2019-20

Algebra and Algebraic Geometry seminar at UBC

Except when noted otherwise, the seminar meets Monday 3-4pm in MATH 225.

Announcements

Talks and schedule changes are announced on the alggeom mailing list. Please email the organizer to be added to the list.

Organizers

Ming Zhang (zhangming@math.ubc.ca)

Schedule

2019/20 Winter Term 2

Date	Speaker	Title/Abstract

2020-01-13	Shuai Wang (Columbia University)	Title: Relative Gromov-Witten theory and vertex operators
		Abstract: We study the relative Gromov-Witten theory on T*P^1 \times P^1 and show that certain equivariant limits give us the relative invariants on P^1\times \P^1. By formulating the quantum multiplications on Hilb(T*P^1) computed by Davesh Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion and tangency operators computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits.

2020-01-20	Dylan G.L. Allegretti (UBC)	Title: Stability conditions and cluster varieties from surfaces
		Abstract: In low-dimensional geometry and topology, there is a classical construction that takes a holomorphic quadratic differential on a surface and produces a PGL(2)-local system. This construction provides a local homeomorphism from the moduli space of quadratic differentials to the moduli space of local systems. In this talk, I will propose a categorical generalization of this construction. In this generalization, the space of quadratic differentials is replaced by a complex manifold parametrizing Bridgeland stability conditions on a certain 3-Calabi-Yau triangulated category, while the space of local systems is replaced by a cluster variety. I will describe a local homeomorphism from the space of stability conditions to the cluster variety in a large class of examples and explain how it preserves the structures of these two spaces.

2020-01-27	Chi-Yu Cheng (University of Washington)	Title: Variation of Instability in Invariant Theory
		Abstract: Mumford's GIT quotient is one way to construct moduli spaces that parametrize classes of algebro-geometric objects. It turns out there is an interesting structure on the set of unstable points discarded in the GIT quotients. In this talk I would aim to describe: 1. the stratification of the unstable points and its variation caused by different choices of linearizations; 2. a wall and chamber decomposition analogous to Variation of Geometric Invariant Theory Quotient; 3. examples and results in the case of projective toric varieties.

2020-02-03	Nguyen-Bac Dang (Stony Brook University)	Title: Spectral gap in the dynamical degrees of tame automorphism preserving an affine quadric threefold
		Abstract: In this talk, I will present the tame automorphisms group preserving an affine quadric threefold. The main focus of my talk is the understanding of the degree sequences induced by the elements of this group. Precisely, I will explain how one can apply some ideas from geometric group theory in combination with valuative techniques to show that the values of the dynamical degrees of these tame automorphisms admit a spectral gap. Finally, I will apply these techniques to study random walks on this particular group.

2020-02-10	Federico Scavia (UBC)	Title: Codimension two cycles on classifying stacks of algebraic tori
		Abstract: We give an example of an algebraic torus T such that the torsion subgroup of the Chow group CH^2(BT) is non-trivial. This answers a question of Blinstein and Merkurjev.

2019-02-17

No seminar (Family Day)

2020-02-24	Dylan G.L. Allegretti (UBC)	Title: Quivers, canonical bases, and categorification
		Abstract: In a famous paper from 2003, Fock and Goncharov conjectured that the algebra of regular functions on a cluster variety has a canonical basis parametrized by the tropicalization of a dual cluster variety. In this talk, I will show how to construct this canonical basis in an interesting class of examples. Using ideas from the representation theory of quivers, I will construct graded vector spaces which categorify the elements of the canonical basis. These graded vector spaces are closely related to spaces of BPS states in supersymmetric quantum field theories.

2020-03-02	Charles Favre (École Polytechnique)	Title: Degeneration of complex manifolds to hybrid spaces and applications
		Abstract: I will discuss the notion of hybrid spaces introduced by Berkovich and further developed by Boucksom and Jonsson in order to understand various problems concerning degenerations of complex manifolds. Applications to complex dynamical systems will be presented.

