It turns out that complex analysis can say much about the interpolation and sampling problem. Especially in the last decade there has been much activity. In this talk, I will take up this point of view. I will begin by explaining the main ideas, and then discuss recent results in the subject.

ABSTRACT: Maximum likelihood estimation is a nonlinear optimization problem that arises in statistics. One way to find a global optimal solution is to solve the critical equations. The maximum likelihood (ML) degree is the number of complex solutions to these critical equations. First, we give formulas for the ML degree in the dense and sparse cases: we show that the ML degree is equal to the degree of the top Chern class of a sheaf of logarithmic differential one-forms. Furthermore, we give algorithms that compute the critical ideal whose roots are the solutions to the critical equation.

We will give examples that illustrate our symbolic-numeric implementation. These will include statistical models  in phylogenetics.

ABSTRACT: In the talk, we will discuss the notion of k-nets which is a discrete analogue of the notion of webs popular in topology. For us  k-nets are special finite configurations of lines and points in the complex projective plane that appear in several areas of mathematics.

Our most general result is the restriction on k - it can be only 3,4, or 5. An interesting class of nets is formed by 3-nets that can be considered as geometric realizations of latin squares and loops. All known examples of 3-nets in the complex projective plane realize finite Abelian groups. We study the problem what groups can be so realized. Our main result is that, except for groups with orders of all elements under 10, realizable groups are isomorphic to subgroups of a 2-torus. This follows from the `algebraization' result asserting that the lines of a net are dual to points lying on a plane cubic.

ABSTRACT: In this talk I will describe a general algebraic basis for arbitrary systems of units such as those used in physical sciences, engineering, and economics. The algebraic basis allows us to treat physical quantities as actual numbers. Nevertheless, physical, engineering, or economic quantities are not simply real or complex numbers, rather they are an ordered pair u ={u, labelu}, that is, u   q = R x WB, a set of "q-numbers", where R is a real (or complex) number and WB are infinite Abelian multiplicative groups of labels with a finite basis. The labels are what we call "units". Extensions to include the possibility of rational powers of labels have been included, as well as the addition of named labels.

Named labels are an essential feature of all practical systems of units. Furthermore, the set q itself is an Abelian multiplicative group, but it is not a ring. q admits decomposition into one-dimensional normed vector spaces over the field C among members with equivalent labels. These properties lead naturally to the concept of well-posed relations, and to Buckingham's theorem of dimensional analysis. A connection is made with a Group Ring structure and an interpretation in terms of the observable properties of physico-chemical systems is given. Finally, the consequences of this rich algebraic structure with respect to the teaching of mathematics will be addressed.

ABSTRACT: DNA topology is the study of geometrical (supercoiling) and topological (knotting) properties of DNA loops and circular DNA molecules. Virtually every reaction involving DNA is influenced by DNA topology, or has topological effects. Site-specific recombinases and topoisomerases are enzymes able to change the topology of circular DNA by breaking the DNA and introducing one or more crossing changes. Mathematical analysis of such changes may provide relevant information about the possible enzymatic pathways, and about DNA conformation at the moment of double-stranded break induction. In this talk I will discuss some of the problems that I am currently interested in, and the topological tools used in their analyses.

First I will talk about Xer recombination and how I applied, and extended, the tangle model for site-specific recombination to propose a unique enzymatic mechanism. I will then present the Java applet TangleSolve that makes the tangle model easily accessible to the interested molecular biologist. Finally, I will talk about the modeling of DNA unknotting by type II topoisomerases.

ABSTRACT: Redundancy is a key to practical and reliable data representation in many settings. The standard example is sampling theory for (bandlimited) audio signals, where oversampling is used to ensure numerical stability. Frame theory provides a mathematical framework for stably representing signals as linear combinations of "basic building blocks" that constitute an overcomplete collection. Finite frames are the piece of this theory that is tailored for finite, but potentially high dimensional, data. We address the problem of how to accurately and efficiently quantize redundant finite frame expansions. We analyze the performance of certain first and second order Sigma-Delta quantization schemes in this setting, and derive rigorous approximation error estimates as well as new stability theorems.

