SCU Mathematics Colloquium Series Schedule

Winter 2006

All talks are Tuesdays at 4:00 pm in O'Connor 106, unless otherwise announced.

January 22: **Larry
Guth**, Stanford University

Title: Ways to describe the size of a surface

Abstract: In this talk, we will discuss different ways to describe how big a surface is. For example, suppose we compare the unit disk to a long thin ellipse. The ellipse looks longer than the disk, but the disk looks thicker than the ellipse. Based on this example, we will give some precise definitions of the length and the thickness of a surface. These definitions apply to any 2-dimensional surface, not just to simple examples like ellipses.

First I will explain Gauss's idea of intrinsic distances on a surface in 3-dimensional space. Using this idea, we will define the diameter of a surface, which corresponds to the word length above.

Second I will give Uryson's definition of the width of a surface. There are several different -inequivalent- definitions that roughly correspond to the intuitive idea of width, and if there is time I will mention another one and compare the two.

After we build up several pieces of vocabulary that describe the size of a surface, we will talk about how they are related to each other. For example, if a surface has small area, can it have a large Uryson width?

January 29: No colloquium scheduled - Math/CS Career Night

February 5: **Scott
Armstrong**, UC Berkeley

Title: Elliptic PDEs and Homogenization

Abstract: Composite materials are created by mixing two or more ingredients on a very fine microscopic scale. If designed with enough cleverness, the result is a material with physical properties that are "better" in some specified way than the average behavior of its constituents. In this talk we will discuss the mathematical theory that describes this phenomena.

We will begin with an elementary exposition on elliptic partial differential equations, before investigating the behavior of solutions of elliptic equations that depend on two scales, usually a `macroscopic' scale of order 1 and a `microscopic' scale of order $\epsilon$. We will be interested in particular in how the solutions behave as $\epsilon$ tends to zero, and explore how interesting and unexpected macroscopic behavior may emerge as the microscopic behavior of the solutions is "homogenized."

February 12:
**Melissa Gilbert**,
Santa Clara University, Department of Education

Title: Applying contemporary views of mathematical proficiency to examine the role of motivation in mathematics achievement

Abstract: In this talk, I will report on a mixed-method study in which I examined how motivation affects pre-algebra students' abilities to demonstrate two important mathematical competencies: solving routine problems and explaining reasoning. The results suggest that different dimensions of motivation correlate with high performance, depending on the type of mathematical knowledge being assessed. For example, mastery-goal endorsement (i.e., students' level of focus on learning and understanding the material) correlated significantly with the ability to explain reasoning successfully, a competency that entails sustained effort and deeper conceptual understanding; it did not correlate with simply solving routine problems. In addition to sharing this research, I will review key motivation constructs and, if time allows, provide a pedagogical framework that supports student motivation.

February 19:
I**zabella Kanaana**,
Sonoma State University

Title: Kirkman's Schoolgirl Problem

Abstract: In 1850, the Reverend Thomas P. Kirkman proposed the following problem which is generally known as "Kirkman's schoolgirl problem": "Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast". In this talk, we analyze this problem (and briefly introduce the topic of combinatorial designs), and construct the solution.

February 26:
**Federico Ardila**,
San Francisco State University

Title: Arrangements of tropical hyperplanes

Abstract: Tropical geometry is an exciting new field which studies geometric objects by "tropicalizing" them. The resulting tropical objects are simpler, and have a very rich discrete structure. I will describe what arrangements of tropical hyperplanes look like, and explain their close relationship with certain tiling problems. In particular, I will explain why an arrangement of tropical lines in the plane is essentially equivalent to a rhombus tiling of an equilateral triangle.

The talk will be mostly self-contained, and will present joint work with Mike Develin.

March 4: **David
Nash**, University of Oregon

Title: The Hook Length Formula

Abstract: The Hook Length Formula is a surprisingly elegant answer to a basic combinatorial question. Namely, how many standard tableaux of shape lambda are there for any given shape lambda? In this talk, we'll discuss one beautiful way to prove the formula using probability as well as some of the important uses for it. Then we'll look at some of Schensted's work on increasing and decreasing subsequences. It was his work in this area that lead to a more complicated bijective proof of the Hook Length Formula. Finally, if there is time, we'll look at a nice application of these two results.

March 11: **Hans
Schepker**, Glass Geometry

Title: Art Meets Geometry

Abstract: Mathematicians and scientist will call mathematics "beautiful, elegant, exciting, stimulating" and a lot more. My work is an effort to make these qualities accessible to the broader public through art. Geometry is the most obvious way to make mathematics visible and comprehensible: everyone can picture a cube, a spiral, and so on. Adding color, transparency, and light to these forms makes them more comprehensible, obvious even, starting points that just about anybody can explore. Once people are drawn in, the step to expressing mathematical concepts through art is a short one; I use this trick to teach students at all levels, using stained glass and paper. Eventually the forms become vehicles to express problems or answers to problems in combinatorics, topology, graphing, and more.

Note: Schepker is a state-juried member of the League of New Hampshire Craftsmen and a roster artist for the New Hampshire State Council on the Arts

There will be refreshments before the talks in O'Connor 31 starting at 3:45pm.

If you have a disability and require a reasonable accommodation, please call or email Frank Farris 408-554-4430 or ffarris@scu.edu .