Abstracts

Mohammed ABOUZAID, Stanford University, USA

Title: Global charts for moduli spaces of pseudo-holomorphic curves

Abstract: In recent work with McLean and Smith, we extended the construction of quantum K-theory for the algebraic to the symplectic setting. The key new idea is the use of the space of sections of positive line bundles on the domain of pseudo-holomorphic curves to construct a finite-dimensional model for the moduli spaces. I will explain the basic idea, and discuss some of its applications.

Richard BAMLER, University of California, Berkeley, USA

Title: Mean curvature flow in R^3 and the Multiplicity One Conjecture, I

Abstract: The Mean Curvature Flow describes the evolution of a family of embedded surfaces in Euclidean space that move in the direction of the mean curvature vector. In recent years, significant progress has been made in the study of mean curvature flow, leading to the resolution of several important conjectures.

We will describe this past work and then focus on the Multiplicity One Conjecture. Broadly speaking, this conjecture asserts that singularities along the flow cannot form through the accumulation of several parallel sheets. We will motivate this conjecture, sketch its proof, and explain how it has led to a much clearer picture of Mean Curvature Flows through singularities.

This is joint work of Richard Bamler and Bruce Kleiner.

Sun-Yung Alice CHANG, Princeton University, USA

Title: On a conformal Einstein fill in problem

Abstract: Given a manifold (M^n,h), when is it the boundary of a conformally compact Einstein manifold (X^{n+1},g), in the sense that there exists some defining function r on X so that r^2 g is compact on the closure of X and r^2 g restricted to M is the given metric h? The model example is the n-sphere as the conformal infinity of the hyperbolic (n+1) ball.

In the special case when n = 3, one can formulate the problem as an Dirichlet to Neumann type inverse problem. In the talk, I will report on some progress made with Yuxin Ge on the issues of the ''compactness", and as an application, the "existence" and "uniqueness" of the fill in problem for a class of metrics of positive scalar curvature defined on the 3-sphere.

Otis CHODOSH, Stanford University, USA

Title: Minimizers in Gamow's liquid drop model

Abstract: Gamow introduced the liquid drop model in 1928 as a model for the nucleus. I will discuss some recent work (with Ian Ruohoniemi) concerning roundness of minimizers.

Tobias COLDING, MIT, USA

Title: TBA

Abstract: TBA

Kenji FUKAYA, Simons Center for Geometry and Physics, USA

Title: Floer homologies, Lagrangian correspondence and Atiyah-Floer conjectures

Abstract: I will survey several works in progress on the Lagrangian and Instanton

Floer homologies related to Lagrangian correspondence and Atiyah-Floer conjecture.

Many parts are joint work with A. Daemi and also partly with M. Lipyanskiy

Wenshuai JIANG, Zhejiang University, China

Title: The Nodal set along Parabolic PDEs

Abstract: Let u(x,t) be a solution of a second order Parabolic PDE in a bounded domain of R^n. When n=1, it is known from Angenent 1982 that the zero set Z_t of u at time t-slice is discrete and nonincreasing with respect to t, which has been widely used in curve-shortening flows and semi-linear heat equations. When n>1, if the coefficients are analytic, it was proved by Lin 1991 that the zero set Z_t at time t-slice has finite (n-1)-Hausdorff measure. Later, it was proved by Han-Lin 1994 that the zero set has a space-time codimension one Hausdorff measure estimate under weak regular coefficients. In this talk, we will study the behaviour of Z_t with weak regular coefficients for n>1. Comparing with n=1, one can see that there exists example showing that the Hausdorff measure of Z_t could be increasing for n>1. We can show that the dimension of Z_t is nonincreasing and the (n-1)-Hausdorff measure of Z_t is finite. This is a joint work with Yiqi Huang.

Bruce KLEINER, New York University, USA

Title: Mean curvature flow in R^3 and the Multiplicity One Conjecture, II

This is joint work of Richard Bamler and Bruce Kleiner.

Chi LI, Rutgers University, USA

Title: Recent progress on the Yau-Tian-Donaldson conjecture for

constant scalar curvature Kahler metrics

Abstract: For any polarized projective manifold (X, L), the Yau-Tian-Donaldson conjecture predicts that the existence of constant scalar curvature Kahler metrics in the first Chern class of L is equivalent to certain algebraic K-stability of (X, L).

We will survey some recent progresses towards this conjecture and how it leads to interesting problems in algebraic geometry.

John LOTT, University of California, Berkeley, USA

Title: Positive scalar curvature on noncompact manifolds

Abstract: There has been much work on using Dirac operators to give obstructions for a compact spin manifold to admit a positive scalar curvature (psc) metric. Recent techniques allow one to extend this to (localized) obstructions for a noncompact spin manifold to admit a psc metric. I'll describe results in both the finite volume case and infinite volume case, along with the relation to a question about simplicial volume for compact manifolds.

Zhiqin LU, University of California, Irvine, USA

Title: The Spectrum of Laplacian on forms over open manifolds

Abstract: We proved the L^p boundedness of certain resolvent of Laplacians by assuming the Ricci lower bound and manifold volume growth. This generalized a result of M. Taylor, in which bounded geometry of the manifold is assumed. This is a joint work of N. Charalambous.

Aaron NABER, Northwestern University, USA

Title: Nonlinear Harmonic Maps and the Energy Identity.

