The Kadison-Singer Problem
Abstract: The Kadison-Singer problem is a problem in the theory of C*-algebras that was motivated by mathematical physics. It has been open for over 50 years. Recently, work of Casazza lead to a resurgence of interest in this problem, when he proved that the Kadison-Singer problem was equivalent to the Feichtinger problem, which arose from the theory of wavelets and signal processing. The Feichtinger problem is a more easily stated problem in the theory of frames. In this talk I will introduce this area, survey some of the recent progress and discuss what I believe to be some of the more approachable unresolved special cases and related problems.

Big Things Are Rarely Amenable
Abstract: We give survey of how the phenomenon of amenability manifests itself for locally compact groups and for Banach algebras, and we will try to convince the audience that amenability is best viewed as a weak finiteness condition.

Broken Symmetries
Abstract: There is a remarkable connection pioneered by Alain Connes between operator algebras, which originated as the mathematical models for quantum mechanical systems, and abstract structures arising from number theory. This connection is based on the fact that systems of numbers and of sub-atomic particles share some common features that make them tractable with the same mathematical tools.

One of these features is the prominent role that symmetries play in both cases; another is the relevance of pairs of operations that do not commute with each other. For quantum systems, the non-commuting operations are the measurements of position and momentum of particles; for number systems, they are addition and multiplication.

I will give a nontechnical overview of the subject, discussing its motivations and implications, and then briefly report some recent developments.

Forbidden Configurations: A Survey
Abstract: Problems in extremal set theory take the form of determining the maximum number of subsets of {1,2, ..., m } you can choose so that the resulting family of subsets has some property. The property I will consider is a trace being forbidden (in hypergraph terms a subhypergraph being forbidden). An incidence matrix encodes the system of subsets as an m-rowed (0,1)-matrix A with no repeated columns. The forbidden trace becomes a 'forbidden configuration' namely for some given (0,1)-matrix F you are forbidding A from having any submatrix which is a row and column permutation of F.

One defines forb(m,F) as the maximum number of columns, over all m-rowed (0,1)-matrices with no repeated column and no submatrix which is a row and column permutation of F. This concept of forbidden configurations appears in a variety of problems of which the study of VC-dimension has been the most notable. I will discuss a number of the bounds obtained and the interesting variety of proofs.

Eigenvalues of Graphs
Abstract: Graph theory is the study of networks. In many situations, the only way we can study key combinatorial parameters of graphs such as edge-distribution, connectivity or expansion, is by using their eigenvalues. In this talk, I will describe some connections between the structure of graphs and their eigenvalues. The talk should be accessible to undergraduate students.

Canonical Structures for Matrix Functions
Abstract: Many problems of mechanics, sound propagation, mathematical biology, etc., can be effectively modelled with matrix eigenvalue problems in which the eigenvalue parameter appears in a nonlinear fashion. Over the last fifty years or so, this has given rise to a comprehensive theory. In particular, canonical structures play an important role, and can be arrived at by either algebraic or analytical methods. We will give a survey of results of this kind in which either Hermitian or unitary symmetry plays an important role. (This talk can be understood with little more than ideas from undergraduate algebra and analysis.)

Entropy in Dynamics
Abstract: The concept of entropy was introduced into ergodic theory by Kolmogorov in the late 1950s. It can be viewed as a measure of the average information gained in learning that the orbit of an unidentified point visits a certain sequence of sets in a given partition of the space. This dynamical version of Shannon's information-theoretic entropy revolutionize the study of measure-preserving actions, which until then had relied on invariants of a spectral, as opposed to combinatorial, nature. Entropy theory as originally conceived by Kolmogorov was ultimately seen to apply most generally to actions of amenable groups, for which one can average over partial orbits in a way that produces a dynamical invariant.

