Georgia Institute of Technology College of Sciences

Allowing formal desuspensions of maps and objects takes the category of topological spaces to the category of spectra, where cohomology is naturally represented. The EHP spectral sequence encodes how far one can desuspend maps between spheres. It's among the most useful tools for computing homotopy groups of spheres. RP^infty has a cell structure with a cell in each dimension and with attaching maps of degrees ...020202... Note that this sequence is periodic. In fact, it is more than the degrees of these maps which are periodic and a map of Snaith relates this periodicity to the EHP sequence.We will develop the EHP sequence, James periodicity and the relationship between the two.

We consider two approaches to address angles between random subspaces: classical random matrix theory and free probability. In the former, one constructs random subspaces from vectors with independent random entries. In the latter, one has historically started with the uniform distribution on subspaces of appropriate dimension. We point out when these two approaches coincide and present new results for both. In particular, we present the first universality result for the random matrix theory approach and present the first result beyond uniform distribution for the free probability approach. We further show that, unexpectedly, discrete uncertainty principles play a natural role in this setting. This work is partially with L. Erdos and G. Anderson.

Probabilistic methods in dynamical systems is a popular area of research. The talk will present the origin of the interplay between both subjects with Poincar\'e's unpredictability and Kolmogorov's axiomatic treatment of probability, followed by two main breakthroughs in the 60es by Ornstein and Gordin. Present studies are concerned with two main problems: transferring probabilistic laws and laws for 'smooth' functions. Recent results for both type of questions are explained at the end.

It is a classical problem in real algebraic geometry to try to obtain tight bounds on the number of connected components of semi-algebraic sets, or more generally to bound the higher Betti numbers, in terms of some measure of complexity of the polynomials involved (e.g., their number, maximum degree, and number of variables or so-called dense format). Until recently, most of the known bounds relied ultimately on the Oleinik-Petrovsky-Thom-Milnor bound of d(2d-1)^{k-1} on the number of connected components of an algebraic subset of R^k defined by polynomials of degree at most d, and hence the resulting bounds depend on only the maximum degree of the polynomials involved. Motivated by some recent results following the Guth-Katz solution to one of Erdos' hard problems, the distinct distance problem in the plane, we proved that in fact a more refined dependence on the degrees is possible, namely that the number of connected components of sign conditions, defined by k-variate polynomials of degree d, on a k'-dimensional variety defined by polynomials of degree d_0, is bounded by (sd)^k' d_0^{k−k'} O(1)^k. Our most recent work takes this refinement of the dependence on the degrees even further, obtaining what could be considered a real analogue to the classical Bezout inequality over algebraically closed fields.