SCIENTIFIC PROGRAMS AND ACTIVITIES

Abramovich, Dan
Brown University

Lecture Notes

The decomposition formula for logarithmic Gromov-Witten invariants

This is joint work of Qile Chen, Mark Gross, Bernd Siebert and me.
A central aim of logarithmic Gromov–Witen theory is to find general formulas relating usual Gromov–Witten invariants of a smooth variety X with appropriate invariants of simpler varieties which appear as components of a degeneration of X. The first step is the decomposition formula, which breaks apart invariants of the singular fiber in combinatorial terms determined by tropical curves or graphs. I will describe our work on the decomposition formula, with examples (at least one example).

Formality of Fukaya categories from disc counts

One of the main problems in computing Fukaya categories is to understand the underlying A-infinity structure even for simple and explicitly constructed Lagrangians. I will explain how one can use counts of holomorphic discs to prove the formality of certain (subcategories of) Fukaya categories, focusing on the example of the Milnor surfaces of type A.
This is joint work with I. Smith.

Hamiltonian local models for symplectic derived stacks

We show that a derived stack with symplectic form of negative degree can be locally described in terms of generalised Darboux coordinates and a Hamiltonian cohomological vector field. As a consequence we see that the classical moduli stack of vector bundles on a Calabi–Yau threefold admits an atlas consisting of critical loci of regular functions on smooth varieties. If time permits, we discuss applications to the categorification of Donaldson–Thomas theory.
This is joint work with subsets of Ben-Bassat, Bussi, Dupont, Joyce, and Szendroi.

${\pi}$-stable pairs and the crepant resolution conjecture in Donaldson-Thomas theory

We construct curve counting invariants for a Calabi–Yau threefold Y equipped with a dominant birational morphism ${\pi: Y \rightarrow X}$. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when ${\pi: Y \rightarrow X}$ is the identity. In the case where ${\pi: Y \rightarrow X}$ is a semi-small crepant resolution, we prove a PT/DT - type formula relating the partition function of our invariants to the Donaldson-Thomas partition function. In the case where X is the coarse space of a Calabi–Yau orbifold, our partition function is equal to the Pandharipande–Thomas partition function of the orbifold.
Our methods include defining a new notion of stability for sheaves which depends on the morphism ${\pi}$. Our notion generalizes slope stability which is recovered in the case where ${\pi}$ is the identity on Y .

Very free curves on Fano complete intersection

The theory of stable log maps are developed for studying the degeneration of Gromov–Witten invariants. In this talk, I will introduce another interesting application of stable log maps to classical birational geometry — we construct very free curves on Fano complete intersections in projective spaces over an algebraically closed field of arbitrary characteristics.
This is a joint work with Yi Zhu.

The geometry of stable quotient spaces in genus one

Stable quotient spaces provide an alternative to stable maps for compactifying spaces of maps. In this talk I will discuss spaces of stable quotients which compactify the space of degree d maps of genus 1 curves to ${P^n}$. I will describe what is known about the geometry of these spaces. I will also discuss the relationship between these spaces and the corresponding spaces of stable maps from the perspective of the minimal model program.

Quantization of BCOV theory on Calabi-Yau manifolds

I’ll discuss some aspects of work in progress with Si Li on quantization of open and closed BCOV theory on Calabi-Yau manifolds. This gives a new formulation of the B-model which is local on the Calabi-Yau. If time permits, I’ll discuss some calculations in both the open and closed versions of this theory.

Parabolic refined invariants and Macdonald polynomials

A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack. Morever, Haiman’s geometric con- styruction for Macdonald polynomials is shown to emerge naturally in the context of geometric engineering.

Refined curve counting and wall-crossing

The tropical vertex group of Kontsevich and Soibelman is generated by formal symplecto-morphisms of the 2-dimensional algebraic torus. It plays a role in many problems in algebraic geometry and mathematical physics. Based on the tropical vertex group, Gross, Pandharipande and Siebert introduced an interesting Gromov–Witten theory on weighted projective planes which admits a very special expansion in terms of tropical counts. I will describe a refinement or “q-deformation” of this expansion, motivated by wall-crossing ideas, using Block–Goettsche invariants. This leads naturally to the definition of a class of putative q-deformed curve counts. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.
Joint work with Jacopo Stoppa.

