In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups. Techniques from equivariant stable homotopy theory are often key to algebraic K-theory computations. In this case we use n-cubes of cyclotomic spectra to compute the topological cyclic homology, and hence K-theory, of truncated polynomial algebras in several variables.

We discuss a result on filling real 2-tori by Levi-flat hypersurfaces in almost complex manifolds and some applications to global rigidity of symplectic structures.

Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

We are now witnessing exciting interactions between computer science and mathematics on one side, and the natural sciences on the other. While the natural sciences are rapidly absorbing notions, techniques and methodologies intrinsic to computer science and mathematics, theoretical computer science is adapting and extending its traditional notion of computation and computational techniques, to account for computation taking part in nature around us.

This talk will outline several of the fruitful directions of theoretical computer science research that originated from the study of DNA. I will describe comma-free codes inspired by the studies into the genetic code, splicing systems, optimal encodings for DNA Computing, sticker systems, Watson-Crick automata, combinatorics on DNA words, cellular computing, and computing by DNA self-assembly.

We introduce fundamental topics involving number fields, including ideal splitting, ramification, and the Frobenius element, along with many motivating questions and examples. We conclude with a discussion of the local global principle, and ask some number theoretic questions which can be easily understood, but require automorphic forms to resolve.