Georgia Institute of Technology College of Sciences

Bézout’s Theorem is the classical statement that generic curves of degree c and d intersect in cd points. However, this theorem requires that we work over an algebraically closed field. Using some tools from A^1-algebraic topology, we will give an arithmetic generalization of Bézout’s Theorem. We will also describe the geometric implications of this generalization over the reals.

A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2+Ω(√m), and this is best possible. We will continue our discussion about recent results on analogues of Edwards’ result and related problems in hypergraphs.

Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions of this type arise in physics as modes of periodic vibration of drums and membranes. They also represent stationary states of a free quantum particle on a Riemannian manifold. In the first part of the lecture, I will give a survey of results which demonstrate how the geometry of $M$ affects the behaviour of these special functions, particularly their “size” which can be quantified by estimating $L^p$ norms. In joint work with Malabika Pramanik (U. British Columbia), I will present in the second part of my lecture a result on the $L^p$ restriction of these eigenfunctions to random Cantor-type subsets of $M$. This, in some sense, is complementary to the smooth submanifold $L^p$ restriction results of Burq-Gérard-Tzetkov ’06 (and later work of other authors). Our method includes concentration inequalities from probability theory in addition to the analysis of singular Fourier integral operators on fractals.

Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects. Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all. Can we characterize when? Can we even formalize what is a "formula"? I will try to answer these questions by presenting many examples, results and open problems. Note: This is an introductory general audience talk unrelated to the colloquium.