Boston Algebraic Geometry Day 

Speakers: Paul Hacking, Siddarth Kannan, Dave Swinarski, Yan Zhou 

Location: Boston University, Center for Computing & Data Science, room 548 (665 Commonwealth Ave).

Parking: If you plan to drive, please contact Yu-Shen Lin (yslin@bu.edu) for parking permits before March 28. 

Schedule: 

9:30-10  Check-in and refreshments
10-11  Siddarth Kannan: Pólya enumeration for moduli spaces
11-11:30  Coffee break
11:30-12:30  Dave Swinarski: The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity
12:30-2:30  Lunch (not provided)
2:30-3:30  Paul Hacking: Homological mirror symmetry for K3 surfaces
3:30-4  Coffee break
4-5  Yan Zhou: Geometric origins of values of the Riemann Zeta functions at positive integers

Abstracts: 

Paul Hacking (UMass Amherst): Homological mirror symmetry for K3 surfaces

Joint work with Ailsa Keating (Cambridge). We prove the homological mirror symmetry conjecture of Kontsevich for K3 surfaces in the following form: The Fukaya category of a projective K3 surface is equivalent to the derived category of coherent sheaves on the mirror, which is a K3 surface of Picard rank 19 over the field of formal Laurent series. This builds on prior work of Seidel (who proved the theorem in the case of the quartic surface), Sheridan, Lekili--Ueda, and Ganatra--Pardon--Shende.

Siddarth Kannan (MIT): Pólya enumeration for moduli spaces

I will discuss a framework for applying classical techniques in graph enumeration and symmetric function theory to compute topological invariants of various moduli spaces, equivariantly with respect to actions of the symmetric group. I will present applications to moduli spaces of stable curves and moduli spaces of stable maps to toric varieties. This is based on joint work in progress with T. Song (Cambridge).

Dave Swinarski (Fordham University): The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity

Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a "worst'' destabilizing 1-parameter subgroup $\lambda$. What are the worst 1-parameter subgroups for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples. This is joint work with Joshua Jackson (Cambridge).

Yan Zhou (Northeastern University): Geometric origins of values of the Riemann Zeta functions at positive integers

Given a Fano manifold, Iritani proposed that the asymptotic behavior of solutions to the quantum differential equation of the Fano should be characterized by the so-called ‘Gamma class’ in its cohomology ring. Later, Abouzaid-Ganatra-Iritani-Sheridan reformulated the ‘Gamma conjecture’ for Calabi-Yau manifolds via the tropical SYZ mirror symmetry and proposed that values of the Riemann Zeta function at positive integers have geometric origins in the tropical periods and singularities of the SYZ geometry. In this talk, we will first review the content of the Gamma conjecture. Then, we will discuss a first step of generalizing AGIS’ approach to Gamma conjecture for the Gross-Siebert mirror families of a Fano manifold in dimension 2 cases, based on joint work with Bohan Fang and Junxiao Wang.