UBC Algebraic Geometry Seminar

The elliptic Hall algebra EHA is a widely studied object in geometric representation theory and combinatorics. This algebra arises as the Hall algebra of the category of coherent sheaves of a smooth elliptic curve. The Bqt algebra arose in the work of Carlsson-Gorsky-Mellit as a certain algebra of operators on the K-theory of parabolic flag Hilbert schemes. In recent work, alongside Gorsky and Simental, we showed that Bqt contains EHA as its spherical subalgebra. In particular, EHA is but one in a family of so-called K-theoretic Hall algebras, which depend on a quiver Q.

In this talk I will discuss work Gorsky and Simental where we define new quiver Bqt algebras whose (conjecturally) spherical subalgebras are precisely the associated K-theoretic Hall algebras. To do, we introduce split parabolic quiver varieties, new spaces which generalize Nakajima quiver varieties, on which we define a family of operators that yield our desired algebra.

In recent work with Felix Thimm and Nick Kuhn, we proved a Joyce-style "universal" wall-crossing formula for certain equivariant moduli problems of 3-Calabi-Yau type, including Donaldson-Thomas theory. An immediate and productive question is how tautological classes like descendents transform under such wall-crossings. I will present an explicit descendent transformation formula for the Donaldson-Thomas/Pandharipande-Thomas wall-crossing of equivariant vertices, explain how the computation works, and speculate on how it may be generalized. This serves as a fairly down-to-earth example of how such wall-crossing formulas may be applied.

We define the algebraic double loop space, and algebro-geometric analogue of the double loop space in topology. We discuss quiver descriptions of the algebraic loop space of various homogeneous spaces. We show how this leads to various new results both in topology and algebraic geometry.

Let F be a number field (say, the field of rational numbers Q) or a p-adic field (say, the field of p-adic numbers Q_p), or a global function field (say, the field of rational functions of one variable over a finite field F_q). Let G be a connected reductive group over F (say, SO(n) ). One needs the first Galois cohomology set H^1(F,G) for classification problems in algebraic geometry and linear algebra over F. In the talk, I will give closed formulas for H^1(F,G) when F is as above, in terms of the algebraic fundamental group \pi_1(G) introduced by the speaker in 1998. All terms will be defined and examples will be given.

The talk is based on a joint work with Tasho Kaletha.

For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behavior of the zeta-function at 1 to other invariants of the surface. We give a version which equates invariants only depending on the Brauer group to invariants only depending on the Neron-Severi group. We estimate the terms appearing in the formula, and discuss the special case of abelian varieties and K3-surfaces. 

We study Quot schemes of rank 0 quotients on smooth projective curves. These Quot schemes exhibit a rich and highly structured geometry, with formal analogies to the Hilbert scheme of points on surfaces. In this talk, I will explain a formula for the twisted Hodge groups of the Quot scheme with values in tautological line bundles pulled back from the symmetric product.

Let G be a split reductive group. For X a projective G-homogeneous variety, there is a description of the decomposition of the Chow motive of X (Chernousov-Gille-Merkurjev, Brosnan). It is also known that these varieties satisfy the so-called Rost nilpotence principle. In classical cohomology theories for algebraic varieties (e.g., singular cohomology, ℓ-adic cohomology), there is a strong relation between the cohomology of a variety and its hyperplane sections, most famously in the form of the Lefschetz Hyperplane Theorem. We present some examples where a motivic version of the Lefschetz theorem holds. More precisely, for Milnor hypersurfaces (incidence varieties of dimension 1 and codimension 1 subspaces) and some twisted forms thereof, we show that hyperplane sections corresponding to regular semisimple elements admit a motivic decomposition of the expected form. As a corollary, we see that the Rost nilpotence principle holds for these hyperplane sections. 

This talk is based on joint work with Kirill Zaynullin and Rui Xiong.

The Cremona dimension of a finite group G is the minimal dimension of a rationally connected variety which admits a faithful action of G.  In this talk, based on joint work with Giulio Bresciani and Angelo Vistoli, I will discuss new lower bounds on the Cremona dimension of a finite p-group.

We present the patching method, a machinery developed by Harbater–Hartmann–Krashen and various other authors, dedicated to the study of arithmetics of linear algebraic groups over function fields of curves over complete discretely valued field such as ℚₚ(T). Then, we present a new result in this direction, which gives a nine 9-term exact sequence for Galois cohomology of 2-term complexes of tori in the patching setting. This relies on the notion of (co-)flasque resolutions of such complexes, generalizing the previous work by Colliot-Thélène–Sansuc. As applications, we show that patching holds for nonabelian second Galois cohomology of reductive groups with a smooth center, as well as a weak local–global principle for this cohomology set. We also rediscover a local–global principle for indices of central simple algebras.

