The cover time of a graph $G$ is the maximum over vertices $v\in V(G)$ of the expected time for a simple random walk to visit all vertices of $G$, starting at $v$. We will review what we know about this question and then focus on two recent results.

{\bf Dense Graphs:} We consider abritrary graphs $G$ with $n$ vertices and minimum degree at least $\delta n$ where $\delta>0$ is constant. If the conductance of $G$ is sufficiently large then we obtain an asymptotic expression for the cover time $C_G$ of $G$ as the solution to some explicit transcendental equation. Failing this, if the mixing time of a random walk on $G$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense subgraphs with high conductance. Failing this we give a deterministic asymptotic 2-approximation of $C_G$.

Joint work with Colin Cooper and Wesley Pegden.

{\bf Emerging Giant:} Let $p=\frac{1+\epsilon}{n}$. It is known that if $N=\epsilon^3n\to\infty$ then w.h.p. $G_{n,p}$ has a unique giant largest component. We show that if in additon, $\epsilon=\epsilon(n)\to 0$ then w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$.

Joint work with Wesley Pegden and Tomasz Tkocz.

Classical models of biological populations, for example, Markov branching processes, typically model population size and possibly the distribution of types and/or locations of individuals in the population. The intuition behind these models usually includes ideas about the relationships among the individuals in the population that cannot be directly recovered from the model. This loss of information is even greater if one employs large population approximations such as the diffusion approximations popular in population genetics. ``Lookdown'' constructions provide representations of population models in terms of countable systems of particles in which each particle has a ``type'' which may record both spatial location and genetic type and a ``level'' which incorporates the lookdown structure which in turn captures the genealogy of the population. The original population model can then be viewed as the result of the partial observation of the more complex model. We will exploit ideas from filtering of Markov processes to make the idea of partial observation clear and to justify the lookdown construction.

Resonances are in some sense analogs of discrete eigenvalues for
certain operators with continuous spectrum. Physically they may
correspond to decaying waves.

For Euclidean scattering in odd dimensions, the resonances are points in
the complex plane; in even dimensions, they lie on the logarithmic cover
of the complex plane. For scattering by an obstacle, we consider the
problem of counting the number of resonances in certain regions. In
particular, we show that surprisingly there is a sharp lower bound on a
resonance-counting function in even dimensions for which the analogous
result is not yet known in odd dimensions.

We extend a model for the morphology and dynamics of a crawling eukaryotic cell to describe cells on micro patterned substrates. This model couples cell morphology, adhesion, and cytoskeletal flow in response to active stresses induced by actin and myosin. We propose that protrusive stresses are only generated where the cells adheres, leading to the cell’s effective confinement to the pattern. Simulated cells exhibit a broad range of behaviors, including steady motion, turning, bipedal motion and periodic migration. We further extensively study the turning instability by simplifying the full PDE model into a minimal one. By using the minimal model, we also study the persistent rotational motion (PRM) of small numbers of mammalian cells crawling on micropatterned substrates.

Given an integer vector $a = (a_1,\dots,a_n)$, let $\rho(a)$ be the number of solutions to $a\cdot x=0$, with $x\in \{\pm 1\}^n$. In 1945, Erdos gave a beautiful combinatorial solution to the following problem that was posed by Littlewood and Offord: how large can $\rho(a)$ be if all the entries of $a$ are non-zero?

Following his breakthrough result, several extensions of this problem have been intensively studied by various researchers. In classical works, Erdos-Moser, Sarkozy-Szemeredi, and Halasz obtained better bounds on $\rho(a)$ under additional assumptions on $a$, while Kleitman, Frankl-Furedi, Esseen, Halasz and many others studied generalizations to higher dimensions. In recent years, motivated by deep Freiman-type inverse theorems from additive combinatorics, Tao and Vu brought a new view to this problem by asking for the underlying structural reason for $\rho(a)$ to be large --this is known as the Inverse Littlewood-Offord problem, which is a cornerstone of modern random matrix theory.

In this talk, we will discuss further extensions and improvements for both forward and inverse Littlewood-Offord problems where combinatorial tools and insights have proved to be especially powerful. We also present several applications in (discrete) matrix theory such as: a non-trivial upper bound on the number of Hadamard matrices, an upper bound on the number of $\pm 1$ normal matrices (improving a result of Deneanu and Vu), and a unified approach for counting the number of singular $\pm1$ matrices from various popular models (improving results of Cook, Nguyen and Vershynin).

Let M\_g be the moduli space of smooth curves of genus g. The tautological ring is a subring of the cohomology of M\_g that was introduced by Mumford in the 1980s in analogy with the cohomology of Grassmannians. Work of Faber and Faber-Zagier in the 1990s led to two competing conjectural descriptions of the structure of the tautological ring. After reviewing these conjectures, I will discuss some of the evidence in recent years favoring one conjecture over the other.