2020-03-09	Zinovy Reichstein (UBC)	Title: On the number of generators of a finite algebra over a ring
		Abstract: Let k be a field, A be a finite-dimensional k-algebra (not necessarily commutative, associative or unital), and R be a commutative ring containing k. An R-algebra B is called an R-form of A if there exists a faithfully flat ring extension S/R such that B and A become isomorphic after tensoring with S. In this talk, based on joint work with Uriya First, I will be interested in the following question: if A can be generated by n elements as a k-algebra, how many elements are required to generate B as an R-algebra? For example, if A is an n-dimensional k-algebra with trivial (zero) multiplication, then an R-form of A is the same thing as a projective R-module. Otto Forster (1964) showed that every projective R-module B can be generated by n + d elements, where d is the Krull dimension of R. Richard Swan subsequently showed that this number is optimal. I will discuss generalizations of Forster's result to other types of algebras, in particular to Azumaya algebras.

2020-03-16	Kuan-Wen Lai (UMass Amherst) [CANCELED]	Title: [CANCELED] - New rational cubic fourfolds arising from Cremona transformations
		Abstract: It is conjectured that two cubic fourfolds are birational if their associated K3 categories are equivalent. We prove this conjecture for very general cubic fourfolds containing a Veronese surface, where the birational maps are induced from a Cremona transformation. Using these birational maps, we find new rational cubic fourfolds. This is joint work with Yu-Wei Fan.

2020-04-06	Nathan Ilten (SFU) [CANCELED]	Title: [CANCELED]
		Abstract:

2019/20 Winter Term 1

Date	Speaker	Title/Abstract

2019-09-09		Title: Organizational meeting
		Abstract:

2019-09-16	Naoki Koseki (University of Tokyo)	Title: Birational geometry of the moduli spaces of coherent sheaves on blown-up surfaces
		Abstract: To study the difference between motivic invariants of the moduli spaces of coherent sheaves on a smooth surface and that on the blown-up surface, Nakajima-Yoshioka constructed a sequence of flip-like diagrams connecting these moduli spaces. In this talk, I will explain birational geometric property of Nakajima-Yoshioka's wall crossing diagram. It turned out that it realizes a minimal model program.

2019-09-23	Ming Zhang (UBC)	Title: K-theoretic quasimap wall-crossing for GIT quotients
		Abstract: For a large class of GIT quotients X=W//G, Ciocan-Fontanine—Kim—Maulik and many others have developed the theory of epsilon-stable quasimaps. The conjectured wall-crossing formula of cohomological epsilon-stable quasimap invariants for all targets in all genera has been recently proved by Yang Zhou. In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants with level structure and prove their wall-crossing formulae for all targets in all genera. In particular, it will recover the genus-0 K-theoretic toric mirror theorem by Givental-Tonita and Givental, and the genus-0 mirror theorem for quantum K-theory with level structure by Ruan-Zhang. It is based on joint work in progress with Yang Zhou.

2019-09-30	Charles Favre (École Polytechnique)	Title: Degree growth of rational maps
		Abstract: The understanding of the growth of degrees of iterates of a rational self-map of a projective variety is a fundamental problem in holomorphic dynamics. I shall review some basic results of the theory and discuss some recent directions of research.

2019-10-07	Sabin Cautis (UBC)	Title: Categorical structure of Coulomb branches of 4D N=2 gauge theories
		Abstract: Coulomb branches have recently been given a good mathematical footing thanks to work of Braverman-Finkelberg-Nakajima. We will discuss their categorical structure. For concreteness we focus on the massless case which leads us to the category of coherent sheaves on the affine Grassmannian (the so called coherent Satake category). This category is conjecturally governed by a cluster algebra structure. We will describe a solution to this conjecture in the case of general linear groups and discuss extensions of this result to more general Coulomb branches of 4D N=2 gauge theories. This is joint work with Harold Williams.

2019-10-14

No seminar (Thanksgiving)

2019-10-21	Jim Bryan (UBC)	Title: K3 surfaces with symplectic group actions, enumerative geometry, and modular forms
		Abstract: The Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the "G-fixed Hilbert schemes” parametrizing G-invariant collections of points on X are related to modular forms of level $|G|$ and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as chi_y genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.

