Abstract: 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.

Abstract: 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.