Georgia Institute of Technology College of Sciences

In this lecture the analog of Riemannian manifold will be introduced through the notion of spectral triple. The recent work on the case of a metric Cantor set, endowed with an ultrametric, will be described in detail during this lecture. An analog of the Laplace Beltrami operator for a metric Cantor set will be defined and studied

For general graphs, approximating the maximum clique is a notoriously hard problem even to approximate to a factor of nearly n, the number of vertices. Does the situation get better with random graphs? A random graph on n vertices where each edge is chosen with probability 1/2 has a clique of size nearly 2\log n with high probability. However, it is not know how to find one of size 1.01\log n in polynomial time. Does the problem become easier if a larger clique were planted in a random graph? The current best algorithm can find a planted clique of size roughly n^{1/2}. Given that any planted clique of size greater than 2\log n is unique with high probability, there is a large gap here. In an intriguing paper, Frieze and Kannan introduced a tensor-based method that could reduce the size of the planted clique to as small as roughly n^{1/3}. Their method relies on finding the spectral norm of a 3-dimensional tensor, a problem whose complexity is open. Moreover, their combinatorial proof does not seem to extend beyond this threshold. We show how to recover the Frieze-Kannan result using a purely probabilistic argument that generalizes naturally to r-dimensional tensors and allows us recover cliques of size as small as poly(r).n^{1/r} provided we can find the spectral norm of r-dimensional tensors. We highlight the algorithmic question that remains open. This is joint work with Charlie Brubaker.

The quest for a suitable geometric description of major analyticproperties of sets has largely motivated the development of GeometricMeasure Theory in the XXth theory. In particular, the 1880 Painlev\'eproblem and the closely related conjecture of Vitushkin remained amongthe central open questions in the field. As it turns out, their higherdimensional versions come down to the famous conjecture of G. Davidrelating the boundedness of the Riesz transform and rectifiability. Upto date, it remains unresolved in all dimensions higher than 2.However, we have recently showed with A. Volberg that boundedness ofthe square function associated to the Riesz transform indeed impliesrectifiability of the underlying set. Hence, in particular,boundedness of the singular operators obtained via truncations of theRiesz kernel is sufficient for rectifiability. I will discuss thisresult, the major methods involved, and the connections with the G.David conjecture.

Enumeration of plane algebraic curves has a 150-year-old history. A combinatorial approach to this problem, inspired by tropical geometry, was recently suggested by Brugalle, Fomin, and Mikhalkin. I will explain this approach and its applications to computing Gromov-Witten invariants (or Severi degrees) of the complex projective plane, and their various generalizations.According to Goettsche's conjecture (now a theorem), these invariants are given by polynomials in the degree d of the curves being counted, provided that d is sufficiently large. I will discuss how to compute these "node polynomials," and how large d needs to be.