Very recently Lewis Bowen showed, quite surprisingly, that the theory of measure entropy can be vastly extended to the realm of actions of countable sofic groups. Soficity is a much weaker kind of finite approximation property than amenability and is satisfied for example by all residually finite groups. The definition of entropy in this case required a completely new strategy that replaces the information-theoretic perspective with the statistical-mechanical idea of counting discrete models. Hanfeng Li and I have subsequently developed an alternative and more general approach to sofic entropy that uses operator algebras in an unexpectedly essential way. I will discuss all of these developments and furthermore indicate some applications of the ideas involved to the structure theory of operator algebras.

Properties of Gabor Multipliers for Physical Modelling
Abstract: We present techniques developed for numerical modeling of wave propagation, and source-signature removal in seismic imaging, based on a class of linear operators known as Gabor multipliers. These operators are localized Fourier multipliers, whose action is selectively localized by an element of a partition of unity. We discuss boundedness and stability properties for these operators, approximations to PDEs and pseudodifferential operators, and an approximate functional calculus.

The Language of Forms
A journey from the Mobius strip, trhough affine Kac-Moody and superconformal algebras, to children's drawings
Abstract: One of the most recurrent themes in both Physics and Mathematics, is the study and construction of objects that locally look the same. I will explain, mainly via examples and in non-technical terms, how the concept of "locally look the same" has evolved through time (mostly through some beautiful ideas of Serre and of Grothendieck).

A Graph of Matrices
Abstract: Free probability is a variation of probability theory for matrix valued random variables. It has many aspects: combinatorial, analytic, theoretical, and applied. I will discuss a problem on a graph of matrices arising from a random matrix problem in free probability.

Let $G = (E, V)$ be a graph and $T$ a map from $E$ to the $N \times N$ matrices. We write the matrix elements of $T(e)$ as $\{ t^{(e)}_{ij} \}$ and let

where $i$ runs over all functions from $V$ to $[N] = \{1, 2, 3, \dots, N\}$. For example if the the graph $G$ is

the corresponding sum is

The question we wish to address is the dependence of $S_G(T)$ on $N$, which as we shall show has a surprisingly simple answer.

Randomized Process of Small Unknowns and Implicitly Enforced Parameter Bounds: New Algorithmic Techniques for Parameterized Computation
Abstract: Parameterized algorithms have witnessed a tremendous growth in the last decade and have become increasingly important in dealing with NP-hard problems that arise from the world of practical computation. In this talk, after a brief review of the basic concepts in parameterized computation, we will be focused on two new algorithmic techniques that have turned out to be useful in the recent development of parameterized algorithms: randomized process of a small unknown subset of a given universal set, and implicitly enforced parameter bounds in a branch-and-search process. These techniques are simple, effective, and have led to significant improved algorithms for a number of well-known NP-hard problems.

Matrix Canonical Forms
Abstract: You are in your office; the door is open. A well-dressed visitor walks in confidently without knocking. He has a single sheet of wrinkled paper in his left hand and a partially open brown envelope in his right hand. Without any introduction, he strides to your desk and drops the paper and envelope on top of the ungraded midterm exams and unfinished referee reports. Glancing at the paper, you see a bold red "Eyes Only" stamp and a complex 7-by-7 matrix; visible in the envelope is a neat stack of crisp $100 bills.

"Tell me about that matrix," he says.

Where to begin? "It has rank five," or "It is nilpotent and has Jordan blocks of sizes three and four," or "Its Hermitian part is positive semidefinite," or "It is congruent to a diagonal matrix." Probably not a good way to begin ... better to ask questions such as "Where do the data come from?", "Are there any relevant symmetries or invariants?", and "Does this matrix interact with others in some way?"

The purpose of your questions is to discover whether there is a natural equivalence relation lurking behind your visitor's matrix; if so, it is likely that reducing it to a corresponding canonical form will be illuminating.

We will give examples of a variety of matrix equivalence relations that arise in practice and will discuss canonical forms corresponding to some of them. We will pay special attention to an alternative to the Jordan canonical form for similarity, and to a recently discovered canonical form for congruence.