# 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, October 29th, 2019

TBA

Tuesday, November 5th, 2019

Chris Kottke, New College of Florida

Tuesday, November 12th, 2019

TBA

Tuesday,November 19th, 2019

TBA

Tuesday, November 26th, 2019

Bo Guan, Ohio State University

Tuesday, December 3rd, 2019

TBA