# Mathematics Colloquium Fall 2019

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.**

**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**

**TBA**