This talk will survey the extremal theory of pattern-avoiding 0-1 matrices, and some of their applications in geometry, combinatorics, and algorithms.  If P is a 0-1 matrix, Ex(P,n) is the maximum number of 1s in an n x n 0-1 matrix that does not contain any submatrix that dominates P.  Every 0-1 pattern P can be regarded as the incidence matrix of a bipartite graph, in which the two sides of the bipartition are ordered.  Thus, this definition can be seen as a generalization of the Turan extremal function (for subgraph avoidance).

Pattern-avoiding 0-1 matrices have been studied since the late 1980s, and yet the precise relationship between 0-1 matrices and Turan theory is still poorly understood.  For many years the foremost open problem has been to characterize the extremal functions of acyclic patterns (those whose graphs correspond to forests).  In 2005 Pach and Tardos conjectured that Ex(P,n) = O(n polylog(n)), for any acyclic P.  We give a simple refutation of the Pach-Tardos conjecture by giving a class of acyclic patterns for which Ex(P,n) > n 2^{sqrt{log n}}.

For a Lie group G, the G-skein module of a 3-dimensional manifold M is a fundamental object in Witten’s interpretation of quantum knot invariants in the framework of a topological quantum field theory. It depends on a parameter q and, when this parameter q is a root of unity, the G-skein module contains elements with a surprising “transparency” property, in the sense that they can be traversed by any other skein without changing the resulting total skein. I will describe some (and conjecturally all) of these transparent elements in the case of the special linear group SL_n. The construction is based on the very classical theory of symmetric polynomials in n variables.  

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

 The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in $n \geq 3$ variables, the image set $Q(Z^n)$  of integral vectors is a dense subset of the real line. Determining the distribution of values of an indefinite quadratic form at integral points asymptotically is referred to as quantitative Oppenheim conjecture. The quantitative Oppenheim conjecture was established by Eskin, Margulis, and Mozes for quadratic forms in $n \geq 4$ variables. In this talk, we discuss the quantitative Oppenheim conjecture for ternary quadratic forms (n=3). The main ingredient of the proof is a uniform boundedness result for the moments of Margulis functions over expanding translates of a unipotent orbit in the space of 3-dimensional lattices, under suitable Diophantine conditions of the initial unipotent orbit.

Between thousand to million times per day, your cellphone calculates the Fourier Transform (FT) of certain functions defined on 0,1,…,N-1, with N large (order of magnitude of thousands and more). The calculation is done using the Fast Fourier Transform (FFT) - discovered by Cooley–Tukey in 1965 and by Gauss in 1805. In the lecture I want to advertise a beautiful way—due to Auslander-Tolimieri—to obtain the FFT as a natural consequence of an answer to the following:

Question: How to think on the space of functions on the set 0,1,…,N-1? 

Engineers tell us that there are two answers for this question: 

(A) as functions on that set, where 0,1,…,N-1 regarded as times; 

and,

(B) as functions on that set, where 0,1,…,N-1 regarded frequencies; 

and then the FT is an operator translating between the two spaces. In the lecture, I will explain that there is another answer, i.e., a not so well-known third space (C), of arithmetic nature, that also gives an answer to the above question, and then the FFT appears simply as the composition of two operators: the one translating between spaces (A) and (C), and the one that translates (C) to (B).  

Remark: The lecture is prepared to be understood to anyone who is familiar with basic linear algebra. In particular, advanced undergraduate students, from computer science, engineering, mathematics, physics, etc, are more than welcome to attend.

The BGG category O plays an important role in the study of representations of semisimple Lie algebras. Its connection to the Hecke category is a starting point of Geometric Representation Theory. In this lecture, I will introduce a version of category O for quantum groups at roots of unity. I will explain a derived equivalence from (the principal block of) quantum category O to the affine Hecke category. Under the equivalence, the highest weight structure of quantum category O provides a categorification of the “periodic Hecke module”.

I will continue in the second lecture by giving basic properties of quantum category O. Then I will introduce the affine Hecke algebra and the periodic Hecke module, and explain a categorification of the former using coherent sheaves on Steinberg variety.