Georgia Institute of Technology College of Sciences

One of the most fundamental issues in fluid dynamics is whether or not an initially smooth fluid flow can evolve over time to arrive at a singularity -- a state for which the classical equations of fluid mechanics break down and the flow field no longer makes physical sense.  While proof remains an open question, numerical evidence strongly suggests that a singularity arises at the boundary of a flow like that found in a stirred cup of tea.  The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity.  A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model generalizes Burger's equation and captures key features in the mechanics of the blowup scenario. 

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in few cases by work of David--Philippon and DeMarco--Krieger--Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov--Gao--Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz--Rabinoff--Zureick-Brown).

Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.

Given a graph G, the clique chromatic number of G is the smallest number of colors needed to color the vertices of G so that no maximal clique containing at least two vertices is monochromatic.
We solve an open question proposed by McDiarmid, the speaker, and Pralat concerning the asymptotic order of the clique chromatic number for binomial random graphs.
More precisely, we find the correct order of the clique chromatic number for most values of the edge-probability p around n^{-1/2}. Furthermore, the gap between upper and lower bounds is at most a logarithmic factor in n in all cases.

Based on joint work in progress with Lyuben Lichev and Lutz Warnke.

(Please note the unusual time from 4-5pm, due to the Virtual Admitted Student Day in the School of Mathematics.)