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Algebra and Algebraic Geometry seminar at UBC

Except when noted otherwise, the seminar meets Monday 4-5pm in MATH 126.


Announcements

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

Organizers

Fei Hu (fhu@math.ubc.ca)

Schedule

2018/19 Winter Term 2

Date Speaker Title/Abstract
2019-01-07 Seidon Alsaody (Alberta) Title: Exceptional Groups and Exceptional Algebras
Abstract: Exceptional groups (over arbitrary rings) are related to octonion algebras, triality and exceptional Jordan algebras. I will talk about recent results of an approach to these objects using certain torsors (principal homogeneous spaces) under smaller exceptional groups, and explain how an explicit understanding of these torsors provides insight into the objects and their interrelations.
2019-01-10 (Special) Farbod Shokrieh (Copenhagen) Title: Heights and tropical geometry
Location: ESB 4133

Time: 3:30pm

Abstract: Given a principally polarized abelian variety A over a number field (or a function field), one can naturally extract two real numbers that capture the ``complexity of A: one is the Faltings height and the other is the N\'eron-Tate height (of a symmetric effective divisor defining the polarization). I will discuss a precise relationship between these two numbers, relating them to some subtle invariants arising from tropical geometry (more precisely, from Berkovich analytic spaces).

(Joint work with Robin de Jong.)

2019-01-14 Kevin Casto (UBC) Title: Representation theory and arithmetic statistics of generalized configuration spaces
Abstract: Church-Ellenberg-Farb introduced the theory of FI-modules to explain the phenomenon of representation stability of the cohomology of configuration spaces. I will explain the basics of how this story goes, and then explain how to extend their analysis to two generalized types of configuration spaces. Furthermore, I will explain how the Grothendieck-Lefschetz formula connects these topological stability phenomena to stabilization of statistics for polynomials and rational maps over finite fields.
2019-01-21 Alexander Neshitov (USC) Title: Motivic decompositions of homogeneous spaces and representations of Hecke type algebras
Abstract: This is a joint work with B. Calmes, V. Petrov, N. Semenov and K. Zainoulline. In the talk I will discuss a connection between direct sum decompositions of the Chow motive with Z-coefficients of a homogeneous space of a group G, and representations of affine nil Hecke algebras defined in terms of root system of G. This connnection can be used in two directions: prove indecomposability of certain motives as well as get some structural results about Hecke algebras.
2019-01-28 Rostislav Devyatov (Alberta) Title: Multiplicity-free products of Schubert divisors
Abstract: Let G/B be a flag variety over C, where G is a simple algebraic group

with a simply laced Dynkin diagram, and B is a Borel subgroup. The Bruhat decomposition of G defines subvarieties of G/B called Schubert subvarieties. The codimension 1 Schubert subvarieties are called Schubert divisors. The Chow ring of G/B is generated as an abelian group by the classes of all Schubert varieties, and is "almost" generated as a ring by the classes of Schubert divisors. More precisely, an integer multiple of each element of G/B can be written as a polynomial in Schubert divisors with integer coefficients. In particular, each product of Schubert divisors is a linear combination of Schubert varieties with integer coefficients.

In the first part of my talk I am going to speak about the coefficients of these linear combinations. In particular, I am going to explain how to check if a coefficient of such a linear combination is nonzero and if such a coefficient equals 1. In the second part of my talk, I will say something about an application of my result, namely, how it makes it possible estimate so-called canonical dimension of flag varieties and groups over non-algebraically-closed fields.

