This talk will be devoted to the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension. In the bistable case, such a pulsating front indeed exists and it also describes the large time dynamics of solutions of the Cauchy problem. However, unlike in the homogeneous case the periodic problem is no longer invariant by rotation, so that the front speed may be different depending on its direction. This in turn raises some difficulties in the spreading shape of solutions of the evolution problem, which may exhibit strongly asymmetrical features. In the general multistable case, that is when there is a finite but arbitrary number of stable steady states, the notion of a single front is no longer sufficient and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states and whose speeds are ordered. The presented results come from a series of work with W. Ding, A. Ducrot, H. Matano and L. Rossi.

Relative Toeplitz algebras of directed graphs were introduced by Spielberg in 2002 to describe certain subalgebras corresponding to subgraphs. They can also be used to describe quotients of graph algebras corresponding to subgraphs. We use the latter relationship to answer a question posed in a recent paper regarding pushout diagrams of graphs that give rise to pullback diagrams of the respective graph C*-algebras. We introduce a new category of relative graphs to this end, and we prove our results using graph groupoids and their C*-algebras. This is joint work with Jack Spielberg.

To each integral vector v in $\mathbb{Z}^n$, we attach several natural objects of geometric/arithmetic nature. For example:

1. The direction of v (i.e., its radial projection to the unit sphere),
2. The orthogonal lattice to v (i.e., the proper rescaling of the lattice of integral points in the orthogonal hyperplane to v),
3. The residue class of v modulo a fixed integer k.

Each of these objects resides in a natural “homogeneous space” which supports a “uniform probability measure”. This allows one to ask statistical questions regarding these objects as v varies in some meaningful set of integral vectors. I will survey some classical and more recent results along these lines where there are limit laws governing the statistics. In some cases one obtains the uniform measure as the limit and in some cases a non-uniform limit. Interesting examples include the integral points on quadratic surfaces and the sequence of “best approximations” of an irrational line. In the talk I will try to explain how homogeneous dynamics can be used to tackle such questions.

The random d-process corresponds to a natural algorithmic model for generating d-regular graphs: starting with an empty graph on n vertices, it evolves by sequentially adding new random edges so that the maximum degree remains at most d.

In 1999 Wormald conjectured that the final graph of the random d-process is "similar" to a uniform random d-regular graph.

We show that this conjecture does not extend to a natural generalization of this process with mixed degree restrictions, i.e., where each vertex has its own degree restriction (under some mild technical assumptions). 
Our proof uses the method of switchings, which is usually only applied to uniform random graph models -- rather than to stochastic processes.

Based on joint work in progress with Mike Molloy and Erlang Surya.

The mapping class group, $\mathrm{Map}(S)$, of a surface $S$, is the group of all isotopy classes of homeomorphisms of $S$ to itself. A mapping class group is a topological group with the quotient topology inherited from the quotient map of $\mathrm{Homeo}(S)$ with the compact-open topology.

For surfaces of finite type, $\mathrm{Map}(S)$ is countable and discrete. Surprisingly, the topology of $\mathrm{Map}(S)$ is more interesting if $S$ is an infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and more importantly, has the structure of a Polish group. In recent literature, this last class of groups is called "big mapping class groups.''

In this talk, I will give a brief introduction to big mapping class groups and explain our results on the topological structure of conjugacy classes. This was a joint work with Jesús Hernández Hernández, Michael Hrušák, Manuel Sedano, and Ferrán Valdez.

For any 3D steady gradient Ricci soliton, if it is asymptotic to a ray we prove that it must be isometric to the Bryant soliton. Otherwise, it is asymptotic to a sector and called a flying wing. We show that all flying wings are O(2)-symmetric. Hence, all 3D steady gradient Ricci solitons are O(2)-symmetric.

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.