover, an explicit description of reflection leads to a categorification of the classical Möbius inversion formula, a Fourier inversion theorem for functions on posets.
November 8, 2021
Essential dimension via prismatic cohomology
Jesse Wolfson (UC Irvine)
Let A be a complex abelian variety. Using prismatic cohomology, we show that for all but finitely many primes p, the multiplication-by-p cover p:A\to A is p-incompressible, as conjectured by Brosnan. As an application, we obtain new p-incompressibility results for congruence covers of Shimura varieties, extending previous work of Farb-Kisin-W, Brosnan-Fakhruddin, and Fakhruddin-Saini. This is joint work with Benson Farb and Mark Kisin.
November 15, 2021
A compactly supported motivic Euler characteristic via the Hochschild complex
Morgan Opie (UCLA)
The motivic Euler characteristic of a smooth, projective variety over a field k is an invariant that takes values in the Grothendieck--Witt group GW(k) of equivalence classes of bilinear forms over k. In this talk, we will show that the motivic Euler characteristic over a field k of characteristic zero can be defined using the Hochschild complex together with a canonical bilinear form. Our definition induces a map from the Grothendieck group of k-varieties to GW(k), extending the definition of the motivic Euler characteristic to all varieties over k. As time permits, we will discuss the possibility of lifting this map to a spectrum-level construction. This is joint work with Niny Arcila-Maya, Candace Bethea, Kirsten Wickelgren, and Inna Zakharevich.
November 22, 2021
Representation Theory on Spaces
Raphaël Rouquier (UCLA)
Geometric representation theory uses geometrical objects to construct linear representations. I will discuss an emerging theory where those geometrical objects are themselves viewed as "representations of a higher group". More general "higher groups” should arise as invariants of algebraic varieties or categories and should act on the moduli spaces of enumerative geometry. I will describe some aspects of the representation theory of the "higher group" associated to a point.
November 29, 2021
Two Lives: Unimodular rows and a new link to Spin groups
Vineeth Chintala (UC San Diego)
Over the decades following the influential Quillen-Suslin theorem (1976), Unimodular rows have been investigated using diverse tools (from A^1 homotopy theory, K-theory and Commutative algebra). In this talk, we will introduce unimodular rows and highlight a new perspective using Clifford algebras.
December 6, 2021
The Hilbert scheme of points on affine space
Burt Totaro (UCLA)
I will discuss the Hilbert scheme of d points on affine n-space, with some examples. This space has many irreducible components for n at least 3 and has been poorly understood. For n larger than d, we determine the homotopy type of the Hilbert scheme in a range of dimensions. Many questions remain. (Partly joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson.)
The previous seminar webpage can be found here.