Mathematics

Let $G$ be a graph with adjacency matrix $A$. A continuous quantum walk on $G$ is determined by the complex unitary matrix $U(t)=\exp(itA)$, where $i^2=−1 and $t$ is a real number. Here, $G$ represents a quantum spin network, and its vertices and edges represent the particles and their interactions in the network. The propagation of quantum states in the quantum system determined by $G$ is then governed by the matrix $U(t)$. In particular, $|U(t)_{u,v}|^2$ may be interpreted as the probability that the quantum state assigned at vertex $u$ is transmitted to vertex $v$ at time $t$. Quantum walks are of great interest in quantum computing because not only do they produce algorithms that outperform classical counterparts, but they are also promising tools in the construction of operational quantum computers. In this talk, we give an overview of continuous quantum walks, and discuss old and new results in this area with emphasis on the concepts and techniques that fall under the umbrella of discrete mathematics.

A zero-free region of the Riemann zeta-function is a subset of the
complex plane where the zeta-function is known to not vanish. In this talk we
will discuss various computational and analytic techniques used to enlarge the
zero-free region for the Riemann zeta-function, when the imaginary part of a
complex zero is large. We will also explore the limitations of currently known
approaches. This talk will reference a number of works from the literature,
including a joint work with M. Mossinghoff and T. Trudgian.

Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.

Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.
Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

how random matrices can be used to describe high-dimensional inference
nonconvex landscape properties
high-dimensional limits of stochastic gradient methods.