Abstract: We will then focus our attention on a type of singularity formulation for nonlinear harmonic maps which result in so called defect measures. Though not typically studied in such generality, these defect measures are a general construction for understanding the loss of energy in limits of H^1 functions. In the case of nonlinear harmonic maps, there is a precise conjecture as to the form of these defect measures called the Energy Identity. This was recently proved together with Daniele Valorta and will be the focus of the talk.

Giulia SACCÀ, Columbia University, USA

Title: Lagrangian fibrations in compact HK geometry

Abstract: Compact HK manifolds are the generalizations of K3 surfaces. They are the building blocks of compact Kahler manifolds with trivial first Chern class, and their role in algebraic geometry has exploded in the last two decades. Lagrangian fibrations are the generalization of elliptic K3 surfaces, and have a fundamental role in the study of compact HK manifolds. In this talk I will survey some recent and less recent results highlighting the role of Lagrangian fibrations in the study of compact HK manifolds.

Nataša ŠEŠUM, Rutgers University, USA

Title: Generalized cylinder limits of Ricci flow singularities

Abstract: We study multiply warped product geometries and show that for an open set of initial data within multiply warped product geometries the Ricci flow starting at any of those develops generalized cylinder as singularity model. More precisely, for any p and q we construct an open set of initial data within multiply warped product geometries whose Ricci flows develops a singularity model S^p x R^q.

We also mention examples of complete noncompact Ricci flows that develop Type I singularities at spatial infinity that are not modelled on shrinking solitons.

Bernd SIEBERT, University of Texas at Austin, USA

Title: Mirror Symmetry via Symplectic Monodromy - Toward HMS for Intrinsic Mirrors

Abstract: Normal crossing degenerations of Calabi-Yau varieties have canonical and universal mirror geometries, their intrinsic mirror partners. These are constructed via logarithmic Gromov-Witten invariants. In the talk I will report on joint work in progress with Tim Perutz showing that the homogenous coordinate ring of the intrinsic mirror equals a Floer-theoretic ring defined by the symplectic monodromy of the degeneration. Homological mirror symmetry can then be deduced by an automatic generation criterion for the Fukaya category and a result of Polishchuk.

Jeffrey STREETS, University of California, Irvine, USA

Title: Pluriclosed flow and the Hull-Strominger system

Abstract: The Hull-Strominger system arose in superstring theory, and is a subject of intense mathematical interest due to its connection to uniformization problems in complex geometry. I will discuss a mild reformulation of this system, and show that the tools of pluriclosed flow/generalized Ricci flow can be used to construct solutions to this system. This point of view leads to a proof of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system. Joint work with M. Garcia-Fernandez and R. Gonzalez-Molina

Jake SOLOMON, Hebrew University, Jerusalem, Israel

Title: The degenerate special Lagrangian equation and Lagrangians with boundary

Abstract: The space of positive Lagrangian submanifolds of a Calabi-Yau manifold is an infinite dimensional Riemannian manifold. There is a functional on this space that is convex along geodesics with critical points at special Lagrangians. Thus, understanding the geodesics of the space of positive Lagrangian submanifolds would shed light on the uniqueness and existence of special Lagrangian submanifolds. The geodesic equation is a degenerate elliptic PDE known as the degenerate special Lagrangian equation (DSL) and large families of solutions have been constructed in arbitrary dimension.

We discuss the space of positive Lagrangian submanifolds with boundary in a fixed Lagrangian. There are now two functionals on the space. One is convex along geodesics while the other is affine. In the case of graphical Lagrangians with boundary, there are existence results for C^0 solutions to the DSL, and also non-existence results for C^2 solutions.

Based on joint works with Rubinstein, Yuval and Kapota.

Jian SONG, Rutgers University, USA

Title: Geometric analysis on singular complex spaces

Abstract: We establish a uniform Sobolev inequality and diameter bound for Kahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular Kahler metrics on Kahler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular Kahler spaces including the spectral theorem, heat kernel estimates, eigenvalue estimates and diameter estimates. Such estimates were only known previously in very special cases such as Bergman metrics. As a consequence, we derive various geometric estimates, such as the diameter estimate and the Sobolev inequality, for Kahler-Einstein currents on projective varieties with definite or vanishing first Chern class.

Brian WHITE, Stanford University, USA

Title: Translating Annuli

Abstract: I will discuss some surprising new examples of translating solutions to mean curvature flow.

I will also mention some challenging open problems. The talk is based on joint work with David

Hoffman and Francisco Martin.

Chenyang XU, Princeton University, USA

Title: Finite generation for valuations beyond divisors

Abstract: One major new birational geometry problem arising in understanding stable degeneration of varieties, which is the algebraic analogue to the compactness of Kähler-Einstein type metrics, is finite generation for valuations of higher rational rank. In the past a few years, we have established finite generation for minimizing valuations of various functionals, by first showing those minimizers are ‘special’; and then proving any special valuation satisfies finite generation. In this talk, I will report results along this direction and its applications to Kähler-Einstein and other similar problems.

Guangbo XU, Rutgers University, USA

Title: Integral Floer Homology and Applications in Hamiltonian Dynamics

Abstract: (Joint with Shaoyun Bai) Floer homology has been a major tool in symplectic geometry. In this talk I will talk about the construction of integral Hamiltonian Floer homology for general compact symplectic manifolds and the applications, including an integer-version of the Arnold conjecture and a version of Hofer-Zehnder conjecture regarding non-contractible periodic orbits.

Weiping ZHANG, Nankai University, China

Title: Deformed Dirac operators and scalar curvature

Abstract: We will describe some applications of deformed Dirac operators on problems concerning metrics of positive scalar curvatures.