Schedule and Abstracts

Saturday, November 12

9:00-9:50Irena Peeva (remote)
10:00-10:50Matt Young
10:50-11:30Coffee & Tea
11:30-12:20Keller Vandebogert
12:20-2:00Lunch break
2:00-2:50Irena Peeva (remote)
3:00-3:50Matt Young
3:50-4:30Coffee & Tea
4:30-5:20Nitin Chadambaram

Sunday, November 13

9:00-9:50Matt Young
10:00-10:50Ana Ros Camacho (remote)
10:50-11:30Coffee & Tea
11:30-12:20James Pascaleff
12:30-1:20Alexei Oblomkov

All talks will take place in Hayes-Healy Center, Room 127. Breakfast and Breaks will take place in Hurley Hall, Room 257. (The two buildings are connected.) The Saturday dinner takes place in the South Dining Hall Hospitality Room.
They will also be broadcast on Zoom (Meeting ID: 92618721919 Passcode: 190534).


Nitin Chidambaram (Edinburgh)
Stratified Mukai via VGIT

The Bondal-Orlov conjecture states that a pair of varieties related by a flopping transformation have equivalent derived categories. An interesting question is then to ask what the Fourier-Mukai kernel of this derived equivalence is. In this talk I will explain a proposal called the Q-construction due to Ballard-Diemer-Favero, that provides a candidate kernel for any flop. The main application that I will discuss is a study of the stratified Mukai flop via the Q-construction applied to a matrix factorization category, and a comparison to the derived equivalence proved by Cautis-Kamnitzer-Licata in this case.

This is based on joint work-in-progress with Matt Ballard and David Favero.

Alexei Oblomkov (UMass Amherst)
Matrix factorizations and knot homology

My talk is based on joint work with Lev Rozansky. I will construct monodical categories of equivariant matrix factorizations that admits a monoidal functor from the braid group. We use this category to associate to a braid $\beta\in Br_n$ a two-periodic complex of coherent sheaves $S_\beta$ on the Hilbert scheme of n points on the plane $Hilb_n(A^2)$. The homology of $S_\beta$ are HOMFLY-PT knot homology of $\beta$. We use this construction knot homology of torus links and some classes of cabled torus knots.

James Pascaleff (UIUC)
Singular support of coherent sheaves and mirror symmetry

The singular support of a coherent complex F is a measurement of the failure of F to be perfect. Thus it is a notion of support that is relevant for the singularity category (= Coh mod Perf) and matrix factorizations. It is also useful for understanding homological mirror symmetry for these categories. In joint work with Nicolo Sibilla, we used this to study singularity categories of normal crossings surfaces and their homological mirrors, which are Fukaya categories of Riemann surfaces or (conjecturally) certain symplectic four-manifolds constructed from Riemann surfaces. The notion of singular support helps clarify the construction of the mirror manifold as well as the homological mirror symmetry equivalence.

Irena Peeva (Cornell)
Background on Free Resolutions and Matrix Factorizations for Hypersurfaces

Motivated by applications in Invariant Theory, Hilbert introduced an approach to describe the structure of modules by free resolutions. Hilbert’s Syzygy Theorem shows that minimal free resolutions over a polynomial ring are finite. Most minimal free resolutions over quotient rings are infinite. We will discuss the properties of such resolutions. The concept of matrix factorization was introduced by Eisenbud 35 year ago, and it describes completely the asymptotic structure of minimal free resolutions over a hypersurface.

Matrix factorizations for complete intersections

In general, the structure of infinite minimal free resolutions can be quite complex; for example, there exist minimal resolutions whose generating functions are not rational. We will discuss the structure of free resolutions over complete intersections. The study of such resolutions started with Tate’s elegant construction of the resolution of the residue field. Eisenbud and I introduced the concept of CI matrix factorization and showed that it suffices to describe the asymptotic structure of minimal free resolutions over complete intersections. We will discuss this concept, with emphasis on the codimension 2 case.

Ana Ros Camacho (Cardiff)
Module tensor categories and the Landau-Ginzburg/conformal field theory correspondence

The Landau-Ginzburg/conformal field theory correspondence is a physics result from the late 80s and early 90s predicting some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. At present we lack an explicit mathematical statement for this result, yet we have examples available. The only example of a tensor equivalence in this context was proven back in 2014 by Davydov-Runkel-RC, for representations of the N=2 unitary minimal model with central charge 3(1-2/d) (where d integer bigger than 2) and matrix factorizations of the potential x^d-y^d. This equivalence was proven back in the day only for d odd, and in this talk we explain how to generalize this result for any d, realising these categories as module tensor categories enriched over $\mathbb{Z}_d$-graded vector spaces. Joint work with T. Wasserman (University of Oxford).

Matt Young (Utah State)
Matrix factorizations and Landau–Ginzburg B-models

The first lecture will introduce matrix factorizations and their categories from a classical perspective following Eisenbud and Buchweitz.

The second lecture will survey work of Carqueville, Dyckerhoff, Murfet, Polishchuk and Vaintrob which constructs from the differential graded category of matrix factorizations of an isolated hypersurface singularity a 2 dimensional open/closed oriented topological field theory, interpreted as the category of boundary conditions in the associated Landau–Ginzburg B-model.

The final lecture will focus on equivariant and Real generalizations of matrix factorizations, which model Landau-Ginzburg orbifolds and orientifolds. Topics will include generalizations of Knörrer periodicity to the equivariant (Hirano), real (Brown) and Real (Spellmann–Young) settings, partial topological field theory constructions and open problems.

Keller Vandebogert
Free Flags, Matrix Factorizations, and Enhancements

Free flag differential modules are objects that almost play the role of classical free resolutions, but in the category of differential modules (that is, modules equipped with a square zero endomorphism). Recent work of Brown-Erman has shown that the classical theory of minimal free resolutions forms an anchor for the homological properties of differential modules, but extending the classical theory to this more general setting has turned out to be a subtle problem. In this talk, I’ll speak on recent progress on this subject, including an “enhancement” process that simultaneously generalizes and unifies the Eisenbud-Shamash construction of matrix factorizations and the construction of any free flag differential module. Much of this material is based on joint work with Maya Banks.