SCU Mathematics Colloquium Series Schedule

Fall 1999

Talks will be at 4:00 Tuesdays in O'Connor 105. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**Sept. 28**Jean Pedersen, Santa Clara UniversityTitle: Paradoxes

The distinction between a genuine contradiction in mathematics and a paradox exhibits the dangers of traditional ways of thinking and is crucial to the understanding of mathematics. We give a number of examples of paradoxes at the frontier between mathematics and the real world.

**Oct. 5**John Etnyre, Stanford UniversityTitle: Knots and Contact Geometry

There have recently been great advances in our understanding of contact geometry in low dimensions and its importance in topology and dynamics. After reviewing some of the basic ideas in contact geometry, I will describe some surprising relations with knot theory and fluid dynamics. I will end the talk with some details concerning the interaction of contact geometry and knot theory.

**Oct. 12**Lesley Ward, Harvey Mudd CollegeTitle:

*A*from dyadic_{p}*A*?_{p}In a celebrated paper, Garnett and Jones showed that the translation average of functions satisfying the bounded mean oscillation (BMO) property on dyadic intervals must actually satisfy the property on all intervals: 'BMO from dyadic BMO.' We ask and answer the analogous question in other settings, notably Muckenhoupt's classes of

*A*weights, and the space of doubling measures._{p}

**Oct. 14 (Note this is a Thursday)**Cindi Thompson, CSLI, Stanford UniversityTitle: Machine Learning: A Research Area within Artificial Intelligence

In this talk, I will present a broad introduction to the goals of teh field of machine learning. In general, practitioners of the field study theories and methods of computational learning. We design, build, and analyze programs that use experience to improve performance at some task.

I will discuss some common methods, approaches, and directions taken by the field. I will then collaborate with the audience in designing solutions for two machine learning problems. First, we will design a program that learns to play checkers; then, a program that learns a dictionary to be used by a natural language parser.

**Oct. 19**Steve Chiappari, Santa Clara UniversityTitle: Proper Holomorphic Mappings from Balls in Complex Euclidean Spaces

Examples and theorems from classical complex analysis serve as motivation for the study of proper holomorphic mappings in complex Euclidean spaces of arbitrary dimension. We present some results from the general theory of such mappings from balls to balls. Then we examine invariance properties of such mappings under certain groups of matrices and describe the basic mappings from which all such rational invariant mappings can be built. This talk combines bits of analysis, algebra, topology, and combinatorics.

**Oct. 26**Frank Farris, Santa Clara UniversityTitle: Seeing a Surface in 4-space

When we look at a surface in 3-space from a point, we see a fold set, where the ray from our eye is tangent to the surface. This fold set is typically a curve on the surface. When the surface lies in 4-dimensional euclidean space, there is generically only a finite set of points where this happens. Similarly, only a finite number of rays from your eye hit the surface straight on. We explain how these finite point sets carry certain characteristic classes of the surface, counting the normal and tangential Euler numbers. Our methods allow us to demonstrate a surprising relationship between these numbers and the number of complex points on the surface. We also present results for surface pairs.

**Nov. 4 (Note this is a Thursday)**Colm Mulcahy, Spelman CollegeTitle: Wavelets -- a new tool for imaging, graphics and Hollywood

Wavelets are a relatively recent arrival on the scene, and have already found numerous applications both within mathematics and in digital imaging, computer graphics, animation, video and audio. The new generation of digital feature films emerging from Hollywood are also poised to use these techniques. Wavelets provide an alternative to classical Fourier methods for one- and multi-dimensional data analysis and synthesis, and are better suited to data which lack periodicity or contain sudden changes. This talk will introduce the basics of wavelets via a popular application to compression of images, data and audio. Prerequisites will be kept to a minimum: anybody who can add, subtract and divide by two should feel quite at home.

**Nov. 9**Robert Bekes, Santa Clara UniversityTitle: Some Classical Results in Representation Theory

**Nov. 16**Tadeusz Januszkiewicz, Wroclaw University, PolandTitle: Elementary Differential Geometry of Surfaces in

*R*and Geometry of the Grassmannian G(2,4).^{4}

If you have a disability and require a reasonable accommodation, please call Rick Scott 554-4460 or 1-800-735-2929 (TTY - California Relay).

The list of talks from previous quarters are available via this archive link.

*Last Updated: 3 January 2000*
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