Mathematics Colloquium Winter 2016

Tuesdays 4:00 p.m.
McHenry Building Room 4130
Refreshments served at 3:30 in the Tea Room (4161)
For further information please contact Professor Torsten Ehrhardt or call the Mathematics Department at 459-2969

Tuesday, January 12, 2016 *SPECIAL COLLOQUIUM*

“Stochastic Growth Processes and KPZ Universality”

Craig A. Tracy, University of California, Davis

In 2010 Takeuchi and Sano, in their now classic experiment, measured the fluctuations of a stochastically growing interface in turbulent liquid crystals.  They compared their measurements with certain distributions functions from random matrix theory.  Why one expected these distribution functions, first discovered in random matrix theory, should be relevant to the Takeuchi-Sano experiment is the story this lecture will tell.  The story involves the KPZ equation, KPZ universality, the asymmetric simple exclusion process and Bethe Ansatz.

Tuesday, January 19, 2016

"Geometry and Analysis on Nilpotent Lie Groups"

Theme: Calculus with a twist

Joseph A. Wolf, University of California, Berkeley

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent  Lie groups.  These have some consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and they have some extensions to (infinite dimensional) locally nilpotent Lie groups.  While this sounds a bit exotic, it mostly is just calculus with some noncommutativity thrown in.

Tuesday, January 26, 2016 *SPECIAL COLLOQUIUM*

"K-stability and CM-stability"

Tian Gang, Princeton University

Both K-stability and CM-stability were first introduced on Fano manifolds in 90s and generalized to any polarized projective manifolds. In this talk, I will introduce both CM-stable and K-stability and show how they are related. I will also discuss their relation to Geometric Invariant Theory and the problem on existence of constant scalar curvature Kahler metrics.

Thursday, 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.

Tuesday, February 2, 2016

"Symplectic embeddings and the Fibonacci numbers"

Daniel Cristofaro-Gardiner, Harvard University- Special Colloquium Speaker

It can be subtle to determine whether or not one symplectic manifold can be embedded into another, even for simple domains in C^n.  For example, McDuff and Schlenk computed when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that if the ellipsoid is close to round, the answer is given by an ``infinite staircase" determined by the odd-index Fibonacci numbers and the Golden Mean.  I will explain recent joint work with Richard Hind, showing that a version of this result holds in all dimensions.   Our proof is connected to some surprising relationships between Seiberg-Witten theory and symplectic geometry that I will also discuss.

Thursday, 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.

Tuesday, February 9, 2016

"The Navier-Stokes equations in the hyperbolic setting"

Magdalena Czubak, Binghamton University - Special Colloquium Speaker

There is an intimate connection between geometry and the methods of partial differential equations.  In this talk we focus on one such connection, the interplay of differential geometry and the Navier-Stokes equations.  The Navier-Stokes equations are one of the fundamental equations of fluid mechanics with applications to many disciplines and with a rich mathematical background. We investigate how the geometry of the underlying domain affects the solutions to the equations.  In particular, we show how the hyperbolic setting can lead to surprising phenomena that is not present in the Euclidean case.  We also present the advantageous properties of the hyperbolic geometry that allow us to obtain results for problems that have not been within reach in the Euclidean setting.  The questions under considerations are uniqueness, Hodge decompositions, and the Liouville problem.

Thursday, February 11, 2016

"Symplectic Normal Crossings and Their Smoothings"

Mohammad Farajzadeh 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.

Tuesday, February 16, 2016

No Colloquium

Thursday, 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 23, 2016

"Bernstein type theorems for the Willmore surface equation"

Jingyi Chen, University of British Columbia and MSRI

A Willmore surface in the 3-dimensional Euclidean space is a critical point of the square norm of the mean curvature of the surface. It is governed by a fourth order non-linear elliptic equation. The round spheres, the Clifford torus and the minimal surfaces are Willmore. A classical theorem of Bernstein says that an entire minimal graph must be a plane. We ask what happens to the entire Willmore graphs. In this talk, I will discuss joint work with Tobias Lamm on the finite energy case and with Yuxiang Li on the radially symmetric case.

March 1, 2016

"Essential spectrum of p-forms on complete Riemannian manifolds"

Zhiqin Lu, University of California, Irvine and MSRI

If the Ricci curvature of a complete noncompact Riemannian manifold is asymptotically nonnegative, then the essential spectrum of the Laplacian on functions is the set of nonnegative real numbers.  When we consider the Laplacians on p-forms, much stronger assumption is needed. We prove that if the manifold is asymptotically flat, then the spectra of p-forms are connected sets of the real line. This is joint work with N. Charalambous.

March 8, 2016

"Phase retrieval, random matrices, and convex optimization"

Thomas Strohmer, University of California, Davis

Phase retrieval is the century-old problem of reconstructing a function, such as a signal or image, from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems - Which arise in numerous areas including X-ray crystallography, astronomy, diffraction imaging, and quantum physics - are notoriously difficult to solve numerically. They also pervade many areas of mathemtics, such as numerical analysis, harmonic analysis, algebraic geometry, combinatorics, and differential geometry. In a recent breakthrough we have derived a novel framework for phase retrieval, which comprises tools for optimization, random matrix theory, and comepressive sensing. In particular, for certain types of random measurements a function, such as a signal or image, can be recovered exactly with high probability by solving a convenient semidefinite program without any assumption about the function whatsoever and under a mild condition on the number of measurements. Our method, known as PhaseLift, is also provably stable in presence of noise. The mathematical tools behind PhaseLift are inspired by Compressive Sensing, a topic that has received enormous attention in recent years. I will conclude with some extensions and open problems. The talk is accessible to a broad audience.