2024

Note the non-standard time! 

In the first part of the talk, I will give an introductory talk to o-minimality, giving the definitions and some of intuition through understanding which Lie groups are amenable to being understood in o-minimal contexts. 

 

In the second part I will state the two theorems which have had the most important applications to algebra and give some intuition of how (at least one of them) has been used.

In 2008, looking to bound the face vectors of tropical linear spaces, Speyer introduced the g-invariant of a matroid. He proved its coefficients nonnegative for matroids representable in characteristic zero and conjectured this in general. Later, Shaw and Speyer and I reduced the question to positivity of the top coefficient. This talk will overview work being written up with Andy Berget (Western Washington) that proves the conjecture. The first half will be introductory, aiming to define the g-invariant, state the theorem, and compare to other recent work in matroid theory. The second half will look at the proof techniques.

Geometrically, the main ingredient in the proof is a Kempf collapsing constructed from the matroid tautological vector bundles of Berget--Eur--Spink--Tseng, and an initial degeneration thereof. Combinatorially, it is an extension of the definition of external activity to a pair of matroids and a way to compute it using the fan displacement rule on tropical Chern classes. The work of Ardila and Boocher on the closure of a linear space in (P^1)^n is a special case.
 

Based on J. Kollár and C. Voisin. Flat pushforwards of Chern classes and the smoothability of cycles below the middle dimension. Ann. of Math. (2), 200(2):771–797, 2024.

Based on preprints https://arxiv.org/abs/2006.07213 and https://arxiv.org/abs/2406.05382.

Mirror symmetry predicts that the symplectic geometry of a manifold is equivalent to the complex geometry of the mirror manifold. In particular, enumerative mirror symmetry relates quantum cohomology (Gromov-Witten invariants) of a manifold with period integrals of the mirror. A fundamental question in mirror symmetry is to construct mirrors. I will explain Gross-Siebert's intrinsic mirror symmetry which gives a mirror construction using relative quantum cohomology. More precisely, one considers a maximally unipotent monodromy degeneration of a Calabi—Yau manifold and take proj of the degree zero part of a relative quantum cohomology ring associated with the degeneration. Theta functions form a canonical basis of the degree zero part of the relative quantum cohomology ring. Theta functions can be defined in terms of punctured Gromov-Witten invariants of the broken line type. I will explain a variant of intrinsic mirror symmetry using orbifold Gromov-Witten invariants of root stacks, theta functions for general snc pairs and possible applications to questions in mirror symmetry.

I will explain how to extend the concept of spin structures from vector bundles to complexes over a scheme (or algebraic stack) X outside of characteristic 2. For a complex E on X with an oriented quadratic structure one obtains an associated Z/2Z-gerbe over X obstructing the existence of a spin structure on E. I will also touch on an expected alternative definition of spin structures on complexes using derived Clifford algebras. One application of the concepts presented is to give a refinement of the virtual structure sheaf of Oh-Thomas in Donaldson-Thomas theory of Calabi-Yau fourfolds.

I will talk about how one can relate intersection theories of Hilbert schemes of points and Fulton-MacPherson compactifications.

Abstract: Although tropical vector bundles have been introduced by Allermann ten years ago, very little has been said about their structure and their relationship to vector bundles on algebraic varieties. I will present recent work with Martin Ulirsch and Dmitry Zakharov that changes exactly this in the case of curves: we prove analogues of the Weil-Riemann-Roch theorem and the Narasimhan-Seshadri correspondence for tropical vector bundles on tropical curves. We also show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to a certain component of the moduli space of semistable tropical vector bundles on its dual metric graph. Time permitting I will also report on work with Inder Kaur, Martin Ulirsch, and Annette Werner and explain some of the difficulties that arise when generalizing beyond the case of curves to Abelian varieties of arbitrary dimension.

Note the non-standard start time!

 
Abstract: We consider mirror pairs of Calabi-Yau hypersurfaces X and X’ in toric varieties associated to dual reflexive polytopes. We will give a proof through tropical geometry that the Hodge numbers of X and X’ are mirror symmetric. The proof goes by considering tropical homology, and works over the ring of integer numbers. In particular, we can use our spectral sequence with Kris Shaw to explore the connections between the topology of the real part of X and cohomological operations on X’.

This is based on joint work with Diego Matessi. 

Donaldson-Thomas and Pandharipande-Thomas theory are two approaches to counting curves on projective threefolds in terms of their moduli spaces of sheaves. An important special case in understanding the DT/PT correspondence the equivariant geometry of affine three-space with the natural coordinate action of the rank 3 torus. I will show how one can use new wall-crossing techniques to prove the equivariant K-theoretic DT/PT correspondence in this situation, which was previously known only in the Calabi-Yau limit.

This is part of an ongoing project with Felix Thimm and Henry Liu in which we aim to prove wall-crossing for virtual enumerative invariants associated to equivariant CY3 geometries by extending a vertex algebra formalism for wall-crossing developed by Joyce.

Gromov—Witten invariants are virtual counts of curves with prescribed conditions in a given algebraic variety. One of the main techniques to study Gromov—Witten invariants is degeneration. The degeneration formula expresses absolute Gromov—Witten invariants in terms of relative Gromov—Witten invariants of algebraic varieties with tangency conditions along boundary divisors.

Relative Gromov—Witten invariants with only one relative marking are relative invariants with maximal contacts along the unique relative marking. The local-relative correspondence proved by van Garrel—Graber—Ruddat states that genus zero relative invariants with maximal contacts are equal to local Gromov—Witten invariants of a line bundle. Local invariants are usually easier to compute. However, The degeneration formula usually involves relative invariants beyond maximal contacts (i.e. with several relative markings). I will explain a generalization of the local-relative correspondence beyond maximal contacts, hence determine all the genus zero relative invariants that appear in the degeneration formula.

This is based on joint work with Yu Wang.