Georgia Institute of Technology College of Sciences

t-Haar multipliers are examples of Haar multipliers were the symbol depends both on the frequency variable (dyadic intervals) and on the space variable, akin to pseudo differential operators. They were introduced more than 20 years ago, the corresponding multiplier when $t=1$ appeared first in connection to the resolvent of the dyadic paraproduct, the cases $t=\pm 1/2$ is intimately connected to direct and reverse inequalities for the dyadic square function in $L^2$, the case $t=1/p$ naturally appears in the study of weighted inequalities in $L^p$. Much has happened in the theory of weighted inequalities in the last two decades, highlights are the resolution of the $A_2$ conjecture (now theorem) by Hyt\"onen in 2012 and the resolution of the two weight problem for the Hilbert transform by Lacey, Sawyer, Shen and Uriarte Tuero in 2014. Among the competing methods used to prove these results were Bellman functions, corona decompositions, and domination by sparse operators. The later method has gained a lot of traction and is being widely used in contexts beyond what it was originally conceived for in work of Lerner, several of these new applications have originated here at Gatech. In this talk I would like to tell you what I know about t-Haar multipliers (some work goes back to my PhD thesis and joint work with Nets Katz and with my former students Daewon Chung, Jean Moraes, and Oleksandra Beznosova), and what we ought to know in terms of sparse domination.

A motivating problem in number theory and algebraic geometry is to find all integer-valued solutions of a polynomial equation. For example, Fermat's Last Theorem asks for all integer solutions to x^n + y^n = z^n, for n >= 3. This kind of problem is easy to state, but notoriously difficult to solve. I'll explain a p-adic method for attacking Diophantine equations, namely, p-adic integration and the Chabauty--Coleman method. Then I'll talk about some recent joint work on the topic.

We discuss the growth of homonoly in finite coverings, and show that the growth of the torsion part of the first homology of finite coverings of 3-manifolds is bounded from above by the hyperbolic volume of the manifold. The proof is based on the theory of L^2 torsion.

There is an increasing trend for robots to serve as networked mobile sensing platforms that are able to collect data and interact with humans in various types of environment in unprecedented ways. The need for undisturbed operation posts higher goals for autonomy. This talk reviews recent developments in autonomous collective foraging in a complex environment that explicitly integrates insights from biology with models and provable strategies from control theory and robotics. The methods are rigorously developed and tightly integrated with experimental effort with promising results achieved.