Colloquium 20042005 
San
Francisco State University Department of Mathematics 
9/22/04
Winfried Bruns
University of Osnabr"uck, Germany
TITLE: Monomial orders, initial ideals and initial algebras. 
ABSTRACT:
The arithmetic of polynomials in one variable relies crucially on division
with remainder. In order to extend it to polynomials in several variables
one has to introduce a monomial order. Therefore monomial orders are the backbone
structure for all algorithms involving polynomials in several variables. They
have very efficiently been implemented in computer algebra systems like Macaulay2
or Singular. However, they are also important for the investigation of structural properties of algebras, through initial ideals and/or initial algebras. The passage to the initial object is wellbehaved, and the latter is essentially a combinatorial object. Therefore monomial orders open an avenue to the application of combinatorial methods in algebraic geometry and commutative algebra. We will discuss such an approach to some classical algebras. 
4:00 PM in TH 211 
refreshments served
in TH 935 at 3:30 PM 
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10/11/04
Stan Wagon
Macalester College, St. Paul, Minnesota
TITLE: From Postage Stamps to Chicken McNuggets: A Fast Solution to an Old Integer Programming Problem  
ABSTRACT:
The Frobenius number is easily described in terms of Chicken McNuggets from
McDonald's. One can purchase packs of 6, 9, or 20 McNuggets. So one can
buy 35 McNuggets, but one cannot buy, say 13, or 43 of them. However,
every number beyond 43 is representable. So 43 is called the Frobenius
number of 6, 9, and 20. In this talk I will show how the Frobenius problem can be reinterpreted as a shortest path problem in a certain symmetric graph. The symmetry can be used to develop algorithms that are very efficient at finding the Frobenius number and solving the equation with a particular target. (Joint work with Dale Beihoffer and Albert Nijenhuis.) 
4:00 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
10/20/04
Dror Varolin
University of Illinois at UrbanaChampaign
TITLE: Interpolation and Sampling in Complex Analysis 
ABSTRACT: The problem of sampling a signal and later reconstructing
it is among the most natural problems in science. The reconstruction,
or interpolation, problem is ubiquitous. The sampling problem was not
seriously considered until Nyquist began studying it, and Shannon used
Nyquists results in his theory of information.
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. 
4:00 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
11/3/04
Serkan Hosten
San Francisco State University
TITLE: The maximum likelihood degree and the likelihood equations 
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 oneforms. Furthermore, we give algorithms
that compute the critical ideal whose roots are the solutions to the
critical equation. 
4:00 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
11/10/04
Sergey Yuzvinsky
University of Oregon and MSRI
TITLE: Nets in the complex projective plane 
ABSTRACT:
In the talk, we will discuss the notion of knets which is a discrete analogue
of the notion of webs popular in topology. For us knets are special finite
configurations of lines and points in the complex projective plane that appear
in several areas of mathematics. 
4:00 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
12/6/04
Sergio Aragon
San Francisco State University
TITLE: Algebraic structure of physical quantities 
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 "qnumbers", 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.

4:00 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
February 4, 2005
Mariel Vazquez
University of California , Berkeley
TITLE: Topological analysis of enzymatic actions: DNA link formation by Xer recombination and DNA unknotting by type II topoisomerase. 
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. Sitespecific 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 doublestranded 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 sitespecific 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. 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
February 7, 2005
Alexander Powell
Princeton University
TITLE: Finite Frames and SigmaDelta Quantization 
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 SigmaDelta quantization schemes in this setting, and derive rigorous approximation error estimates as well as new stability theorems. 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
February11, 2005
Federico Ardila
Microsoft Corporation
TITLE: "The Catalan Matroid" 
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, ... 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
February 14, 2005
Tara Holm
University of California , Berkeley
TITLE: Morse theory in real symplectic geometry 
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. 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
March 9 , 2005
Matthias Beck
San Francisco State University
TITLE: An invitation to Finite Fourier Analysis 
ABSTRACT: Finite Fourier Analysis studies the vector space of all complexvalued 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 partialfraction 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. 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
April 13, 2005
Javier Arsuaga
University of California San Francisco
TITLE: Quantitative Analysis of radiation induced chromosome aberrations 
ABSTRACT: Exposure of mammalian cells to ionizing radiation induces chromosome breaks that, upon misrepair, create exchangetype chromosome aberrations, that is a reshuffling and misrejoining of the chromosome fragments. Large (>1Mb) rearranged chromosome fragments between nonhomologous human chromosomes can be identified by multiple fluorescence insitu 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 preselected 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 endjoining like phenomenon) and Recombinational Misrepair (a mechanism based on sequence homology). Using MonteCarlo 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 radiationinduced chromosome aberrations as well as an excellent framework to investigate new mathematical properties of such abe. 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
May 4, 2005
Jamylle Carter
Mathematical Sciences Research Institute ( MSRI )
TITLE: A Dual Method for Total VariationBased Image Restoration 
ABSTRACT:
This talk will describe a computational method for the inverse problem of
edgepreserving image restoration. Total Variation (TV) regularization
removes noise from an image while retaining its edges. We solve an 
4:10 PM in TH 211 
refreshments
served in TH 935 at 3:30 PM 
_______________________________________________________________________________________ 
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