March 7-8, 2020 @ USaskatchewan

Location and Directions

All talks will take place in room ESB3 (Edwards School of Business, room 3). The poster session will take place in McLean Hall Room 201 (our math lounge). Maps with walking routes from the hotel to ESB and then from ESB to McLean are embedded below.

For lunch, there are a few food options: There is an all you can eat buffet at Marquis Hall on campus. This would be the closest option. There are also a handful of off-campus restaurants at the corner of Cumberland and College.

Speakers and Abstracts

Nils Bruin (SFU): Quasi-hyperbolicity of nodal surfaces

A surface is called algebraically quasi-hyperbolic if it only contains finitely many curves of genus zero or one. One way to show that a projective surface is quasi-hyperbolic is to show that the m-th symmetric power of the sheaf of differentials on the surface has global sections. Various results on hyperbolicity of sufficiently general surfaces in certain families have been obtained this way.

Bogomolov and Oliveira observed that the presence of nodal singularities on a hypersurface contributes to the existence of m-th symmetric differentials on the desingularization. We refine and generalize their results to complete intersections, giving bounds rather than asymptotic results.

We also outline a way to explicitly determine a locus containing all the non-hyperbolic curves on the surface. As an application, we show that any genus zero curve on the so-called rational cuboid surface needs to pass through singularities of the surface (whether the surface is quasi-hyperbolic remains open).

This is joint work with Anthony Várilly-Alvarado and Jordan Thomas.

Charles Doran (Alberta): Gluing Periods for DHT Mirrors

Let X be a Calabi–Yau manifold that admits a Tyurin degeneration to a union of two quasi-Fano varieties X_1 and X_2 intersecting along a smooth anticanonical divisor D. The “DHT mirror symmetry conjecture” implies that the Landau–Ginzburg mirrors of (X_1,D) and (X_2,D) can be glued to obtain the mirror of X. In this talk, I will explain how periods on the Landau-Ginzburg mirrors of (X_1,D) and (X)2,D) are related to periods on the mirror of X. The relation among periods relates different Gromov–Witten invariants via their respective mirror maps. This is joint work with Fenglong You and Jordan Kostiuk.

Susan Cooper (Manitoba): Pruning Fat Points

The Hilbert function of a point set in projective space encodes a surprising amount of data, and hence plays a central role in numerous interesting problems in Algebraic Geometry and Commutative Algebra. In this talk we will investigate the challenging problem of characterizing Hilbert functions of point sets when multiplicities of points are considered.

Cameron Franc (Saskatchewan): Classification of some vertex operator algebras

Vertex operator algebras (VOAs) are vector spaces endowed with infinitely many bilinear operations satisfying many intricate identities. They are of importance in mathematics in fields as diverse as finite group theory and moduli of curves and vector bundles. Likewise, VOAs find importance in physics via conformal field theory and string theory. There are surprising and deep connections between the theory of VOAs and the theory of modular forms that have connected these subjects from the very beginning of VOA theory. In this talk we will survey some of these ideas and explain some recent results that use modular forms and number theory to help classify certain VOAs.

David Favero (Alberta): Exceptional Collections for Factorizations of Invertible Polynomials with Maximal Symmetry

Homological mirror symmetry (which conjectures an equivalence between categories from algebraic and symplectic geometry) is often proven by comparing generating objects for the two categories in question. The simplest class of such generating sets are called full exceptional collections. I will discuss recent joint work with Daniel Kaplan and Tyler Kelly wherein we prove that the equivariant derived category of factorizations of an invertible polynomial with its maximal symmetry group has a full exceptional collection (this is the mirror affine space together with the transpose polynomial). In the Gorenstein case, this full exceptional collection is also strong. The former was conjectured by Hirano-Ouchi and the latter was conjectured (in general) by Takahashi and Lekili-Ueda. This should be of use in proving homological mirror symmetry for Berglund-Hubsch-Krawitz pairs.

Elana Kalashnikov (Harvard): Finding mirrors for Fano quiver flag zero loci

One interesting feature of the classification of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersection, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I’ll discuss finding mirrors of four dimensional Fano quiver flag zero loci via finding degenerations of the ambient quiver flag varieties. These degenerations generalise the Gelfand-Cetlin degeneration, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.

Matthew Satriano (Waterloo): Interpolating Between the Batyrev-Manin and Malle Conjectures

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.

Adam Topaz (Alberta): Incidence structures in anabelian geometry

This talk will discuss the classical approach to projective geometry via incidence structures (abstract points, lines etc.) and some of its applications in modern arithmetic geometry. One such application highlights the connection between such incidence structures, valuation theory, and K-theory, and plays a fundamental role in several recent results from birational anabelian geometry. Time permitting, I will also discuss some work in progress which relates incidence geometry with number theory via the structure of étale fundamental groups of configuration spaces.

Schedule

SATURDAY:

9:00 - 9:50 David Favero

10:15 - 11:05 Elana Kalashnikov

11:30 - 12:20 Matthew Satriano

LUNCH

2:30 - 3:20 Charles Doran

3:45 - 4:35 Susan Cooper

4:45 - 6:00 POSTERS

6:30 DINNER

SUNDAY

9:15 - 10:05 Nils Bruin

10:30- 11:20 Cameron Franc

11:30 - 12:20 Adam Topaz

Registration

You may register and submit poster abstracts here: https://www.pims.math.ca/scientific-event/200307-cwags

We have some funding to cover local expenses of students and postdocs. Anyone who is interested in receiving funds should email cawagsymposium@gmail.com by Feb. 3. Preference will be given to those presenting a poster.

Lodging

If your lodging has not been arranged by the organizers, you might consider staying at the Holiday Inn Express.

Organized by Charles Doran, David Favero, Nathan Ilten, and Jenna Rajchgot. To contact the organizers, please email cawagsymposium@gmail.com.

Support for CWAGS is generously provided by the Pacific Institute for the Mathematical Sciences.