Georgia Institute of Technology College of Sciences

Immersion is a containment relation between graphs (or digraphs) which is defined similarly to the more familiar notion of minors, but is incomparable to it. Of particular interest is to find conditions on a graph (or digraph) G which guarantee that G contains a clique (or bidirected clique) of order t as an immersion. This talk will begin with a gentle introduction, and will then share two new results of this form, one for graphs and one for digraphs. In the former case, we find that minimum degree 200t is sufficient, and in the later case, we find that minimum degree t(t-1) is sufficient, provided that G is Eulerian. These results come from joint work with Matt DeVos, Jacob Fox, Zdenek Dvorak, Bojan Mohar and Diego Scheide.

Computational models of the Earth system lie at the heart of modern climate science. Concerns about their predictions have been illegitimately used to undercut the case that the climate is changing and this has put dynamical systems in an awkward position. I will discuss ways that we, as a community, can contribute by highlighting some of the major outstanding questions that drive climate science, and I will outline their mathematical dimensions. I will put a particular focus on the issue of simultaneously handling the information coming from data and models. I will argue that this balancing act will impact the way in which we formulate problems in dynamical systems.

Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 By the classical Weierstrass theorem, any function continuous on a compact set can be uniformly approximated by algebraic polynomials. In this talk we shall discuss possible extensions of this basic result of analysis to approximation by homogeneous algebraic polynomials on central symmetric convex bodies. We shall also consider a related question of approximating convex bodies by convex algebraic level surfaces. It has been known for some time time that any convex body can be approximated arbirarily well by convex algebraic level surfaces. We shall present in this talk some new results specifying rate of convergence.

I will discuss a natural elliptic obstacle problem that arises in the study of the Abelian sandpile. The Abelian sandpile is a deterministic growth model from statistical physics which produces beautiful fractal-like images. In recent joint work with Wesley Pegden, we characterize the continuum limit of the sandpile processusing PDE techniques. In follow up work with Lionel Levine and Wesley Pegden, we partially describe the fractal structure of the stable sandpiles via a careful analysis of the limiting obstacle problem.