Georgia Institute of Technology College of Sciences

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.

A well-known problem in geometry, which may be traced back to the Renaissance artist Albrecht Durer, is concerned with cutting a convex polyhedral surface along some spanning tree of its edges so that it may be isometrically embedded, or developed without overlaps, into the plane. We show that this is always possible after an affine transformation of the surface. In particular, unfoldability of a convex polyhedron does not depend on its combinatorial structure, which settles a problem of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embeddings among immersed planar disks.

Fourier PCA is Principal Component Analysis of the covariance matrix obtained after reweighting a distribution with a random Fourier weighting. It can also be viewed as PCA applied to the Hessian matrix of functions of the characteristic function of the underlying distribution. Extending this technique to higher derivative tensors and developing a general tensor decomposition method, we derive the following results: (1) a polynomial-time algorithm for general independent component analysis (ICA), not requiring the component distributions to be discrete or distinguishable from Gaussian in their fourth moment (unlike in the previous work); (2) the first polynomial-time algorithm for underdetermined ICA, where the number of components can be arbitrarily higher than the dimension; (3) an alternative algorithm for learning mixtures of spherical Gaussians with linearly independent means. These results also hold in the presence of Gaussian noise.