We will work through a specific example of the homogenization of a nonlinear PDE to give a gentle yet realistic introduction to what this stuff is all about.

The talk will survey the basic definitions and results of the emerging theory of cluster algebras, viewed from a combinatorial perspective. Joint work with Andrei Zelevinsky.

Contour trees and Morse complexes as embedded graphs with non-degenerate critical points (zero gradient of functions), together with linking manifolds, capture the topology and the invariant geometry of smooth functions. Viruses (macromolecules of proteins and nucleic acids), are one of the smallest parasitic nano-organisms that are agents of human disease. They have no systems for translating RNA, ATP generation, or protein, nucleic acid synthesis, and therefore need the subsystems of a host cell to sustain and replicate. I shall show how these mathematical graphs provide a useful mechanism for elucidating the structure of proteins such as hemoglobin and the ultra-structure of viruses.

The p-adic numbers can be defined analytically as a completion of the rational numbers with the p-adic norm. They can also be constructed using the language of categories and inverse limits. We will discuss both approaches and use this as motivation to talk about categories.

We'll discuss what an open book is and construct some interesting examples on T^3.

We study the optimal consumption / portfolio choice problem for an isoelastic utility agent reluctant to let her wealth fall below a fraction of its historical maximum (drawdown constraint). When the wealth is close to a new height, there is a risk of making the constraint more stringent by raising the minimum floor. Consequently, the investor may have incentive to reduce her risky investment to mitigate the ratcheting effect (and the associated cost) of the constraint. The fraction of wealth invested in the stock can be increasing or hump-shaped (possibly inverted U-shaped) in wealth. Due to hedging motives, risk aversion in wealth rises, which leads to lower risky investment relative to the unconstrained case. Finally, the drawdown constraint induces habit formation as optimally consumption never falls below an endogenous fraction of its historical maximum. See http://www.ma.utexas.edu/Seminars/MathFin/ for more information.

I will begin by giving a quick overview of the nice properties of cohomology and then, following Eilenberg-Steenrod, put them into a set of axioms that we would like any `cohomology theory' to satisfy. It turns out that these axioms are quite rigid, but if they are relaxed slightly they provide a very colorful spectrum of theories. The talk should be accessible to someone who has taken the algebraic topology course or has seen some version of homology or cohomology.