SCU Mathematics Colloquium Series Schedule

Fall 1997

Talks will be at 4:00 Tuesdays in O'Connor 107. There will be refreshments before all talks in O'Connor 31 starting around 3:30pm.

**Oct. 7**Neville Robbins, San Francisco State UniversityTitle: Partitions

Let n be a natural number. A partition of n is a representation of n as a sum of one or more natural numbers. For example, the partitions of 6 are the following: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1. In general, we denote the number of partitions of n by p(n), which is called the unrestricted partition function. The above enumeration shows that p(6)=11. I will discuss: (1) how to estimate p(n) for large n; (2) how to compute p(n); (3) some arithmetic properties of p(n); (4) (if time permits) some other partition functions, namely q(n), the number of partitions of n into distinct parts, and q_{0}(n), the number of partitions of n into distinct odd parts.

**Oct. 14**Howard Swann, San Jose State UniversityTitle: The cell discretization algorithm for solving partial differential equations

We see how the Galerkin method can transform the task of approximating a solution of a differential equation to a linear algebra problem. We then describe the Cell Discretization algorithm, which gives new theoretical power to the Galerkin Method by using non-conforming finite element methods and a `virtual mortar' argument.

**Oct. 21**Ron Kimmel, Lawrence Berkeley National LaboratoryTitle: Images as embedding maps and minimal surfaces: A unified approach for image diffusion

The speaker will introduce a new geometrical framework for image processing. This framework finds a seamless link between the TV-L1 and the L2 norms that are often used in image processing, based on the geometry of the image and its interpretation as a surface. It unifies most of the current `scale space' models for images by a simple selection of one parameter, yet more important, it enables to introduce new methods to deal with images in a simple and natural way.

A functional called "Polyakov action", borrowed from high energy physics, is shown to be useful for image enhancement in color, texture, volumetric medical data, movies, and more.

The idea is to consider images as surfaces rather than functions. Then, minimize the area of the surface in a special way. E.g. a gray level image is consider to be a 2D surface given by the graph I(x,y) in the 3D space (x,y,I). Similarly, a color image is a 2D surface that is given by the three graphs: R(x,y), G(x,y), and B(x,y), in the 5D space (x,y,R,G,B).

The results that will be presented in this talk are based on joint collaboration with Nir Sochen (Tel Aviv Univ. and Technion Israel) and Ravi Malladi (LBNL, UC).

**Oct. 28**Jim Wright, M.S.R.I.Title: Differentiation of integrals with respect to families of rectangles

An old problem in the theory of differentiation of integrals is to consider for every point p=(x,y) in the plane, a given family of rectangles F(p) which contain the point p. We then ask for which functions f(x,y) of two variables do the averages with respect to these rectangles converge to f. We'll discuss some recent results in this area.

**Nov. 4**Geoff Mason, U.C. Santa CruzTitle: A brief history of the Monster

The Monster is one of the most interesting of all groups. Discovered around 1973, it exhibits a number of remarkable connections with mathematics of the 19th century (abelian functions), the 20th century (classification of finite simple groups), and the 21st century (conformal field theory). In this talk we will survey some of these connections. No specialized knowledge is required.

**Nov. 11**Lily Khadjavi, U.C. BerkeleyTitle: Heights and effective bounds for a theorem of Belyi's

A surprising theorem of Belyi's, giving covering maps from curves to the projective line with certain properties, has been used in a wide variety of deeper settings, including progress on the Inverse Galois Problem. More recently, Noam Elkies used this theorem to connect an effective ABC conjecture to an effective Mordell's Theorem--leading one mathematician to comment, "Mordell is as easy as ABC".

We will introduce these theorems and then look at the details of Belyi's algorithm. Belyi's construction will provide us with a natural motivation to define heights of algebraic numbers as a general tool for computing bounds, which we can then use to prove bounds on the degree and height of the coefficients of such covering maps. If time permits, we will discuss some alternate algorithms and other interesting applications of Belyi's Theorem.

**Nov. 18**David Gauld, University of AucklandTitle: Non-metrisable manifolds

Manifolds are topological spaces which are locally like euclidean space. Most of the study of manifolds has been concentrated on compact, or at least metrisable, manifolds, where there have been huge advances. Less effort has gone into the study of non-metrisable manifolds, the simplest example being the long line. The main topic addressed in this talk is the question when is a manifold metrisable. Even this leads to some interesting complications. An old question of Alexandroff will be addressed, the answer being yes for some models of set theory and no for others. Quite recent work relating this topic to a generalisation of fibre bundles will also be addressed.

If you have a disability and require a reasonable accommodation, please call Ed Schaefer 554-6899 or 1-800-735-2929 (TTY - California Relay).

*Last Updated: 10 November 1997*
Maintainer: dsmolarski@scu.edu