# Undergraduate Mathematics SymposiumSaturday, October 1, 2011University of Illinois at Chicago

Organized by David Dumas and Steven Hurder

### Plenary lectures

• Yvonne Lai (Michigan)
Decomposing polyhedra   (email to request slides)
Abstract: Suppose you are given two polygons of equal area. Can you slice up one polygon with scissors, rearrange the pieces, and obtain the other polygon? What if we ask the analogous problem of polyhedra? How we answer this question depends on how we define slicing and rearranging. Starting from natural but different definitions leads to radically disparate answers. We will examine these questions from several perspectives, winding our way from Euclid's Elements to foundational mathematics of the 20th century.
• Christopher Leininger (UIUC)
Geometry and dynamics of surface homeomorphisms (PDF slides)
Abstract: Surfaces are fundamental objects in mathematics arising in a number of different fields. Although compact surfaces were classified 150 years ago, they continue to be central to many areas of mathematics. In this talk, I'll give some indication of Thurston's classification theorem for surface homeomorphisms, primarily through examples. Then I'll focus on the "generic" type, the pseudo-Anosov homeomorphisms. I will explain some of the connections between geometry, topology and dynamics for this class of homeomorphisms.
• Dhruv Mubayi (UIC)
From Ramsey theory to arithmetic progressions and hypergraphs (PDF slides)
Abstract: I will begin by giving a brief introduction to Ramsey theory, in layman's terms, and describing one of the most well-known open problems in this area. This will naturally lead to a similar question about arithmetic progressions in the integers. Finally, we will see a glimpse of how these problems about arithmetic progressions are solved by modern combinatorial methods using hypergraphs.

### Schedule of events

⇒ Full schedule with abstracts of plenary and student lectures (PDF)

 8:15 - 8:55am Sign-in and coffee in SEO 300 Morning Session - Plenary lectures, 50 minutes each, in SEO 636 8:55am Opening remarks 9:00am Yvonne Lai (U Michigan) — Decomposing polyhedra 10:00am Dhruv Mubayi (UIC) — From Ramsey theory to arithmetic progressions and hypergraphs 11:00am Christopher Leininger (UIUC) — Geometry and dynamics of surface homeomorphisms 12:00pm Lunch in SEO 300 Afternoon Session 1 — Student lectures, 20 minutes each, in SEO 636 1:30pm Sean McAfee (UIC) — On improving Cayley's theorem for groups of order p4 2:00pm Jonathan Gleason (U Chicago) — The F*-algebraic Formulation of Quantum Mechanics 2:30pm Christopher Perez (CalTech) — Siegel modular varieties: arithmetic invariants and cusps 3:00pm Coffee break in SEO 300 Afternoon Session 2 — Student lectures, 20 minutes each, in SEO 636 3:30pm In-Jee Jeong (Brown) — Bipartite Graphs, Quivers, and Cluster Variables 4:00pm Anita Thomas (IIT) — Sampling Within k-Means Algorithm to Cluster Large Datasets 4:30pm Jahan Claes (U Chicago) — Spectral Rigidity on Flat Tori 5:00pm Joshua Jay Herman (UIC) — Introduction to topological quantum computation

### For further information

Contact David Dumas (ddumas@math.uic.edu).