Department of Mathematics
                       Colloquium
                        2003-2004

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9/24/03

Francisco Santos

MSRI and University of Cantabria

TITLE: The space of triangulations of a finite point set in R^d.
ABSTRACT: I will introduce the concept of triangulation of a point set or a polytope, and that of local move ("geometric bistellar flip") between two triangulations. My goal will be to describe an example of a point set (in dimension 5, but yet "visualizable") whose graph of flips is not connected. Time permitting, some algebraic-geometric consequences of this

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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10/8/03

Neville Robbins

San Francisco State University

 

TITLE: The theory of partitions
ABSTRACT: If n is a natural number, then a partition of n is a representation of n as the sum of one or more natural number summands. For example, 7 = 3+2+2 is a partition of 7. Let p(n) denote the number of partitions of n. My talk will concern exact formulas and estimates for p(n), and will also discuss some other partition functions

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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10/22/03

James Smith

San Francisco State University

TITLE: Finding mathematical information
ABSTRACT: Some students and colleagues have asked for help in finding information about specific areas of mathematics in which I'm not expert. Of course, I've talked with many over the years about general sources on technical, social, and organizational aspects of our discipline. And recently, I've needed to look for details in the history of my own subject, foundations of geometry. This informal colloquium will consider old and new methods of digging. Have you noticed that techniques are changing fast? It's based partially on an instructive website I wrote a year ago, and--I hope--on questions, tips, and comments from students and colleagues in the audience.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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11/5/03

Sergei Ovchinnikov

San Francisco State University

TITLE: Media Theory
ABSTRACT: In many empirical situations, subjects are repeatedly asked to provide judgments concerning commodities or individuals. It is also typical that these judgments take the form of relations such as rank orders, quasi orders, choice functions, or other relations. Quantitative modeling of the temporal evolution of such judgments requires a thorough understanding of the combinatorics structure of the particular class of relations considered. Indeed, it is reasonable to assume that the judgment at time t will much resemble that at time t+d if d is small, and will tend to wander away from it if d grows larger. Accordingly, if we want to understand the details of this evolution, we must research the structure of relevant families of relations from the standpoint of the resemblance between particular relations. The media theory is concerned with the evolution of subjective relations under the influence of a stream of elementary events. In this lecture, we overview the main concepts of the media theory and discuss some existing and potential applications.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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11/24/03

Peter Casazza

University of Missouri

TITLE: Applications of Hilbert space frames
ABSTRACT: Hilbert space frames have traditionally been used in signal/image processing. Recently, there have arisen a variety of new applications to wireless communication, internet coding, speech recognition technology, physics, Engineering, Financial Mathematics, Communciation Theory, and much more. We will look at some of the new applications of frame theory and how frame theory has also begun to impact some of the most famous unsolved problems in ``pure'' mathematics.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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12/03/03

Matthias Beck

San Francisco State University

TITLE: A tale of seven polynomials
ABSTRACT: We follow a common thread of chromatic and flow polynomials of graphs and signed graphs, a polynomial counting function for magic squares, Ehrhart polynmials enumerating integer points in polytopes, characteristic polynomials of hyperplane arrangements. We will introduce the theory of "inside-out polytopes", an attempt to unify all of the above polynomials. This is joint work with Thomas Zaslavsky (Binghamton University, SUNY).

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

 

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1/30/04

Fred J. Hickernell

Hong Kong Baptist University

TITLE: What Do You Need to Solve Problems with Many Variables?
ABSTRACT: In many practical problems the number of independent variables is large compared to the number of data. For example, to determine the fair price of an exotic option one must compute the average or integral of the payoff function, which may depend on hundreds of variables. In laboratory or computer experiments one may wish to estimate or optimize the response as a function of a number of input variables or parameters. It is normally prohibitively expensive to perform the experiment for all possible combinations of the different levels of the input variables. The numerical solution of problems with many variables requires: i) a good design, ii) a good algorithm, and iii) an easy problem. This talk describes recent interdisciplinary research on numerical methods for solving problems with many variables. This research involves linear and abstract algebra, numerical analysis, probability and statistics, computational complexity, software design, parallel computing, and various application areas.

