Great Lakes Geometry Conference 2011: Abstracts of Talks

Great Lakes Geometry Conference 2011: Abstracts of Talks:

Logarithmic stable maps
Dan Abramovich

The problem of generalizing relative stable maps and the degeneration formula in Gromov--Witten theory has been a significant challenge. Recent work of Qile Chen, Abramovich-Chen and Abramovich-Chen-Gillam-Marcus provides a route towards a degeneration formula in some generality; another closely related approach is given by Gross and Siebert. The main construction is that of logarithmic stable maps, as proposed by Bernd Siebert in a 2001 lecture.

I will discuss the background and main ideas involved in such a construction.

Differential graded Banach algebras and Lie infinity-groupoids (joint work with Kai Behrend)
Ezra Getzler

We develop a generalization of Kuranishi's construction of the analytic germ of deformations of a holomorphic vector bundle where the vector bundle is replaced by a complex of vector bundles. We introduce a natural infinity-category whose objects are the Maurer-Cartan elements of a differential graded Banach algebra. We show that the quasi-isomorphisms in this category form a Lie infinity-groupoid: this is a topological version of a result of Joyal. In particular, we explain what a Lie infinity-groupoid is: it generalizes of a definition due to André Henriques.

Minimal surface asymptotics in AdS/CFT correspondence
Robin Graham

The asymptotics of minimal surfaces in products of an asymptotically hyperbolic manifold with a compact manifold will be discussed, together with some examples. This is joint work with Andreas Karch.

Open and closed Gromov-Witten invariants of toric Calabi-Yau 3-folds
Chiu-Chu Melissa Liu

Open/closed Gromov-Witten invariants count holomorphic maps from Riemann surfaces with/without boundaries to a Kahler manifold. We will describe results and conjectures on generating functions of open and closed Gromov-Witten invariants of toric Calabi-Yau 3-folds.

Singularities of Mean Curvature Flow
William Minicozzi II

I will describe joint work with Toby Colding on singularities of mean curvature flow, including the dynamics near a singularity.

Relative volume rigidity in Alexandrov geometry
Xiaochun Rong

We will report a recent progress on the relative volume rigidity problem in Alexandrov geometry: let $f: X\to Y$ be a distance non-increasing onto map between two compact Alexandrov $n$-spaces with curvature bounded below. The relative volume rigidity conjecture asserts that if the volume of $Y$ achieves the maximal possible value i.e., equal to the volume of $X$, then $Y$ is isometric to $X$ module a boundary identification (which is allowed to be trivial). This is a joint work with Nan Li.

Sharp eigenvalue estimates for manifolds with boundary
Richard Schoen

In this lecture we will describe some results and questions concerning sharp bounds on Steklov eigenvalues (eigenvalues of the Dirichlet to Neumann map). It turns out that for surfaces, there is an interesting connection between this problem and the study of minimal surfaces in the ball satisfying a free boundary condition. We will describe some results on this problem which are joint work with A. Fraser.

Cohomological Field Theories on the Lattice
A. J. Tolland

I'll discuss some work in progress, which aims at rigorously constructing the Euclidean path integral for various topologically-interesting quantum field theories.