Great Lakes Geometry Conference 2012: Abstracts of Talks:
Some Estimates for Gradient Ricci solitons
Bennett Chow
Gradient Ricci solitons (shrinkers, steadies, and expanders) often model singularities of Ricci flow. We discuss joint work with Peng Lu and Bo Yang where we consider aspects of the geometry of gradient Ricci solitons such as their asymptotic volume ratio (AVR), scalar curvature, and Ricci curvature. In particular, we discuss a condition for shrinkers to have positive AVR, a lower bound for the scalar curvature of nonexpanders, and an integral Ricci curvature bound along modified geodesics.
Regularized determinants and conformally invariant operators
Matthew J. Gursky
In this talk I will give an overview of some geometric and analytic issues related to the regularized determinant of an elliptic operator. I will begin with a quick overview of the work of Osgood-Phillips-Sarnak on the determinant of the laplacian for surfaces, then move to four dimensions, where the starting point is a formula of Branson-Orsted for conformal variations of the determinant. I will talk about a question posed by Connes concerning the determinant of the Paneitz operator and the half-torsion, and describe some variational properties, including multiplicity results.
SYZ mirror symmetry transformation
Naichung Conan Leung
In this talk, I will discuss recent progress on SYZ proposal which explains mirror duality between complex geometry and symplectic geometry via a Fourier-Mukai transformation along Lagrangian fibrations.
On the Tian-Yau-Zelditch expansion on Riemann surfaces
Zhiqin Lu
It is well-known that the convergence of the TYZ expansion depends on the geometry of the manifold. In particular, the convergence heavily depends on the injectivity radius. In this talk, we give a preliminary report on the TYZ expansions on a degeneration family of compact Riemann surfaces. This is joint work with Chiung-ju Liu.
Geometric surgery by partial differential equations
Jian Song
We will discuss recent developments on how geometric PDEs can perform canonical geometric surgery. We propose the analytic minimal model program with Ricci flow to classify algebraic varieties via geometric surgeries in Gromov-Hausdorff topology equivalent to birational surgeries such as contractions and flips. This approach can also be applied to the study of degeneration of Calabi-Yau metrics. As an application, we prove a conjecture of Candelas and de la Ossa for conifold flops and transitions.
The evolution of a Hermitian metric by its Chern-Ricci curvature
Valentino Tosatti
I will discuss the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci curvature. This is an evolution equation which coincides with the Ricci flow if the initial metric is Kahler, and was first studied by M.Gill. I will describe the maximal existence time for the flow in terms of the initial data, and then discuss the behavior of the flow on complex surfaces and on some higher-dimensional manifolds. This is joint work with Ben Weinkove.
Polytopes, Skeleta, and Homological Mirror Symmetry
Eric Zaslow
I will describe some relations betweeen constructible sheaves and the categories of homological mirror symmetry, then apply this perspective to affine Calabi-Yau hypersurfaces. Given a function on a complex torus with reflexive Newton polytope, I will build a skeleton which is homotopic to the affine hypersurface the function describes. I will also explain the link to homological mirror symmetry at the large complex/large volume limit. This talk is based on work with several groups, including my current collaborators Helge Ruddat, Nicolò Sibilla, and David Treumann.
Paley-Wiener, nodal sets and ergodicity on analytic Riemannian manifolds
Steve Zelditch
According to the physics' heuristic random wave model, the nodal (zero) sets of eigenfunctions of the Laplacian of a Riemannian manifold (M, g) with ergodic geodesic flow should become uniformly distributed with respect to the volume form. This conjecture is way beyond current and foreseeable technology, but it turns out to be provable when $(M, g)$ is real analytic if one analytically continues eigenfunctions to the complexification of M and considers complex zero sets.
The relation between real and complex zero sets becomes closer if one intersects nodal sets with geodesics. I will concentrate on recent work and work in progress on the distribution of these intersection points. It is based in part on joint work with J. Toth on quantum ergodic restriction theorems.