4:10 PM in TH 211
refreshments served in TH 935 at 3:30 PM

 

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2/6/04

Tatyana Foth

University of Michigan

TITLE:
ABSTRACT: A moduli space is a set (a manifold or a variety)whose points parametrize certain structuresor isomorphism classes of structures.Among well-known and widely studied examples there are moduli spaces of complex structures, the moduli space M g of compact Riemann surfaces of genus g>1, moduli spaces of polarized Kahler manifolds,and moduli spaces of Kahler metrics.Many problems related to moduli spaces have a physical motivation or interpretation. I shall talk in details about M g and its universal cover T g . I shall state a recent result (joint with A. Uribe) aboutthe curvature of a connection in a certain vector bundle over T g .

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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2/9/04

Yitwah Cheung

Northwestern University

TITLE: Ergodic Theory of Billiards in Polygons
ABSTRACT: Consider a billiard ball as it moves on a frictionless (not necessarily rectangular) billiard table. Does it eventually reach every corner of the table? Will its trajectory be uniformly distributed? In this talk I will describe some answers to these questions and explain how they relate to other areas of mathematics, such as interval exchange transformations.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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2/12/04

Jaimyoung (Jaimie) Kwon

Department of Statistics and Institute of Transporations Studies

University of California

TITLE: Application of Statistics to Transportation Data: Challenges and Some Success Stories
ABSTRACT: Transportation data poses unique challenges to scientists and engineers, due to the complexity of the traffic phenomena and the sheer size of the data. I will first give a brief overview of three data sources - loop detectors, video and tag readers - and practical application of the spatio-temporal data obtained from these sensors. Statistics plays a key role at various stages in the process, including sensor malfunction detection, imputation of bad samples and travel time prediction. Among others, I will present two success stories: (1) Web-Of-Evidence (WOE) model class that is developed to detect sensor malfunctions in correlated sensor networks. Gibbs sampler is used to compute the posterior probability of sensor malfunction given confounded scores, and (2) Method of Moment (MM) algorithm for estimating time-varying Origin-Destination (OD) traffic matrices from partially observed vehicle trajectories. These two methods also carry over to other areas, general sensor networks and network tomography, respectively.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

 

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2/13/04

Tao Li

Oklahoma State

TITLE: Topology of 3-manifolds
ABSTRACT: Three-manifolds are objects modeled on the 3-dimensional space that we are living in. However, many questions, which have been solved for higher dimensional manifolds, remain open in the 3-dimensional case. In recent years, there has been tremendous progress in the research on the topology of 3-manifolds using geometric methods. I will give an overview of these methods, and discuss some recent results and open questions.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

 

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2/26/04

Alice Silverberg

Ohio State

TITLE: Some Applications of Number Theory and Algebraic Geometry to Cryptography
ABSTRACT: Public key cryptography is about 25 years old, and relies on number theory. We will discuss Diffie-Hellman key exchange and ElGamal encryption, and some recent improvements on them. We show how number theory and algebraic geometry, and in particular the rationality of certain algebraic tori, can be used to give a deeper understanding of these improvements, and to give new cryptosystems.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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4/12/04

Michel Deza

Ecole Normale Superieure, Paris, France

TITLE: FULLERENES AND GENERALIZATIONS: INTERPLAY BETWEEN GEOMETRY AND CHEMISTRY
ABSTRACT: Since the discovery of molecule of  C60 (truncated icosahedron), the fullerenes, i.e. simple polyhedra with only pentagonal and hexagonal faces, became the main object in Organic Chemistry; the synthesis of C60  was marked by the Nobel prize 1996. We present fullerenes from their origins (isoperimetric problem in M.Golgberg's paper 1933) till their modern use i n Chemistry, Virology, Architecture etc. We review some generalizations and relatives of them, including:
1) analogues on surfaces;
2) icosahedral polyhedra with only pentagonal     and m-gonal faces;
3) plane partitions with only pentagonal and hexagonal     faces;
4) tetrahedral close packings of 3-space by four    Frank-Kasper polyhedra (t.c.p. phases of metallic alloys).
It will be an expositary lecture on new connections of Discrete Geometry and Chemistry.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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4/21/04

Frank Sottile

MSRI and Clay Mathematical Institute

TITLE: Certificates of Algebraic Positivity.