Perturvation of constant maps, String topology and Perturbative Chern–Simons Theory

In this talk I will explain the way to obtain a solution of certain master equation onthe cyclic bar complex of the de-Rhamcohomology. This solution is a constant map part of Lagrangian Floer theory and is related to the two stories in the title. Something new in this talk, which I will explain, is the way how to do it without using pseudoholomorphic curvein the cotangent bundle. So everything works in the story of finite dimensional spaces. However ‘virtual technique’ is used much.

Introduction to Logarithmic Gromov–Witten invariants

Log Gromov–Witten invariants are a generalization of relative Gromov–Witten invariants.
They can be used to define the notion of curve counts with specified tangencies along normal crossings divisors, or curve counts in normal crossings target spaces. I will outline the basic definitions of these invariants as developed by myself and Siebert, on the one hand, and Abramovich and Chen, on the other.

Distinguished Lecture 1, What is tropical mathematics?

In tropical mathematics the usual laws of algebra are changed, the subtraction is forbidden, the division is always permitted, and 1+1 is equal to 1. Analogs of usual geometric shapes like lines, circles etc. are replaced by new figures composed of pieces of lines. I will try to explain basics of tropical algebra and geometry, its relation with more traditional domains, and its role in mirror symmetry which is a remarkable duality originally discovered in string theory about 20 years ago.

I’ll describe a new approach to cluster varieties and mutations based on scattering diagrams and wall-crossing formalism. The central role here is played by certain canonical transformation (formal change of coordinates) associated with arbitrary quiver. Also, a complex algebraic integrable system under some mild conditions produces a quiver, and the associated canonical transformation is a birational map.

Bridgeland’s notion of stability in triangulated categories is believed to be a mathematical encoding of D-branes in string theory. I’ll argue (using physics picture) that partially degenerating categories with stability should be described as a mixture between symplectic geometry and pure algebra. Spectral networks of Gaiotto, Moore and Neitzke appear as an example.

A mathematics theory of gauged linear sigma model

Several years ago, we (Fan, Jarvis and myself) developed a theory for so called Landau-Ginzburg model. It has a variety of applications in integrable hierarchy, LG/CY correspondence and modularity. LG-model is a limit of so called gauged linear sigma model. In the talk, I will discuss a construction to generalize our ”classical” theory to the general situation of gauged linear sigma model. Some potential applications will be discussed.

Speculations on Mirror Symmetry for Riemann surfaces

There has been quite some evidence that some form of mirror symmetry is valid for curves of higher genus. In known constructions, the dual geometry is derived from a higher-dimensional Landau–Ginzburg model. We present some ideas of how an intrinsic form of the mirror construction could be fomulated.

3-dimensional Calabi-Yau manifolds and Hitchin integrable systems

I am going to discuss the relationship between two topics mentioned in the title from the point of view of theory of Donaldson-Thomas invariants and wall-crossing formulas developed by Kontsevich and myself.

Mirror theorem, Seidel representation, and holomorphic disks

The quantum cohomology ring QH*(X) of a projective toric manifold X can be computed in several ways. A presentation of QH*(X) can be derived from the toric mirror theorem of Givental, Lian-Liu-Yau, and Iritani. McDuff-Tolman use d Seidel representations to derive a presentation of QH*(X). More recently, Fukaya-Oh-Ohta-Ono showed that QH*(X) is isomorphic to the Jacobian ring of the Lagrangian Floer superpotential of X, which is defined in terms of counting of holomorphic disks in X. The purpose of this talk is to explain the geometric reason underlying the equivalence of these three seemingly very different approaches, when X is semi-Fano.

Mirror Symmetry for Stable Quotients Invariants

I will describe a mirror formula for the direct analogue of Givental’s J-function in the SQ theory. It is remarkably similar to the mirror formula in the Gromov–Witten theory, but the former does not involve a change of variables. This suggests that the mirror map relating the GW-invariants to the B-model of the mirror is more reflective of the choice of curve counting theory on the A side than of mirror symmetry. The proof of the mirror formula in the Fano case is as in the GW–theory. On the other hand, the proof in the Calabi–Yau case consists of showing that it is a consequence of the Fano case.
This is joint work with Y. Cooper.

Legendrian knots and constructible sheaves

Given a Legendrian knot, we construct a category, invariant under Legendrian isotopies up to equivalence. Rank-one objects of our category play a special role. On the one hand, they define a subcategory which we conjecture to be equivalent to the bilinearized Legendrian contact homology of the knot. On the other hand, the moduli of these objects is an interesting space which, for positive braid closures, enables one to recover the Khovanov–Rozansky categorified invariant of the topological type of the knot. I will try to explain all this by working through simple examples.
This work is joint with Vivek Shende and David Treumann