ABSTRACT: The concept of dependence has many manifestations in different branches of mathematics. Three examples are linear dependences in a vector space, algebraic dependences in a field extension, and cycles in a graph. A matroid is a combinatorial object which captures the essence of dependence, generalizing these three notions simultaneously.

Consider the paths which start at the origin, and take 2n steps, each of which is a unit step northeast or southeast. What might it mean for such a path to be dependent? I will present an answer to this question which fits naturally in the context of matroids, and gives us new insight into the ubiquitous sequence of Catalan numbers: 1, 1, 2, 5, 14, 42, ...

ABSTRACT: Much as real algebraic geometry is the study of the real points of complex varieties, real symplectic geometry examines the real loci of symplectic manifolds. While arbitrary submanifolds of a manifold can be fiendishly complicated, real loci display a surprising and beautiful structure. The relationship between the topology of the real locus and the topology of the ambient symplectic manifold can be best understood using Morse theory. In this lecture, I will first review the basics of Morse theory in the concrete setting of the height function on the torus. I will then explain several facts from symplectic geometry and examine their analogues in the context of real loci, giving examples along the way.

ABSTRACT: Finite Fourier Analysis studies the vector space of all complex-valued periodic functions on the integers, with period p. It turns out that every such function f(n) can be written as a polynomial in the pth root of unity e^{2 pi i n/b}. Such a representation for f(n) is called a finite Fourier series. Like their infinite counterparts, finite Fourier series are an elegant tool to study periodic functions; however, the underlying theory turns out to be somewhat simpler in the finite case (for example, one never has to worry about convergence). Finite Fourier Analysis has had an important impact, e.g., on Number Theory, Representation Theory, and Discrete Geometry. The general philosophy here is tht finite sums of rational functions of roots of unity are basic ingredients in many mathematical structures. We will outline a development of the finite Fourier theory by using rational functions and their partial-fraction decomposition. We then define the Fourier transform and the convolution of finite Fourier series, and show how one can use these ideas to prove identities on trigonometric functions, as well as find connections to the classical Dedekind sums.

ABSTRACT: Exposure of mammalian cells to ionizing radiation induces chromosome breaks that, upon misrepair, create exchange-type chromosome aberrations, that is a reshuffling and misrejoining of the chromosome fragments. Large (>1Mb) rearranged chromosome fragments between non-homologous human chromosomes can be identified by multiple fluorescence in-situ hybridization (mFISH). Quantitative analysis extracts valuable information from these data. I will here present two projects to illustrate this. The first project investigates the relative position of chromosomes in the early stages of the cell cycle (G0/G1). During G0/G1 chromosomes occupy subnuclear regions called chromosome territories. The relative positions of these territories are believed to be associated with a number of biological processes including the formation of chromosome aberrations found in some cancers. We developed a new statistical method that tests for significance of pre-selected chromosome clusters. We found an overall random organization of chromosomes together with two groups of chromosomes ([1,16,17,19,22] and [13,14,15,21,22]) whose members form aberrations more often than what randomness would predict. In a second project we investigated pathways of aberration formation. There exist two pathways of aberration formation during G0/G1: Breakage and Reunion (a nonhomologous end-joining like phenomenon) and Recombinational Misrepair (a mechanism based on sequence homology). Using Monte-Carlo computer simulations of misrepair processes as well as a graph theoretical approach we characterized both pathways of aberration formation. We concluded that Breakage and Reunion is the predominant mechanism of aberration formation during G0/G1. Later we introduced "Aberration Multigraphs", which provide a new nomenclature to describe radiation-induced chromosome aberrations as well as an excellent framework to investigate new mathematical properties of such abe.

ABSTRACT: This talk will describe a computational method for the inverse problem of edge-preserving image restoration. Total Variation (TV) regularization removes noise from an image while retaining its edges. We solve an
equivalent dual version of the TV problem. Images restored using this dual approach will have crisp edges (discontinuities), whereas images recovered under earlier primal methods may contain blurred edges. Joint work with Tony F. Chan, Pep Mulet, and Lieven Vandenberghe.