The Stanford University Mathematical Organization recently began a speaker series. The goal of the series is to let students listen to professors discuss their backgrounds and research, as well as to let students hear some interesting problems or ideas.
"Elliptic curves form one of the hottest topics in arithmetic algebraic geometry. Applications of elliptic curves range from a proof of Fermat's Last Theorem to the design of secure cryptosystems. In the lecture we present, as a novel application of elliptic curves, a mathematical analysis of Escher's lithograph `Print Gallery'."
"The Grassmannian G(k,n), parametrizing k-planes in n-space, naturally arises as an important tool in many fields. Questions like the one in the title are really questions about the geometry (and topology) of the Grassmannian. Although in some sense the Grassmannian is just about linear algebra, it has an incredibly rich and subtle structure. (For example, the answer to the above question is neither 0, 1, nor infinity!) We'll investigate some of this structure, and see how it links geometry, topology, and combinatorics." (No background will be assumed.)
"The immune system's response to disease is somewhat of a mystery, and mathematics has recently been used to gain insights into this. Starting with simple predator-prey type systems of differential equations, interactions between immune system cells and invading pathogens have been modeled for a number of diseases, including HIV and leukemia. I'll show and explain some of these models, as well as mathematical analyses and results that have led to dramatically improved treatments for patients."
"Algebraic topology is a subject which allows one to distinguish between topological spaces by assigning groups to each of the spaces, and checking whether those groups are isomorphic. It turns out that one can use these techniques to identify and locate interesting features in geometric objects, such as singular points. This kind of feature identification can be useful in various kinds of applications. For instance, it allows one to distinguish between a printed letter A and a printed letter B. We will talk about how this works, and do some interesting examples."