2019-02-04 Federico Scavia (UBC) Title: Motivic classes of algebraic groups
Abstract: The Grothendieck ring of algebraic stacks was introduced by Ekedahl in 2009. It may be viewed as a localization of the more common Grothendieck ring of varieties. If $G$ is a finite group, then the class $\{BG\}$ of its classifying stack $BG$ is equal to $1$ in many cases, but there are examples for which$\{BG\}\neq 1$. When $G$ is connected, $\{BG\}$ has been computed in many cases in a long series of papers, and it always turned out that $\{BG\} \cdot \{G\}=1$. We exhibit the first example of a connected group $G$ for which $\{BG\} \cdot \{G\}\neq 1$. As a consequence, we produce an infinite family of non-constant finite étale group schemes $A$ such that $\{BA\}\neq 1$.
2019-02-11 Fei Hu (UBC) Title: Cohomological and numerical dynamical degrees on abelian varieties
Abstract: In 2013, Esnault and Srinivas proved that as in the de Rham cohomology over the field of complex numbers, the algebraic entropy of an automorphism of a smooth projective surface over a finite field $\mathbb{F}_q$ is taken on the span of the Néron–Severi group inside of $\ell$-adic cohomology. Later, motivated by this and Weil's Riemann Hypothesis, Truong asked whether the spectral radius $\chi_{2k}(f)$ of the pullback $f^* \colon H^{2k}(X, \mathbb{Q}_\ell) \to H^{2k}(X, \mathbb{Q}_\ell)$ is the same as the spectral radius $\lambda_k(f)$ of the pullback $f^* \colon N^k(X)_\mathbb{R} \to N^k(X)_\mathbb{R}$, where $f \colon X \to X$ is a surjective self-morphism of a smooth projective variety $X$ of dimension $n$ defined over an algebraically closed field $\mathbb{k}$ and $N^k(X)$ denotes the finitely generated abelian group of algebraic $(n-k)$-cycles modulo the numerical equivalence. He has shown that $\displaystyle \max_{0\le i\le 2n} \chi_{i}(f) = \max_{0\le k\le n} \lambda_{k}(f)$. We give an affirmative answer to his question in the case of abelian varieties and $k=1$.
2019-02-18 No seminar (Family Day)
2019-02-25 Kai Behrend (UBC) Title: The motivic weight of the stack of bundles
Abstract: I will talk about a new approach to computing the motivic weight of the stack of $G$-bundles on a curve. The idea is to associate a motivic weight to certain ind-schemes, such as the affine Grassmannian and the scheme of maps $X \to G$, where $X$ is an affine curve, using Bittner's calculus of $6$ operations. I hope that this will eventually lead to a proof of a conjectural formula for the motivic weight of the stack of bundles in terms of special values of Kapranov's zeta function.
2019-03-04 Xi Chen (Alberta) Title: On a conjecture of Voisin
Abstract: C. Voisin proved that no two distinct points on a very general surface of degree $\ge 7$ in ${\mathbb P}^3$ are rationally equivalent. She conjectured that the same holds for a very general sextic surface. We settled this conjecture by improving her method which makes use of the global jet spaces. This is a joint work with James D. Lewis and Mao Sheng.
2019-03-11 no seminar
2019-03-18 Sebastian Casalaina-Martin (Colorado) Title: Distinguished models of intermediate Jacobians
Abstract: In this talk I will discuss joint work with J. Achter and C. Vial showing that the image of the Abel--Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers. The main focus will be on applications to topics such as: descending cohomology geometrically, a conjecture of Orlov regarding the derived category and Hodge theory, and motivated admissible normal functions.
2019-03-25 Abhishek Kumar Shukla (UBC) Title: Minimal number of generators of an étale algebra
Abstract: O. Forster proved that over a ring $R$ of Krull dimension $d$ a finite module $M$ of rank at most $n$ can be generated by $n+d$ elements. Generalizing this in great measure U. First and Z. Reichstein showed that any finite $R$-algebra $A$ can be generated by $n+d$ elements if each $A\otimes_R k(\mathfrak{p})$, for $\mathfrak{p}\in \mathrm{MaxSpec}(R)$, is generated by $n$ elements. It is natural to ask if the upper bounds can be improved. For modules over rings R. Swan produced examples to match the upper bound. Recently B. Williams obtained weaker lower bounds in the context of Azumaya algebras. In this paper we investigate this question for étale algebras. We show that the upper bound is indeed sharp. Our main result is a construction of universal varieties for degree-2 étale algebras equipped with a set of $r$ generators and explicit examples realizing the upper bound of First & Reichstein. This is joint work with Ben Williams.
2019-04-01 Wenhao Ou (UCLA) Title:
Abstract:

2018/19 Winter Term 1

Date Speaker Title/Abstract
Special: 2018-08-20 Zheng Hua (Hong Kong) Title: Noncommutative Mather-Yau theorem and its applications
Abstract: We prove that the right equivalence class of a super potential in complete free algebra is determined by its Jacobi algebra and the canonical class in its 0-th Hochschild homology represented by the super potential, assuming the Jacobi algebra is finite dimensional. This is a noncommutative version of the famous Mather-Yau theorem in isolated hyper surface singularities. As a consequence, we prove a rigidity theorem for Ginzburg dg-algebra. I will discuss some applications of these results in three dimensional birational geometry. This is a joint work with Guisong Zhou (1803.06128).
2018-09-10 Elana Kalashnikov (Imperial College London) Title: Four dimensional Fano quiver flag zero loci
Abstract: The classification of Fano varieties is unknown beyond dimension 3; however, many Fano fourfolds are expected to be GIT theoretic subvarieties of either toric varieties or quiver flag varieties. Quiver flag varieties are a generalization of type A flag varieties and are GIT quotients of vector spaces. In this talk, I will discuss my recent work on quiver flag varieties, including a proof of the Abelian/non-Abelian correspondence for quiver flag varieties, and its application in the large scale computer search for Fano fourfolds that I have carried out in joint work with T. Coates and A. Kasprzyk. We find 139 new Fano fourfolds. I will also discuss the importance of these subvarieties as a testing ground for the conjectures of Coates, Corti, Galkin, Golyshev, Kasprzyk and Tveiten on mirror symmetry for Fano varieties.
Special: 2018-09-13 Burt Totaro (UCLA) Title: Hodge theory of classifying stacks
Abstract: The goal of this talk is to create a correspondence between the representation theory of algebraic groups

and the topology of Lie groups. The idea is to study the Hodge theory of the classifying stack of a reductive group over a field of characteristic p, the case of characteristic 0 having been studied by Behrend, Bott, Simpson and Teleman. The approach yields new calculations in representation theory, motivated by topology.

