Neural networks are highly non-linear functions often parametrized by a staggering number of weights. Miniaturizing these networks and implementing them in hardware is a direction of research that is fueled by a practical need, and at the same time connects to interesting mathematical problems. For example, by quantizing, or replacing the weights of a neural network with quantized (e.g., binary) counterparts, massive savings in cost, computation time, memory, and power consumption can be attained. Of course, one wishes to attain these savings while preserving the action of the function on domains of interest. 
We discuss connections to problems in discrepancy theory, present data-driven and computationally efficient stochastic methods for quantizing the weights of already trained neural networks and we prove that our methods have favorable error guarantees under a variety of assumptions.  

Problems in extremal graph theory typically aim to maximize some graph parameters under local restrictions. In order to prove lower bounds for these kinds of problems, several techniques have been developed. The most popular one, initiated by Paul Erdős, is the probabilistic method. While this technique has enjoyed tremendous success, it does not always provide sharp lower bounds. Frequently, algebraically and geometrically defined graphs outperform random graphs. We will show how historically, geometry over finite fields has been a rich source of such graphs. I will show a broad class of graphs defined from the geometry of finite fields, which has found several recent applications in extremal graph theory. Often, certain interesting families of graphs had in fact already been discovered and studied, years before their value in extremal graph theory was realized. I will demonstrate some instances of this phenomenon as well, which indicates that there might still be uncharted territory to explore.

The sum-of-squares framework converts proofs into algorithms so long as those proofs can be formulated as sum-of-squares proofs—proofs that demonstrate their conclusions via the non-negativity of square polynomials. This is a powerful framework that captures many inequalities known in analysis. We formulate sum-of-squares proofs for the trace power method by giving sum-of-squares proofs for non-commutative (matrix) inequalities including the Araki-Lieb-Thirring inequality and Holder's inequality for Schatten norms, and sketch how one might use these proofs to obtain algorithms to constructively find solutions to random optimization problems whose upper bounds can be proven by techniques in random matrix and spin glass theory. Based on work in progress with Juspreet Singh Sandhu.