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.