John Pardon has recently proved the Gromov-Witten Donaldson-Thomas correspondence using a new homology/cohomology machinery for 1-cycles in 3-folds.  We study a much simplified version, namely for 0-cycles (points) in curves.  In this case the theory is quite simple to explain, but still exhibits some interesting features of Pardon’s theory.

Starting from celebrated work of Kazhdan-Lusztig, rich geometric and spectral pictures of affine Hecke algebras have emerged as a special case of the Local Langlands Correspondence for p-adic groups. I will report on work, initiated by Braverman-Kazhdan, to extend these pictures to a slightly larger algebra, Lusztig's asymptotic Hecke algebra. I will give some formulas for the latter in terms of its action on the equivariant K-theory of the flag variety and the geometry of the nilpotent cone. Similar functions appear in work of Casselman-Cely-Hales.

We will discuss a class of ideals in a polynomial ring studied by Mark Haiman in his work on the Hilbert scheme of points and discuss how they are related to homology of affine Springer fibers, Khovanov-Rozansky homology of links, and to a conjecture by Oblomkov, Rasmussen, and Shende.

Given an abelian variety G defined over a field of characteristic 0, the intersection between a subvariety X of G with another subvariety Y of G for which dim(X) + dim(Y) < dim(G) is generally expected to be empty. Rooted in this basic observation, Pink-Zilber and Bombieri-Masser-Zannier formulated the following conjecture regarding unlikely intersections in arithmetic geometry (over a field of characteristic 0). So, let X be an irreducible subvariety of dimension d of the abelian variety G, and assume X is not contained in any proper algebraic subgroup of G. Then the intersection of X with the union of all algebraic subgroups of G of codimension d+1 is predicted not to be Zariski dense in X. Motivated by this conjecture, we present a couple of related questions in arithmetic dynamics.

I will discuss how to formulate and prove a localization theorem for the virtual fundamental class of a moduli space with a relative perfect obstruction theory over a singular base.  In the motivating example of the moduli space of stable log maps, I will explain how this leads to sums over types of tropical curves of cycle classes on moduli spaces of curves. Time permitting I will discuss some work in progress with Hu and Zaslow doing computations of counts on toric log K3 surfaces where we find agreement between these invariants and counts of constructible sheaves.

Tableaux are fundamental objects in representation theory and combinatorics, and variations of the Schensted algorithm have endowed them with rich algebraic structures. In this talk I will discuss a naive monoid structure on the set of semistandard Young tableaux that does not arise as an insertion algorithm, and the good properties inherited by its associated algebra. We will then mention two applications of our work; one to algebraic geometry, and another to representation theory. For the first, we will show that the tableaux algebra is a flat degeneration of the algebra of global sections of the partial flag variety. For the second, we will show that this naive monoid structure of semistandard Young tableaux induces a crystal embedding when applied to the highest (or lowest) weight vectors.

This talk provides a gentle introduction to Arthur packets, a less gentle introduction to generalized local Arthur packets, and concludes with an application to a problem in quantum circuit design. 

The Briançon-Skoda theorem is a comparison relating the integral closure of powers of a finitely generated ideal with its ordinary powers. Originally proved using analytic methods for coordinate rings of smooth varieties over the complex numbers in 1974, it took until 1981 for Lipman and Sathaye to provide an algebraic proof for all regular local rings, regardless of characteristic. Since then, there have been other proofs and generalizations to mild singularities, most notably using tight closure theory in positive characteristic and reduction mod p. In this talk, we prove a general Briançon-Skoda containment for pseudo-rational singularities in all characteristics. Our method is quite simple, and it recovers and unifies many previously known results while also extending them to mixed characteristic. It also yields some new results on F-pure and Du Bois singularities (as well as a characteristic-free analog) and settles a conjecture of Huneke on uniform bounds regardless of the singularities of the ring. This is based on joint work with Linquan Ma, Rebecca R.G., and Karl Schwede.

I will talk about several results on Hecke algebras attached to Bernstein blocks of arbitrary reductive p-adic groups, and their applications to the local Langlands program. One such application is an explicit understanding of the (classical) arithmetic Local Langlands correspondence with explicit L-packets. Another such application involves various categorical "upgrades", for example, an equivalence between module categories of Hecke algebras arising from both the automorphic and the spectral sides. If time permits, I will talk about a categorical local Langlands correspondence featuring certain coherent Springer sheaves on moduli spaces of Langlands parameters. Moreover, I will try to explain why such results are of interest to number theorists.

