Georgia Institute of Technology College of Sciences

I will discuss several classical problems in number theory about representation of numbers and forms by quadratic forms and related counting results. We will describe several applications such as pair correlations and eigenvalue statistics for quantum systems.
Then we will move to related problems about irrational forms, such as the Oppenheim conjecture and explain how homogeneous dynamics can help to tackle such problems.
The talk will be self contained and hopefully accessible.

In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space.

Ice sheets are fascinating dynamical systems that flow, fracture and melt on a wide range of time scales, presenting a range of challenging prediction problems with important implications for how coastal communities plan for sea level rise. In this talk, I will introduce a few outstanding problems concerning the evolution of Earth’s ice sheets under climate change.

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

We will discuss some surprising rigidity phenomena for Anosov flows in dimension 3. For example, in the context of generic transitive 3-dimensional Anosov flows, any continuous conjugacy is either smooth or reverses the positive and negative SRB measures.

This is joint work with Martin Leguil and Federico Rodriguez Hertz

Abstract: Classifying the invariant measures for a given dynamical system represents a fundamental challenge.

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09