Mathematics Colloquium Fall 2019

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

Tuesday, October 8th, 2019

Frederic Faure, Joseph Fourier University

Emergence of the quantum wave equation in classical deterministic hyperbolic dynamics

In the 80's, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos ("Ruelle resonances"). For example, a geodesic flow on a strictly negative curvature Riemannian manifold is chaotic: each trajectory is strongly unstable and its behavior is unpredictable. A smooth probability distribution evolves also in a complicated way since it acquires higher and higher oscillations. Nevertheless this evolution is predictable in the sense of distributions and converges towards equilibrium. Following this approach and use of microlocal analysis, one obtains that long time fluctuations of classical probabilities are described by an effective quantum wave equation. This may be surprising because there is no added quantization procedure. We will explain the concepts and results using different simple models. Joint work with Masato Tsujii.

Tuesday, October 15th, 2019

Eduardo Fuertes

Some new constructions for loops of Legendrians in the standard contact $3$--sphere.

There is a rich literature about the study of the inclusion map from the space of Legendrian embeddings in a contact $3$--manifold into the space of formal Legendrian embeddings at the $\pi_0$--level but almost nothing is known for higher homotopy groups. In particular, it is an open question if the induced map at $\pi_k$--level is injective or not. For loops of Legendrians the only known non trivial construction of a loop of Legendrians, due to T. K\'alm\'an in 2005, turns out to be formally non--trivial. In this talk we introduce the parametric sum of Legendrians embeddings in the standard contact  $\mathbb{S}^3$ (and in the standard $\mathbb{R}^3$) which gives rise to an effective way of manipulating the formal invariants (and hopefully, this is work in progress, produce candidates of formally trivial loops which potentially are non--trivial).

Tuesday, October 22nd, 2019

Nikhil Savale, University of Cologne

Spectrum and Abnormals in Sub-Riemannian geometry: the 4D quasi-contact case

We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry

Tuesday, November 5th, 2019

Chris Kottke, New College of Florida

You'll look sweet, Upon the seat, Of a bigerbe made for two

Gerbes are geometric objects on a space which represent degree 3 integer cohomology, in the same way that complex line bundles (classified by the Chern class) represent cohomology in degree 2. Among other settings, they arise naturally as an obstruction to lifting the structure group of a principal bundle to a central extension. Higher versions of gerbes, representing cohomology classes of degree 4 and up, are typically complicated by the need to use higher categorical concepts (2-morphisms and so on) in their definition. In contrast, bigerbes (and their higher cousins) admit a simple, geometric, non-higher-categorical description, and provide a satisfactory account of the relationship between so-called `string structures' on a manifold and `fusion spin structures' on its loop space. This is based on recent joint work with Richard Melrose.

Tuesday, November 19th, 2019

Mikko Salo, MSRI & University of Jyväskylä

Inverse problems for PDEs

Inverse problems research concentrates on the mathematical theory and practical implementation of indirect measurements. Applications are found in numerous research fields involving scientific, medical or industrial imaging; familiar examples include X-ray computed tomography and ultrasound imaging. Inverse problems have a rich mathematical theory employing modern methods in partial differential equations (PDEs), harmonic analysis, and differential geometry. In this colloquium talk we give an introduction to mathematical inverse problems, and outline a recent approach to develop general theory for inverse problems for PDEs. We will also explain some basic ideas of phase space (or microlocal) analysis that will be useful for this. The talk is based on joint work with Lauri Oksanen (UCL), Plamen Stefanov (Purdue) and Gunther Uhlmann (Washington / IAS HKUST.)

Tuesday, November 26th, 2019

Bo Guan, Ohio State University

Conformal deformation of Chern-Ricci curvatures and fully nonlinear elliptic equations on complex manifolds.

Fully nonlinear PDEs play important roles in complex geometry. It goes back to the study of complex Monge-Ampere equations by S.T. Yau and Aubin, and their proof of Calabi conjectures in Kaehler geometry. In recent years there have been increasing interests in more general equations beyond the Monge-Ampere, often non-Kaehler complex manifolds. In our talk we shall report some recent progresses in the effort to solve these equations, focusing on the Dirichlet problem and equations on closed manifolds as well. In the second part of the talk (if time allows) we shall discuss problems and results on equations related to conformal deformation Chern-Ricci curvatures on a non-Kahler Hermitian manifold.

Tuesday, December 3rd, 2019