Geometry & Analysis Seminar Winter 2016

McHenry Building
Thursdays 4:00pm
Room 4130
For further information please contact Professor Longzhi Lin or call 831-459-2400

January 7, 2016

The Nirenberg problem and its generalizations: A unified approach

Tianling Jin, The Hong Kong University of Science and Technology

The classical Nirenberg problem asks for which functions on the sphere arise as the scalar curvature of a metric that is conformal to the standard metric. In this talk, we will discuss similar questions for fractional Q-curvatures. This is equivalent to solving a family of nonlocal nonlinear equations of order less than n, where n is the dimension of the sphere. We will give a unified approach to establish existence and compactness of solutions. The main ingredient is the blow up analysis for nonlinear integral equations with critical Sobolev exponents.

January 14, 2016

Period integrals and tautological systems

Ruifang Song, University of California, Davis

We introduce a system of differential equations associated to a smooth algebraic variety X with the action of a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is regular holonomic if G acts on X with finitely many orbits. When X is a homogeneous space, our construction gives the Picard-Fuchs system governing the period integrals on Calabi-Yau hypersurfaces in X. When X is a toric variety, our construction recovers certain generalized hypergeometric systems introduced by Gelfand, Kapranov and Zelevinsky. Such systems have played important roles in the study of mirror symmetry.

January 21, 2016

Skew Mean Curvature Flow

Chong Song, Xiamen University

The skew mean curvature flow(SMCF) or binormal flow, which origins from fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. In this talk, I will show the basic properties of the SMCF and the short-times existence of SMCF in Euclidean space R^4 . If time permits, I will also talk about recent progress on the uniqueness of SMCF in arbitrary dimension.

January 28, 2016

Continuous symplectic topology and area-preserving homeomorphisms

Sobhan Seyfaddini, Massachusetts Institute of Technology- Special Colloquium Speaker

After briefly discussing symplectic diffeomorphisms, I will speak about  continuous symplectic topology and the notion of symplectic homeomorphisms.  We will first see how this notion arises as a natural consequence of a celebrated theorem of Eliashberg and Gromov and then proceed to discuss the behavior of symplectic homeomorphisms with respect to the underlying symplectic structure.

In the final part of my talk, I will focus on the case of surfaces where symplectic homeomorphisms are the area and orientation preserving  maps and we will discuss an application of continuous symplectic topology to the field of area-preserving dynamics.

February 4, 2016

Rigidity phenomena in random point sets

Subhroshekhar Ghosh, Princeton University - Special Colloquium Speaker

In several naturally occurring (infinite) random point processes, we establish that the number of the points inside a bounded
domain can be determined, almost surely, by the point configuration outside the domain. This includes key examples coming from random matrices and random polynomials. We further explore other random processes where such ''rigidity'' extends to a number of moments of the mass distribution. The talk will focus on particle systems with such curious  "rigidity" phenomena, and their implications.  We will also talk about applications to natural questions in stochastic geometry and harmonic analysis.

February 11, 2016

Symplectic Normal Crossings Configurations and Their Smoothings

Mohammad Farajzaden Tehrani, Simons Center, Stony Brook - Special Colloquium Speaker

Normal crossings (NC) divisors and configurations have long played a central role in algebraic geometry. For example, they appear in A-side of mirror symmetry. I will first introduce symplectic (topological) notions of NC divisors and configurations which generalize the notion of NC in algebraic geometry. We show that symplectic NC divisors/configurations are morally equivalent to almost Kahler NC divisors/configurations. This equivalence gives rise to a multifold version of Gompf's symplectic sum construction and related smoothing of NC configuration which fit naturally with some aspects of the Gross-Siebert program for a direct proof of mirror symmetry. This is a joint work with M. McLean and A. Zinger.

February 18, 2016

Geodesic X-ray transforms on surface and tensor tomography

Francois Monard, University of Michigan - Special Colloquium Speaker

In this talk, we will study what can be reconstructed about a function (or a tensor) on a surface, from knowledge of its integrals along a given family of geodesic curves, that is, its X-ray transform. The "straight-line" version of this question was first answered by J. Radon in 1917 and its solution forms the theoretical backbone of Computerized Tomography since the 1960's. In practice, variations of the refractive index do occur and bend photon paths in optics-based imaging, and this requires
that the problem be studied for general curves. 

In a geometric setting beyond that of the Radon transform, examples of situations impacting the qualitative invertibility and stability of these transforms are (i) the case of "simple" surfaces, (ii) the presence of conjugate points/caustics, and (iii) the presence of trapped geodesics. We will discuss positive and negative theoretical results occurring when one considers each of the scenarios above, established using energy methods, microlocal methods and Fourier analysis for PDEs posed on the unit tangent bundle. Numerical illustrations will be presented throughout the talk.

February 25, 2016

A local regularity theorem for mean curvature flow with triple edges

Felix Schulze, University College London

We consider the evolution by mean curvature flow of surface clusters, where along triple edges three surfaces are allowed to meet under an equal angle condition. We show that any such smooth flow, which is weakly close to the static flow consisting of three half-planes meeting along the common boundary, is smoothly close with estimates. Furthermore, we show how this can be used to prove a smooth short-time existence result. This is joint work with B. White.

March 3, 2016

2:00-3:00pm *Note the time change**

A nonlocal diffusion problem on manifolds

Maria del Mar Gonzalez Nogueras, Universitat Politècnica de Catalunya

We consider a nonlocal diffusion problem on a manifold. This kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. After briefly considering existence and uniqueness of solutions, we prove that, for a convenient rescaling the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior: while on compact manifolds the asymptotics are given by the spectral properties of the operator; in the model of hyperbolic space we find a different and interesting behavior. This is joint work with C. Bandle, M. Fontelos and N. Wolanski.


Moduli space of Fano K\"haler-einstein manifolds

Xiaowei Wang, Rutgers University

In this talk, we will discuss our construction of compact Hausdorff Moishezon moduli spaces parametrizing smoothable K-stable Fano varieties. The solution relies on the recent solution of the Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun and Tian. In particular, we prove the uniqueness of the degeneration of Fano Kahler-Einstein manifolds and more algebraic properties that are needed to construct an algebraic moduli space. (This is a joint work with Chi Li and Chenyang Xu).

March 9, 2016 **Note the day change**

Riemannian manifolds with positive Yamabe invariant and Paneitz operator

Yueh-Ju Lin, University of Michigan

For a compact Riemannian manifold of dimension at least three, we know that positive Yamabe invariant implies the existence of a conformal metric with positive scalar curvature. As a higher order analogue, we seek for similar characterizations for the Paneitz operator and Q-curvature in higher dimensions. For a smooth compact Riemannian manifold of dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q-curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator. In addition, we also study the relationship between different conformal invariants associated to the Q-curvature. This is joint work with Matt Gursky and Fengbo Hang.