Given a graded Hopf algebra $A$, one wants to compute the stable representation ring $Stab(A)$. By work of Margolis, computing all possible Adams spectral sequence $E_2$-terms for finite module spectra over certain commutative ring spectra amounts to computing the cohomology of A with coefficients in each generator for Stab(A), when is a subalgebra of the Steenrod algebra. However, actually computing $Stab(A)$ is (in Margolis' words) "a very difficult problem in general."

In this talk we describe this relationship between Stab(A) and Adams spectral sequences, and we describe a new approach to the computation of Stab(A) which uses a twisted version of the deformation theory of modules. While untwisted first-order deformations of an A-module M are classified by the Hochschild cohomology group $HH^1(A, End(M))$, our twisted deformations instead are classified by a nonabelian (that is, with coefficients in a nonsymmetric module) version of the "higher-order Hochschild cohomology" of Pirashvili. We discuss existence and uniqueness results for these nonabelian higher-order Hochschild cohomologies, and the relative difficulty of actually making these computations (in particular, when they do and do not run up against of the unsolvability of the word problem!).

We will discuss some recent results on the $L^2$-theory of the $\overline{\partial}$-equation on the domain $\{\vert z_1 \vert < \vert z_2\vert <1\}$ in $\mathbb{C}^2$. This is joint work with Mei-Chi Shaw.

The rational numbers are dense in the reals, so we can always approximate a real as closely as we'd like to by a rational number. But how close can the rational get if we restrict the size of the denominator? If only a few lucky ones can ever get close, the number will be mighty lonely. Many great mathematicians have worked on this problem, and in this talk we'll give a historical survey of their results, along with examples of transcendental and Liouville numbers - the least lonely of them all.

A fundamental problem in mathematics is to approximate a given function on some region R with a nice function, such as a polynomial. In order to get a good approximation, the standard strategy is to subdivide R into smaller regions Δi, approximate f on those regions, and require compatibility conditions on Δi∩Δj. In the most studied case, the Δi are simplices, and the compatibility condition is Cr-smoothness. The set of piecewise polynomial functions of degree at most k and smoothness r on a triangulation Δ is a vector space, and even when Δ ⊆ R2, the dimension of Crk(Δ) is unknown. Work of Alfeld-Schumaker provides an answer if k≥3r+1, and Billera earned the Fulkerson prize for solving a conjecture of Strang for the case r=1 and a generic triangulation Δ. I will discuss recent progress on the dimension question using tools of algebraic geometry, when Δ is a polyhedral complex. I will also touch on a beautiful connection to toric geometry, provided by work of Payne on the equivariant Chow cohomology of toric varieties.

We study the linear syzygies of a homogeneous ideal $I$ in a polynomial ring $S = k[x_0..x_n]$, focussing on the graded betti numbers \[ b_i = {\textrm{dim}}_k {\textrm{Tor}}_i(S/I, k)_{i+1}. \] For any projective variety $X$ in $P^n$ and divisor $D$, what conditions on $D$ ensure that $b_i$ is nonzero? Eisenbud has shown that a decomposition $D=A+B$ such that $A$ and $B$ have at least two sections give rise to determinantal equations (and corresponding syzygies) in $I_X$ and conjectured that if the quadratic component of $I$ is generated by quadrics of rank at most four, then the last nonvanishing $b_i$ is a consequence of such a decomposition. We describe obstructions to the conjecture and prove a variant. The obstructions arise from toric specializations of the Rees algebra of Koszul cycles, and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This leads to a number of interesting open questions. (joint work with M. Stillman).