As Sunada originally thought of his famous theorem using zeta functions and trace formulae, we will outline how he proved that the almost conjugacy of subgroups implies isospectrality of the manifolds. We will then continue with some important theory and examples of Riemannian orbifolds and construct Buser's example from 1987 of two isospectral, nonisometric flat (not planar) two-dimensional manifolds with boundary, embedded in $\mathbb{R}^{3}$. Finally, we will construct the important planar domain examples by Gordon, Webb and Wolbert in 1992. In addition, we will provide explicit pictures of these domains and subsequent isometric plane domains as constructed by Buser and Conway in 1994.

I will introduce an extension of Levine and Morel's algebraic cobordism to (quasi-projective) derived schemes. This extension will have the additional utility of naturally encompassing virtual fundamental classes and virtual Gysin homomorphisms. Time permitting, I will discuss the isomorphism with a different (obvious) extension of algebraic cobordism to derived schemes that shows any quasi-smooth projective variety is cobordant to a smooth projective variety.

Mixed-norm L^P spaces were described by Benedek and Panzone in 1961, but are connected to Littlewood's earlier 4/3 inequality and recent work on the Bohnenblust-Hille inequality. Among these properties are mixed-norm versions of Holder's inequality and Minkowski's integral inequality, which can work together to simplify certain proofs. Minkowski's integral inequality already has a mixed-norm character, and its general mixed-norm extension (described by Fournier in 1987) allows embeddings among various mixed norm spaces on a product measure space. Much of the talk should be clear with a basic background in integration and conventional L^p spaces

For any manifold $M$ with a spin structure $P_{spin}(X)$, we can define a canonical first order operator, known as the Dirac operator, acting on a spinor bundle. The spinor bundle is a bundle associated to the spin structure and the representation of spin group, coming from an irreducible representation of the Clifford algebra. The existence of an irreducible real Clifford module is equivalent to the existence of a spin structure on the manifold; however, we can always construct a Dirac bundle, which is (not necessarily irreducible) a $Cl(X)$-module with compatible metric and connection. Similarly, we may define a differential operator known as the generalized Dirac operator. In this talk, after some general discussion about Clifford bundles and Dirac bundles, we will focus on two important examples of the generalized Dirac operator: the de Rham complex and, in the $4k$-dimensional case, the signature complex.

Teichmuller space is a moduli space of Riemann surfaces, where two Riemann surfaces are equivalent if they are biholomorphic and are homotopically related in a certain sense. It can be thought of as the space of local deformations of Riemann surfaces. Conformal field theories are quantum mechanical or statistical field theories which are invariant under infinitesimal rotations and rescalings, and thus in two dimensions they are closely tied to complex analysis. A mathematical model of conformal field theory was sketched by Segal and Kontsevich. Attempts to realize this model rigorously has spawned a great deal of deep mathematics.

A certain moduli space in conformal field theory turns out to be the quotient of Teichmuller space by a discrete group action. This relation between the two moduli spaces leads to the solution of analytic problems in the rigorous formulation of conformal field theory, and new results in Teichmuller theory. In this talk, I will give a non-technical introduction to the ideas of Teichmuller theory, and sketch the correspondence between the moduli spaces. Joint work with David Radnell.

We study the image of an additive polynomial $f$ over a field $k$ of characteristic $p > 0$. We define the additive rank of $f$ over $k$ to be the smallest positive integer $r$ such that there exists an additive polynomial $g$ in $r$ variables with coefficients in $k$ which generates the same image as $f$ does. We show that over perfect fields the additive ranks of (non-zero) additive polynomials are always 1. We also show that for every positive integer $r$, there is some additive polynomial over a certain field with additive rank $r$.

In this talk first I will give some examples of vector bundles on $S^2$ which appears to be very important in complex topological K-theory and show how to imagine them. Then I will define clutching map of vector bundles on $S^n$ which will be used to prove the Bott periodicity. Then we are prepared to give a definition of K-group and ring structure on it. At last is the statement of Bott periodicity which I will prove in the next week.