SCU Mathematics Colloquium Series Schedule
Title: Taming Chaos, or How to Construct a Cross-section to an Anosov Flow
Abstract:Hyperbolic systems are prime examples of chaotic dynamical systems. The Smale horseshoe, the "cat map", and the geodesic flow of a negatively curved surface are just a few examples of such systems. In this talk, I will give an elementary introduction to so called Anosov systems, which are globally hyperbolic in the sense that they exponentially contract and exponentially expand at every point. I will then talk about the history, background, and (time-permitting) a recent proof of a thirty-year old conjecture of Verjovsky concerning the classification of codimension one Anosov flows in dimensions greater then three.
Title: Using Mathematics to Improve Medical Treatments
Abstract: How does HIV interact with the human immune system? Mathematics has been used to shine light on this mystery. In particular, the predator-prey models taught in some calculus courses have been very successful in capturing the struggle between the immune system and diseases. I'll talk about a few simple mathematical results for models of HIV and leukemia, resulting treatment recommendations, and some of the remaining urgent questions for these diseases.
Title: PageRank, Markov chains, and Search on the Web
Abstract: One feature that Google uses in determining search results is the PageRank algorithm. This has a nice mathematical formulation in terms of random walks on graphs, and our understanding of these random walks can provide insight into how the web is structured, and how to improve the search ranking algorithm.
Title: Right-Angled Mock-Reflection Groups
Abstract: A useful testing ground for classification efforts in the theory of infinite groups is the class of right-angled Coxeter groups. Examples of these groups include groups generated by orthogonal reflections across the faces of a convex polyhedron (spherical, Euclidean, or hyperbolic) whose dihedral angles are all 90 degrees. We will show that any such group acts naturally on a certain cubical complex, and generalize this type of action to obtain the notion of a mock reflection group. We will provide interesting examples and give a combinatorial description in the same spirit as the well-known presentation for Coxeter groups.
Title: Mathematics Unravels the Negatives of Arrow's and Sen's Theorems
Abstract: Probably the best known academic theorems, Arrow's and Sen's Theorems, are unknown to most mathematicians. I'll introduce these results, which have had a significant impact on several disciplines, saying that what we want to do with voting, decisions, society, and even statistics, cannot be done. The theorems are correct, they have been proved with filters and ultrafilters, combinatorics, geometry, and even algebraic topology. But do they mean what we think they mean? In this expository lecture, I will show how basic mathematics can lead to radically different interpretations.
Title:From Kepler's Monsters to Thurston's Doughnut
Abstract: Filling out a plane with shapes is called a tesselation or tiling. Brick walls, tile floors, Escher prints and bathroom drains are examples of tilings. This talk gives a brief history of tilings and a survey of unsolved problems in this field of study.
Title: Systematic Paper Folding and a Related Number-Theoretic Conjecture
Abstract: I will present joint work with Peter Hilton and myself involving a conjecture that grew out of paper-folding. The conjecture is easily understood and we are convinced that it is true, but so far we have no proof. (Participants are kindly asked not to give their proof until I have finished showing examples after the statement of the conjecture!)
Unless noted otherwise, talks will be at 4:00pm Tuesdays. Room O'Connor 206. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.