Georgia Institute of Technology College of Sciences

A sum of squares of real numbers is always nonnegative. This elementary observation is quite powerful, and can be used to prove graph density inequalities in extremal combinatorics, which address so-called Turan problems. This is the essence of semidefinite method of Lov\'{a}sz and Szegedy, and also Cauchy-Schwartz calculus of Razborov. Here multiplication and addition take place in the gluing algebra of partially labelled graphs. This method has been successfully used on many occasions and has also been extensively studied theoretically. There are two competing viewpoints on the power of the sums of squares method. Netzer and Thom refined a Positivstellensatz of Lovasz and Szegedy by showing that if f> 0 is a valid graph density inequality, then for any a>0 the inequality f+a > 0 can be proved via sums of squares. On the other hand, Hatami and Norine showed that testing whether a graph density inequality f > 0 is valid is an undecidable problem, and also provided explicit but complicated examples of inequalities that cannot be proved using sums of squares. I will introduce the sums of squares method, do several examples of sums of squares proofs, and then present simple explicit inequalities that show strong limitations of the sums of squares method. This is joint work in progress with Annie Raymond, Mohit Singh and Rekha Thomas.

Non-compact hyperbolic surfaces serve as a model case for quantum scattering theory with chaotic classical dynamics. In this talk I’ll explain how scattering resonances are defined in this context and discuss our current understanding of their distribution. The primary focus of the talk will be on some recent conjectures inspired by the physics of quantum chaotic systems. I will introduce these and discuss the numerical evidence as well as recent theoretical progress.

In this talk I will discuss the Mikhlin-H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. I will show that this theorem does not hold in the limiting case $|1/p - 1/2|=s/n$. I will also present a sharp variant of this theorem involving a space of Lorentz-Sobolev type. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos.

In 1665, Huygens discovered that, when two pendulum clocks hanged from a same wooden beam supported by two chairs, they synchronize in anti-phase mode. Metronomes provides a second example of oscillators that synchronize. As it can be seen in many YouTube videos, metronomes synchronize in-phase when oscillating on top of the same movable surface. In this talk, we will review these phenomena, introduce a mathematical model, and analyze the the different physical effects. We show that, in a certain parameter regime, the increase of the amplitude of the oscillations leads to a bifurcation from the anti-phase synchronization being stable to the in-phase synchronization being stable. This may explain the experimental observations.