Georgia Institute of Technology College of Sciences

The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open. In this talk, I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics around a Bose-Einstein condensate, within the quantum field theoretic description of an interacting Boson gas. This is based on joint work with Michael Hott.

Join Zoom Meeting at https://gatech.zoom.us/j/92873362365

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will illustrate how to understand the long-time behavior solutions to dispersive waves via various results I obtained over the years.

I'll talk about some 2D billiards, the most visual class of dynamical systems, where orbits (rays) move along straight lines within a billiard table with elastic reflections off the boundary.  Elliptic flowers are built “around" convex polygons, and the boundary of corresponding billiard tables consists of the arcs of ellipses. It will be explained why some classes of such elliptic flowers demonstrate a never expected before dynamics, and why it raises a variety of (seemingly new) questions in geometry (particularly in 3D), in bifurcation theory (particularly about singularities of wave fronts and creation of wave trains), in statistical mechanics,  quantum chaos, and perhaps some more. The talk will be concluded by showing a free movie. Everything (including various definitions of ellipses) will be explained/reminded.

The phylogenetic birth-death process is a probabilistic model of evolution that
is widely used to analyze genetic data. In a striking result, Louca & Pennell
(Nature, 2020) recently showed that this model is statistically unidentifiable,
meaning that an arbitrary number of different evolutionary hypotheses are
consistent with any given data set. This grave finding has called into question
the conclusions of a large number of evolutionary studies which relied on this
model.

In this talk, I will give an introduction to the phylogenetic birth-death
process, and explain Louca and Pennell's unidentifiability result. Then, I will
describe recent positive results that we have obtained, which establish that, by
restricting the evolutionary hypothesis space in certain biologically plausible
ways, statistical identifiability is restored. Finally, I will discuss some
complementary hardness-of-estimation results which show that, even in identifiable
model classes, obtaining reliable inferences from finite amounts of data may be
extremely challenging.

No background in this area is assumed, and the talk will be accessible to a
mathematically mature audience. This is joint work with Brandon Legried.

Zoom link:  https://gatech.zoom.us/j/99936668317