ABSTRACT: Positivity is a distinguishing property of the field of real numbers.  Writing
a  polynomial as  a sum  of squares  gives a  certificate that  it  is positive. Hilbert showed that a positive homogeneous quartic polynomial in three variables (ternary quartic) is  a sum of three squares of  quadratic polynomials.  He also showed that  there are  positive polynomials of  every higher degree  or greater number of  variables with no such  sum of squares representations.   This led to his  17th problem---to determine  whether a  positive  polynomial is  a sum  of squares of rational functions.  This was answered in the affirmative by Artin in 1926.
Recently, positive  polynomials have  have undergone a revival.  In the 1990's Lasserre realized  that recent theoretical results from  real algebraic geometry and semi-definite programming could be combined to give effective algorithms for solving a  class of relaxations  of hard optimization problems.   The relaxation replaces positivity by sum of squares representation.
I will  briefly survey  the history of  positive polynomials and  these modern applications, and  then discuss a  recent strengthening of Hilbert's  Theorem on ternary quartics:  a positive ternary quartic is  a sum of squares  in exactly 8 inequivalent ways.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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5/5/04

Ingileif B. Hallgrimsdottir

UC Berkeley, Department of Statistics

 

TITLE: Resultants in Genetic Linkage Analysis

ABSTRACT: The field of Genetic Linkage Analysis, or gene mapping, is concerned with finding the chromosomal location of disease genes. Over 1200 disease genes have been successfully mapped, most of them for Mendelian (one gene) disorders.  However, most common diseases are caused not by one but by many interacting genes and it has proved to be very hard to map the genes for these complex diseases.  Much of the difficulty is due to the complex underlying biology, but it is also important to develop the statistical models used.

The statistical models used in genetic linkage analysis of k-locus (gene) diseases are k-dimensional subvarieties of a (3^k-1)-dimensional probability simplex.  We have determined the algebraic invariants of these models with general characteristics for k=1 and k=2, using the Bezout resultant.

Joint work with Bernd Sturmfels.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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5/12/04

Monica Moreno Rocha

Tufts University

 

TITLE: Rational Maps and Sierpinski Gaskets

ABSTRACT: Sometimes the terms ``fractal" and ``Julia set"  are used in a undistinguished  manner to describe certain sets that arise from iterative methods in dynamical systems or topology. This is not always correct.

Consider the Sierpinski gasket, which is a classical example of a fractal set (broadly speaking, is a self-similar set with a fractal dimension). Many are the methods to produce such set, some of them are topological in nature (as the iterative process of removing open middle equilateral triangles) and some are more dynamical (e.g. the ``chaos game").

On the other hand, the Julia set of a map can be defined to be the closure of the set of points that under iteration exhibit an unpredictable behavior. For example, the Julia set of the map z\to z^2 is no other than the unit circle, since all other points either map toward infinity or toward the zero under iteration of the quadratic map. Clearly, the circle is a set of dimension one.

Then, not all Julia sets are fractals and not all fractals arise as Julia sets of some map. Well, not quite.

In this talk we present two families of rational maps of degree three and four acting on the Riemann sphere. We will show that the Julia sets for many elements of these families are generalized Sierpinski gaskets and that one, and only one element in the degree three family has a Julia set homeomorphic to the Sierpinski gasket.

This talk is based on the paper ``Rational Maps with Generalized Sierpinski Gasket Julia Sets'', written by R. L. Devaney, M.
Moreno Rocha and S. Siegmund, 2004.

4:00 PM in TH 211
refreshments served in TH 935 at 3:30 PM

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