Past Events
A knot in S^3 is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally…
The Abelian sandpile is a diffusion process on the integer lattice which produces striking, kaleidoscopic patterns. Why do these patterns appear? How robust are the patterns to noise? What happens in dimensions higher than two? I will discuss recent progress towards answering these questions.…
In positive characteristic, there are two different notions of rational connectedness: a variety can be rationally connected or separably rationally connected (SRC). SRC varieties share many of the nice properties that rationally connected varieties have in…
The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and each piece is realizable as a finite volume quotient…
Abstract: Given a curve $\Gamma$, what is the surface $T$ that has least area among all surfaces spanning $\Gamma$? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or…
Toeplitz asked in 1911 whether every Jordan curve in the Euclidean plane contains the vertices of a square. The problem remains open, but it has given rise to many interesting variations and partial results. I will describe some of these and the proof of a result which is best possible when the…
We will talk about classification of ancient solutions in geometric flows. In particular, we will show the only closed ancient noncollapsed Ricci flow solutions are the shrinking spheres and Perelman's solution. We will talk about the higher dimensional analogue of this result under…
Macdonald polynomials are a remarkable family of functions. They are a common generalization of many different families of special functions arising in the representation theory of reductive groups, including spherical functions and Whittaker functions.
In turn, Macdonald polynomials can…