Algebra and Algebraic Geometry seminar at UBC
Except when noted otherwise, the seminar meets Monday 45pm 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

20190107

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.

20190110 (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\'eronTate 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.)

20190114

Kevin Casto (UBC)

Title: Representation theory and arithmetic statistics of generalized configuration spaces



Abstract: ChurchEllenbergFarb introduced the theory of FImodules 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 GrothendieckLefschetz formula connects these topological stability phenomena to stabilization of statistics for polynomials and rational maps over finite fields.

20190121

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 Zcoefficients 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.

20190128

Rostislav Devyatov (Alberta)

Title: Multiplicityfree 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 socalled canonical
dimension of flag varieties and groups over nonalgebraicallyclosed
fields.

20190204

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 nonconstant finite étale group schemes $A$ such that $\{BA\}\neq 1$.

20190211

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 selfmorphism 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 $(nk)$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 $k=1$.

20190218

No seminar (Family Day)


20190225

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 indschemes, 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.

20190304

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.

20190318

Sebastian CasalainaMartin (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 AbelJacobi 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.

20190325

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 degree2 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.

2018/19 Winter Term 1
Date

Speaker

Title/Abstract

Special: 20180820

Zheng Hua (Hong Kong)

Title: Noncommutative MatherYau 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 0th Hochschild homology represented by the super potential, assuming the Jacobi algebra is finite dimensional. This is a noncommutative version of the famous MatherYau theorem in isolated hyper surface singularities. As a consequence, we prove a rigidity theorem for Ginzburg dgalgebra. I will discuss some applications of these results in three dimensional birational geometry. This is a joint work with Guisong Zhou (1803.06128).

20180910

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/nonAbelian 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: 20180913

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.

20180917

Matthew Satriano (Waterloo)

Title: Interpolating between the BatyrevManin and Malle Conjectures



Abstract: The BatyrevManin 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 ZureickBrown.

20180924

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.

20181001

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, CohenMacaulay, 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.

20181008

No seminar (Thanksgiving Day)


20181015

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.

20181022

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 Ktheory and to intersection theory.

20181105

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.

20181112

No seminar (Remembrance Day)


SPECIAL: 20181113

Fenglong You (Alberta)

Title: Relative and orbifold GromovWitten 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 GromovWitten invariants of (X,D) and orbifold GromovWitten invariants of the rth root stack X_{D,r}. For sufficiently large r, AbramovichCadmanWise 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 GromovWitten invariants of X_{D,r} are polynomials in r and the constant terms are exactly higher genus relative GromovWitten 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 GromovWitten 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.

20181119

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 nonplanar curves.

20181126

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.