2019-10-28	Alex Weekes (UBC)	Title: Coulomb branches of 3d N=4 theories
		Abstract: The Coulomb branches of 3d N=4 gauge theories were recently given a mathematical definition by Braverman, Finkelberg, and Nakajima. These very interesting algebraic varieties were already discussed in Sabin Cautis's talk a few weeks ago, but since they may be unfamiliar I will overview their definition and properties, and discuss some interesting examples. Finally, I will discuss my joint work with Nakajima where we give a generalization of the definition of Coulomb branches, which allows us to realize affine Grassmannian slices of all finite types.

2019-11-04	Stephen Scully (University of Victoria)	Title: On a generalization of Hoffmann's separation theorem for quadratic forms over fields
3:00-4:00 PM		Abstract: The problem of determining conditions under which a rational map can exist between a pair of twisted flag varieties plays an important general role in the study of linear algebraic groups and their torsors over arbitrary fields. A non-trivial special case arising in the algebraic theory of quadratic forms amounts to studying the extent to which an anisotropic quadratic form can become isotropic over the function field of a quadric. To this end, let $p$ and $q$ be a pair of anisotropic non-degenerate quadratic forms over a field, and let $k$ be the dimension of the anisotropic part of $q$ over the function field of the quadric $p=0$. We then make the general conjecture that the dimension of $q$ must lie within $k$ of an integer multiple of $2^{s+1}$, where $s$ is such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$. This generalizes a well-known "separation theorem" of D. Hoffmann, bridging the gap between it and some other classical results on Witt kernels of function fields of quadrics. Note that the statement holds trivially if $k \geq 2^s - 1$. In this talk, I will discuss recent work that confirms the claim in the case where $k \leq 2^{s-1} + 2^{s-2}$, and more generally when $\mathrm{dim}(p) > 2k - 2^{s-1}$.
2019-11-04	Clifton Cunningham (University of Calgary)	Title: The geometry of Arthur packets for p-adic groups
4:10-5:10 PM		Abstract: Using an example to illustrate the process, I will explain how an Arthur parameter $\psi$ for a p-adic group $G$ determines a category $P_\psi$ of equivariant perverse sheaves on a moduli space $X_\psi$ of Langlands parameters for $G$ and then how the microlocal perspective on $P_\psi$ reveals the local Arthur packet $\Pi_\psi$ attached to $\psi$. This talk will not assume you already know how to compute Arthur packets for p-adic groups but rather will show how to compute these things directly using geometric tools -- that's really one of the main points of this perspective. Joint with Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu.

2019-11-11

No seminar (Remembrance Day)

2019-11-18	Shamil Asgarli (UBC)	Title: Biregular Cremona transformations of the plane
		Abstract: We study the birational self-maps of the projective plane that induce bijections on the k-rational points for a given field k. These form a subgroup BCr_2(k) inside the Cremona group. The elements of BCr_2(k) are called Biregular Cremona transformations. We show that BCr_2(k) is not finitely-generated under a mild hypothesis on the field k. When k is a finite field, we study the possible permutations induced on the k-rational points of the plane. This is joint work with Kuan-Wen Lai, Masahiro Nakahara and Susanna Zimmermann.

2019-11-25	Toni Annala (UBC)	Title: Projective bundle formula in derived cobordism theory
		Abstract: I will introduce the universal precobordism theory, which generalizes algebraic cobordism of Levine-Morel to arbitrary quasi-projective schemes over a Noetherian base ring A. In the main part of the talk I will outline the proof of projective bundle formula for this new cohomology theory. The usual proof techniques based on resolution of singularities and weak factorization break down in this generality, so we have to use an alternative approach based on carefully studying the structure of precobordism rings of varieties with line bundles, which were inspired by a paper of Lee-Pandharipande. The talk is based on joint work with Shoji Yokura.

2019-12-02	Yu-Hsiang Liu (UBC)	Title: Donaldson-Thomas theory for quantum Fermat quintic threefolds
		Abstract: In this talk, I will define Donaldson-Thomas type invariants for non-commutative projective Calabi-Yau-3 schemes whose associated graded algebras are finite over their centers. As an example, I will discuss the local structure of Hilbert schemes of points on the quantum Fermat quintic threefold, and compute some of its invariants.