Springer fibers are certain subvarieties of the flag variety that provide a geometric construction of representations of the symmetric group. For instance, the graded Frobenius character of their cohomology are the modified Hall–Littlewood symmetric functions. Many recent works have studied two “double” generalizations of this classical story in terms of Borel-Moore homology of affine Springer fibers or global sections of coherent sheaves on the Hilbert scheme of points on the plane, both of which turn out to be closely related to Khovanov-Rozansky homology of algebraic links. In this talk, I will introduce these double generalizations and their connections in the case of torus links. Based on joint work in progress with Pablo Boixeda Alvarez and Thomas Hameister.

If X is a smooth complete variety, then the alternating sums of dimensions of sheaf cohomology groups gives an additive map from the Grothendieck group of vector bundles on X to the integers. If X is not complete, then sheaf cohomology groups are generally infinite-dimensional, so we shouldn’t expect such an additive map to exist. Nonetheless, motivated by recent results in matroid theory, we introduce and study a class of incomplete toric varieties for which an analogue of this additive map still exists. I will discuss structural results about these special toric varieties, as well as generalizations of the Hirzebruch-Riemann-Roch formula and Brion’s localization formula. This talk covers joint work (some ongoing) with subsets of Matt Beck, Melody Chan, Emily Clader, and Carly Klivans.

Learning linear causal models via algebraic constraints: The main task of causal discovery is to learn direct causal relationships among observed random variables. These relationships are usually depicted via a directed graph whose vertices are the variables of interest and whose edges represent direct causal effects. In this talk we will discuss the problem of learning such a directed graph for a linear causal model. I will specifically address the case where the graph may have hidden variables or directed cycles. In general, the causal graph cannot be learned uniquely from observational data. However, in the special case of linear non-Gaussian acyclic causal models, the directed graph can be found uniquely. When cycles are allowed the graph can be learned up to an equivalence class. We characterize the equivalence classes of such cyclic graphs and we propose algorithms for causal discovery. Our methods are based on using algebraic relationships among the second and higher order moments of the random vector. We show that such algebraic relationships are enough to identify the graph.

The Ceresa period from tropical homology: The Jacobian of a very general curve of genus at least 3 contains an algebraic cycle called the Ceresa cycle that is homologically trivial but algebraically nontrivial.  It is straightforward to show that hyperelliptic curves have trivial Ceresa cycle, but understanding whether the converse holds has been a subject of interest for decades.  I will give an overview of the classical story before explaining how (and why) to extend the Ceresa cycle to the setting of tropical geometry.

Mathematics for AI: What Algebraic Geometry Brings to the Table?: In this talk, I will give an overview of (neuro)algebraic geometry, an emerging field analogous to algebraic statistics that uses algebraic geometry to study the theory of deep learning. I will focus on feedforward neural networks. The universal approximation theorem tells us that, with sufficiently large hidden width, a shallow neural network can approximate any continuous function on a compact set arbitrarily well. Rather than taking width to infinity, we fix a neural network architecture and ask about its expressivity. To study this question, we associate a geometric object called a neuromanifold to a fixed architecture: the image of its parameter space inside an ambient space of functions. When the activation function is algebraic, this ambient space can be chosen to be finite-dimensional. This allows us to take the Zariski closure of the neuromanifold over the real or complex numbers, thereby associating an algebraic variety to the architecture. I will explain how geometric invariants of this variety, such as dimension, degree, defining equations, and singularities, can reveal structural properties of neural networks and shed light on their black-box behavior. More broadly, the talk illustrates how algebro-geometric tools can contribute to the mathematical foundations of deep learning alongside more familiar statistical and probabilistic methods.

Generic curves non-coprime Catalans: Given a plane curve singularity with a parametrization (x(t),y(t)), one can define its compactified Jacobian as the moduli space of (x(t),y(t))-submodules in the ring of power series in t. Affine pavings of compactified Jacobians for various families of curves have been extensively studied because of their connections to representation theory, combinatorics, and invariants of algebraic knots.

The case when the degrees of the leading terms of x(t) and y(t) are relatively prime goes back to Lusztig and Smelt, and later Piontkowski. Back in 2013, joint with Eugene Gorsky, we related Piontkowski's results to rational Catalan combinatorics. More recently, together with Eugene Gorsky and Alexei Oblomkov, we studied the class of generic curves with non-coprime lead exponents. We constructed affine pavings in this case, and provided a connection to non-coprime Catalan combinatorics, extending our previous work.

In this talk, I will introduce compactified Jacobians of plane curve singularities, explain the most natural approach to subdividing it into simpler pieces, and formulate our results for the generic curves, providing plenty of examples.

Asgarli, Ghioca, and Reichstein recently proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. In this talk, I will discuss how to address this question. More generally, I will show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. Joint work with Shamil Asgarli and Jonathan Love.