The study of Legendrian and transverse knots has a long and rich history in contact topology, but in the last few years we been able to understand quite a lot of the qualitative features once can expect to see while studying such knots. I will survey what is known about the classification of these knots and discuss several outstanding problems in the area.

In the 60's VI Arnol'd remarked that the space of real symmetric matrices has an interesting stratification, defined by eigenvalue multiplicity. Since these spaces parametrize families of harmonic oscillators, this stratification has applications to resonance phenomena in mechanics.

In a posting at arXiv:1009.3224 (currently undergoing perestroika), Satyan Devadoss and I propose a kind of resolution of Arnol'd's stratification, relating compactified configuration spaces of eigenvalues to certain smooth aspherical spaces of trees [see arXiv:math/0507514 in one direction, and work of Vogtmann, Billera and Holmes in another]. We hope these will have useful applications in genetics; in any case they are a source of lots of nice picture.

Viewing modules over rings as spectra in the category of spaces over the classifying space of the units of the ring may have technical advantages. The eventual goal is a better understanding of characteristic features of smooth manifolds. Parts will be quite speculative.

Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. In this talk we will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.

We report on a joint project with Mike Hopkins and introduce Hodge filtered complex cobordism groups for complex manifolds. These groups combine integral topological and complex geometric information in analogy to differential cohomology groups for smooth manifolds. By taking the Hodge filtration into account these groups form a natural generalization of Deligne cohomolgy and become especially interesting for Kaehler manifolds.

A classic problem in invariant theory, often referred to as Hilbert's 14th problem, asks, when a group acts on a finitely generated commutative algebra by algebra automorphisms, whether the ring of invariants is still finitely generated. The question extends to finite generation of the cohomology algebra of a group. We'll present recent progress and generalizations with a stress on the case of a general base ring.

The study of polynomial functors (in the sense of Goodwille) and of nilpotent groups share strong similarities. We provide motivation of the development of Goodwillie's Taylor tower from its relationship to the lower central series of a group. As shown by Biedermann and Dwyer, the Goodwillie polynomial approximations to a functor naturally take values in spaces with an interesting nilpotence condition. We recover this result by establishing a stronger property on the finite limits used to construct the polynomial approximations.

In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups. In particular, we consider the K-theory of truncated polynomial algebras in several variables. 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 the rings in question.

I'll describe an axiomatic characterization of factorization homology (a.k.a. topological chiral homology) of topological manifolds, in a sense analogous to (and generalizing) the Eilenberg-Steenrod axioms for usual homology. I'll then present some classes of computations of factorization homology, conjecture on its relation to previously studied invariants, and speculate just how refined of a manifold invariant it is -- for example, it's not homotopy invariant.

The K(2)-localization of the sphere spectrum admits a conjectural small resolution built from TMF and "TMF with level structures" --- the evaluation of the TMF sheaf on the stack of elliptic curves equipped with an order l subgroup. In this talk, I will use variations on Tate normal form to describe several Hopf algebroids that stackify to elliptic curves with level structure. These Hopf algebroids lead to computations of the Behrens-Lawson spectrum Q(l), a height 2 analogue of the image of J. This is current work with Mark Behrens, Nat Stapleton, and Vesna Stojanoska.