Mathematics Colloquium Fall 2017

Tuesdays - 4:00 p.m.
McHenry Library 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, October 3, 2017

No Colloquium

SPECIAL TIME 3:30 PM Tuesday, October 10, 2017

Daniel Cristofaro-Gardiner, University of California, Santa Cruz

Size and Time in Symplectic Geometry

A landmark result in symplectic topology is the discovery, due to Gromov, that there are obstructions to symplectic embeddings that are fundamentally different from the classical volume obstruction. The modern point of view on this phenomenon uses the language of symplectic capacities. Symplectic capacities are measurements of "symplectic size". For symplectic manifolds with contact type boundary they are often defined as the periods of certain sets of closed orbits of the Reeb vector field on the boundary, and so connect embedding theory with dynamics.

For a large class of symplectic four-manifolds, I showed in joint work that there is a sequence of symplectic capacities, called ECH capacities, that naturally recover the classical volume in their asymptotic limit. In the first part of my talk, I will explain how these capacities are defined, and introduce some of the ideas behind the proof of this asymptotic formula. I will then discuss some applications of this formula to two problems in dynamics: i) showing the existence of closed orbits of the Reeb vector field on closed three-manifolds; and ii) establishing closing lemmas in high orders of smoothness.

Tuesday, October 17, 2017

Joel Hass, University of California, Davis

Comparing the Shape of Surfaces 

Almost everything we encounter in our 3-dimensional world is a surface - the outside of a solid object. Comparing the shapes of surfaces is, not surprisingly, a fundamental problem in both theoretical and applied mathematics. Deep mathematical results are now being used to study objects such as bones, brain cortices, proteins and biomolecules. This talk will discuss this and recent joint work with Patrice Koehl that introduces a new metric on the space of genus-zero surfaces.

Tuesday, October 24, 2017

Aleksandr Koldobsky, University of Missouri

Fourier Analysis in Geometric Tomography

Geometric tomography is the area of mathematics where one investigates geometric properties of solids based on data about sections and projections of these solids. Recently, a new Fourier analytic approach has been developed where geometric parameters are expressed in terms of the Fourier transform which allows to treat geometric questions as problems from harmonic analysis. In particular, this approach has led to an analytic solution of the Busemann-Petty problem asking whether convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. In this talk, we recall the main features of the Fourier approach to sections of convex bodies and present recent developments including quantitative versions of the Busemann-Petty problem, volume difference inequalities designed to estimate the error in volume computations, slicing inequalities for measures of convex bodies.

Tuesday, October 31, 2017

Colloquium canceled

Tuesday, November 7, 2017

Victor Reiner, University of Minnesota

Finite General Linear Groups and Symmetric Groups

In recent years we have seen surprising counting phenomena related to the symmetric group, with striking analogues for general linear groups over finite fields. Often the explanations come from invariant theory. We will give examples, and pose some intriguing conjectures that come from pursuing the analogy further. 

Tuesday, November 14, 2017

Xiangwen Zhang, University California Irvine

The Anomaly Flow and Hull-Strominger system

We discuss the development on geometric and analytic aspects of the Anomaly flow. Such flow naturally arises in the study of a system of equations for supersymmetric vacua of superstrings proposed independently by C. Hull and A. Strominger in 1980s. The system allows non-vanishing torsion and they incorporate terms which are quadratic in the curvature tensor. As such they are also particularly interesting from the point of view of both non-Kaehler geometry and the theory of nonlinear partial differential equations. While the complete solution of the system seems out of reach at the present time, we describe progress in developing a new general approach based on geometric flows. It turns out that the corresponding flow shares some features with the Ricci flow and preserves the conformally balanced condition of Hermitian metrics. In particular, on toric fibrations, the flow exists for all time and converges, recovering in this way the well-known solution obtained by J. Fu and S.T. Yau in 2006 by solving a delicate elliptic equation of Monge-Ampere type.  This is joint work with D. Phong and S. Picard.

Tuesday, November 21, 2017

Werner Bley, Ludwig Maximillan University Munich

The equivariant analytic class number formula

The relevant case of the equivariant Tamagawa Number Conjecture in the number field case is an equivariant refinement of the analytic class number formula. Whereas the analytic class number formula is a theorem covered in in many courses on Algebraic Number Theory, the equivariant refinement is widely open. 

Starting from the class number formula we will sketch the formulation of the equivariant refinement and describe a possible strategy of a proof. Finally we will report on known results.

Tuesday, November 28, 2017

Andras Vasy, Stanford University

Boundary rigidity and the local inverse problem for the geodesic X-ray transform
on tensors

In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points. Applications include travel time tomography for waves of isotropic elasticity, such as seismic waves, as long as the anisotropy of the elastic materials, such as crystals, can be neglected.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under a convexity assumption on the boundary, one can invert the local  geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner. I will also explain how the analogous result can be achieved on one forms and 2-tensors up to the natural obstacle, namely potential tensors (forms which are differentials of functions vanishing at the boundary, respectively tensors which are symmetric gradients of one-forms vanishing at the boundary).

Here the local transform means that one would like to recover a function (or tensor) in a suitable neighborhood of a point on the boundary of the manifold given its integral along geodesic segments that stay in this neighborhood (i.e. with both endpoints on the boundary of the manifold). Our method relies on microlocal analysis, in a form that was introduced by Melrose.

I will then also explain how, under the assumption of the existence of a strictly convex family of hypersurfaces foliating the manifold, this gives immediately the solution of the global inverse problem by a stable `layer stripping' type construction. Finally, I will discuss the relationship with, and implications for, the boundary rigidity problem.


Tuesday, December 5, 2017

Richard Montgomery, University of California, Santa Cruz

Mechanics and Mnev's Universality Theorem

The Jacobi-Maupertuis [JM] principle asserts that solving Newton's equations at energy H is equivalent to finding geodesics for a conformally flat metric with conformal factor depending linearly on H. This principle, combined with reduction, led to a metric with negative Gauss curvature for a planar three-body problem which solved an outstanding problem. Connor Jackman showed that the corresponding curvature can be positive or negative in case of four bodies. Switching gears, I teach a Classical Geometries class pretty much every year, and every year most of the students leave that class with essentially no clue what hyperbolic geometry is. SoI search and search for mechanical models which might make hyperbolic geometry more intuitive. I will describe a strange 3-body potential for which the JM metric, after reduction, is isometric to the hyperbolic plane. When I tried, a la Jackman, to generalize hyperbolicity to the analoguous N-body problem, I ran smack into Mnev's Universality Theorem. Call an N-tuple of points in the real projective plane "generic" if no three are collinear. The open set of generic points in the N-fold copy of the projective plane falls into many copies if N is large. Mnev's Universality theorem implies that every possible homotopy type of every possible compact connected manifold is realized by such a component.