SCIENTIFIC PROGRAMS AND ACTIVITIES

(1) The Tropical Vertex Group, Scattering Diagrams and Quivers with Man-Wai Cheung

The tropical vertex group $\mathbb{V}$ introduced by Kontsevich-Soibelman plays a role in many problems in algebraic geometry and mathematical physics. The group itself can be understood in very different ways. In the approach due to Gross, Pandharipande and Siebert, a central role is played by tropical curves in $\mathbb{R}^2$ and their enumerative invariants. This approach leads to a number of applications. On the one hand, correspondence theorems connect factorizations in $\mathbb{V}$ with Gromov-Witten theory. On the other hand, these tropical methods when combined with results of Reineke, allow to relate Gromov-Witten theory to the topology of moduli spaces of quiver representations. First we will describe the tropical vertex group and in particular scattering diagrams. Then we will sketch the connection with tropical curves and, if time permits, with moduli spaces of quiver representations.

A toric degeneration is (roughly speaking) a family of varieties whose central fiber is a union of toric varieties glued pairwise torically along toric prime divisors. It is possible to encode all information about the degenerating variety into certain combinatorial data, namely an affine manifold with singularities together with a compatible piecewise-linear function. We will introduce singular affine manifolds and the construction of toric degenerations and discuss the scattering process.

The Tropical Vertex Group, Scattering Diagrams and Quivers

The tropical vertex group V introduced by Kontsevich-Soibelman plays a role in many problems in algebraic geometry and mathematical physics. The group itself can be understood in very different ways. In the approach due to Gross, Pandharipande and Siebert, a central role is played by tropical curves in R2 and their enumerative invariants. This approach leads to a number of applications. On the one hand, correspondence theorems connect factorizations in V with Gromov-Witten theory. On the other hand, these tropical methods when combined with results of Reineke, allow to relate Gromov-Witten theory to the topology of moduli spaces of quiver representations. First we will describe the tropical vertex group and in particular scattering diagrams. Then we will sketch the connection with tropical curves and, if time permits, with moduli spaces of quiver representations.

(1) Survey of Donaldson-Thomas and Pandharipande-Thomas theory

This talk is a survey of the definition and properties of Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants for a Calabi-Yau threefold $X$. The focus will be on overviewing some of the modern developments of the theory. The weighted Euler characteristic approach will be mentioned. It will be explained how PT invariants yield a (conjectural) construction of integer-valued BPS state counts.
Time permitting, it will be discussed how DT and PT invariants are naturally realized as counts of objects in the bounded derived category of coherent sheaves on $X$. In that setting, the wall-crossing formula for going from DT to PT corresponds to a change of stability condition.

We give an introduction to logarithmic geometry which will be fundamental knowledge for the conference talks by Abramovich, Chen and Gross.
We define log stable maps and explain why a stable curve is a smooth curve in the logarithmic sense.

Logarithmic Gromov-Witten (GW) invariants are a generalization of GW invariants to a logarithmically smooth situation. One major advantage is a clarification of the degeneration formula, although its definitive form is still work in progress. In this talk, we define these invariants and motivate them from the perspective of the degeneration formula.

(1) Tropical Curves and Disks

I will present a few perspectives on tropical geometry, emphasizing concrete descriptions and properties of so called "parametrized" tropical curves, disks, and trees. These objects will play a central role in the discussion of mirror symmetry for $\mathbb{P}^2$.

I will give a sketch of Gross's construction of mirror symmetry for $\mathbb{CP}^2$. The presentation will rely heavily on the tools introduced in the week's earlier discussion of tropical geometry.

(1),(2): Geometric Quantization and its applications to Gromov--Witten theory

In the first talk I will try to motivate the methods that are employed in geometric quantization, such as Feynman diagrams and Givental's formalism. In the second talk I will introduce the methods more explicitly and try to give a few examples of how it relates to GW theory. 

(1),(2): Physics of Mirror Symmetry: The Basics

I will review the ideas that lead physicists to Mirror Symmetry: $N=2$ superconformal field theories, their chiral rings and moduli spaces. Then I will discuss some simple examples and applications.

An introduction to Gromov-Witten theory

We will go over (quickly!) the motivation and ideas behind Gromov-Witten theory, focusing in particular on the case of $\mathbb{P}^2$. Heavy emphasis will be on examples and concrete computations as much as possible.

We define Lagrangian Floer homology and the Fukaya category.
We give some examples and explain the idea of the proof of homological mirror symmetry for the elliptic curve.

We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of the corresponding tropical curves. The proof relies on degeneration techniques and log deformation theory and is a precursor to log Gromov-Witten theory.