2018-09-17 Matthew Satriano (Waterloo) Title: Interpolating between the Batyrev--Manin and Malle Conjectures
Abstract: The Batyrev--Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown.
2018-09-24 Alexei Oblomkov (UMass) Title: Knot invariants, Hilbert schemes and arc spaces
Abstract: In my talk I will explain (partially conjectural) relation between

1) Homology of Hilbert scheme of points on singular curves

2) Knot homology of the links of curve singularities

3) Space functions on the moduli space of maps from the formal disc to the curve singularities.

I will center my talk around discussion of the case of cuspidal curve $$ x^m=y^n $$ and its singularity. In this case it is now known that 1) 2) and 3) are essentially equal. Talk is based on the joint projects with Gorsky, Rozansky, Rasmussen, Shende and Yun.

2018-10-01 Rahul Singh (Northeastern) Title: The Conormal Variety of a Schubert Variety
Abstract: Let N be the conormal variety of a Schubert variety X. In this talk, we discuss the geometry of N in two cases, when X is cominuscule, and when X is a divisor.

In particular, we present a resolution of singularities and a system of defining equations for N, and also describe certain cases when N is normal, Cohen-Macaulay, and Frobenius split. Time permitting, we will also illustrate the close relationship between N and orbital varieties, and discuss the consequences of the above constructions for orbital varieties.

2018-10-08 No seminar (Thanksgiving Day)
2018-10-15 Nicolas Addington (Oregon) Title: Exoflops
Abstract: The derived category of a hypersurface is equivalent to the

category of matrix factorizations of a certain function on the total space of a line bundle over the ambient space. The hypersurface is smooth if and only if the critical locus of the function is compact. I will present a construction through which a resolution of singularities of the hypersurface corresponds to a compactification of the critical locus of the function, which can be very interesting in examples. This is joint work with Paul Aspinwall and Ed Segal.

2018-10-22 Toni Annala (UBC) Title: Bivariant Theories and Algebraic Cobordism of Singular Varieties
Abstract: I will outline the construction of a natural bivariant theory extending algebraic bordism,

which will yield an extension of algebraic cobordism to singular varieties. I will also discuss the connections of this theory to algebraic K-theory and to intersection theory.

2018-10-29 Sebastian Casalaina-Martin (Colorado) CANCELED
2018-11-05 Jeremy Usatine (Yale) Title: Hyperplane arrangements and compactifying the Milnor fiber
Abstract: Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement's Milnor fiber. This talk is based on joint work with Max Kutler.
2018-11-12 No seminar (Remembrance Day)
SPECIAL: 2018-11-13 Fenglong You (Alberta) Title: Relative and orbifold Gromov-Witten theory
Location: MATX 1102 Abstract: Given a smooth projective variety X and a smooth divisor D \subset X, one can study the enumerative geometry of counting curves in X with tangency conditions along D. There are two theories associated to it: relative Gromov-Witten invariants of (X,D) and orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, Abramovich-Cadman-Wise proved that genus zero relative invariants are equal to the genus zero orbifold invariants of root stacks (with a counterexample in genus 1). We show that higher genus orbifold Gromov-Witten invariants of X_{D,r} are polynomials in r and the constant terms are exactly higher genus relative Gromov-Witten invariants of (X,D). If time permits, I will also talk about further results in genus zero which allows us to study structures of genus zero relative Gromov-Witten theory, e.g. Givental formalism for genus zero relative invariants. This is based on joint work with Hisan-

Hua Tseng, Honglu Fan and Longting Wu.

2018-11-19 Dori Bejleri (MIT) Title: Motivic Hilbert zeta functions of curves
Abstract: The Grothendieck ring of varieties is the target of a rich invariant associated to any algebraic variety which witnesses the interplay between geometric, topological and arithmetic properties of the variety. The motivic Hilbert zeta function is the generating series for classes in this ring associated to a certain compactification of the unordered configuration space, the Hilbert scheme of points, of a variety. In this talk I will discuss the behavior of the motivic Hilbert zeta function of a reduced curve with arbitrary singularities. For planar singularities, there is a large body of work detailing beautiful connections with enumerative geometry, representation theory and topology. I will discuss some conjectural extensions of this picture to non-planar curves.
2018-11-26 Dan Edidin (Missiouri) Title: Saturated blowups and canonical reduction of stabilizers
Special: 3pm Abstract: We introduce a construction call the {\em saturated blowup} of an Artin stack with good moduli space. The saturated blowup is a birational map of stacks which induces a proper birational map on good moduli spaces. Given an Artin stack ${\mathcal X}$ with good moduli space $X$, there is a canonical sequence of saturated blowups which produces a stack whose rigidification is a DM stack. When the stack is smooth, all of the stacks in the sequence of saturated blowups are also smooth. This construction generalizes earlier work of Kirwan and Reichstein in geometric invariant theory and the talk is based on joint work with David Rydh.