Topologies of nodal sets of random band limited functions

Peter Sarnak
Institute for Advanced Study; Faculty, School of Mathematics
March 3, 2014
We discuss various Gaussian ensembles for real homogeneous polynomials in several variables and the question of the distribution of the topologies of the connected components of the zero sets of a typical such random real hypersurface. For the "real -Fubini -Study ensemble" and at the other end the "monochromatic wave ensemble ", one can show that these have universal laws. Some qualitative features of these laws are also established. Joint work with I. Wigman.

Fast matrix multiplication

Yuval Filmus
Institute for Advanced Study; Member, School of Mathematics
March 4, 2014
How many arithmetic operations does it take to multiply two square matrices? This question has captured the imagination of computer scientists ever since Strassen showed in 1969 that \(O(n^{2.81})\) operations suffice. We survey the classical theory of fast matrix multiplication, starting with Strassen's algorithm and culminating with the state-of-the-art Coppersmith-Winograd algorithm and its recent improvements. We also describe Coppersmith's \(O(n^2 log^2 n)\) algorithm for multiplying rectangular matrices, used by Ryan Williams in his separation of ACC and NEXP.

Contact invariants in sutured monopole and instanton homology

Steven Sivek
University of Warwick
March 5, 2014
Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. In this talk, I will construct an invariant of a contact structure on a 3-manifold with boundary as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps and an exact triangle associated to bypass attachment, and explain how this construction leads to an invariant in the sutured version of instanton Floer homology as well.

Princeton/IAS Symplectic Geometry Seminar

Keon Choi
University of California, Berkeley
March 7, 2014
Embedded contact homology is an invariant of a contact three-manifold, which is recently shown to be isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the contact form on the manifold and the almost complex structure on its symplectization. This fact can be used to extract symplectic geometric information (e.g. ECH capacities) but explicit computation of the chain complexes has been carried out only on a few cases.

Two Structural Results for Low Degree Polynomials and Applications

Avishay Tal
Weizmann Institute
March 10, 2014
We give two structural results concerning low degree polynomials over the field \(\mathbb{F}_2\). The first states that for any degree d polynomial f in n variables, there exists a subspace of \(\mathbb{F}_2^n\) with dimension \(\Omega(n^{1/(d-1)})\) on which f is constant. This result is shown to be tight. Stated differently, a degree d polynomial cannot compute an affine disperser for dimension smaller than \(\Omega(n^{1/(d-1)})\).