Thinking in terms of Waldhausen categories, we therefore get two cofibration sequences for THH, THH(k) → THH(R) → THH(F|R) (first constructed by Hesselholt and Madsen) and THH( Spec(R) on Spec(k) ) → THH(R) → THH(F) (generalizing to THH a well-known exact sequence in Hochschild homology). The first arises by looking at enrichments by connective spectra and the second by looking at enrichments in non-connective spectra.

For cotangent bundles these finite dimensional approximations exists canonically - but are not natural. I will explain this in more detail for a nearby Lagrangian, and describe how this lead to new insights into the coherent orientations in Floer homology. If time permits I will talk about some generalizations of the finite dimensional approximations and relations to complex periodic cobordism.

Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory of the tensor algebra T_R(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected).

Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie Taylor towers of the two (as functors of M, with R fixed), and allows us to give a formula for the suspension of the invariant W(R;M) (which can be thought of as Witt vectors with coefficients in M, and is what the Goodwillie Taylor tower of the source functor converges to) as the inverse limit, as n goes to infinity, of the reduced algebraic K-theory of T_R(M)/ (Mⁿ).

If $\Gamma$ has no elements of order two, then we obtain equivariant topological rigidity of the pair $(X, \Gamma)$. Hence, if $\Gamma$ is torsion-free, then we generalize a recent theorem of A. Bartels and W. Lück, which validates the classical Borel Conjecture for $\mathrm{CAT}(0)$ fundamental groups. Otherwise, if $\Gamma$ has elements of order two, we show how to parameterize all possible counter-examples, in terms of Cappell's $\mathrm{UNil}$ summands of the L-theory of infinite dihedral groups. In certain cases, these are detected along hypersurfaces in the orbifold $X / \Gamma$ by generalized Arf invariants.

This question was studied by Dupont and Sah, who assigned groups of scissors congruence types on manifolds and analyzed many structures on these groups. In particular, it turns out that in the case of $E^n$ and $S^n$, the groups assemble into a graded ring. In this talk we give a different perspective on scissors congruence groups by showing that they arise as the 0-th K-group of a particular type of Waldhausen category, and use Dupont and Sah's observations to construct these ring structures directly on the K-theoretic level.

We explain how to obtain the $E_2$-page of the short resolution spectral sequence for $(E_2)_*V(0)$ and compute the complete spectral sequence for $v_1^{-1}(E_2)_*V(0)$. This gives the $E_2$-page of the Adams-Novikov spectral sequence, namely $H^*(\mathbb{G}_2, v_1^{-1}(E_2)_*V(0))$. We expect the differentials to follow classical patterns and explain what this would imply for $v_1^{-1}\pi_*L_{K(2)}V(0)$.

In this talk on joint work with Peter Teichner, I will present a conjectural picture of TMF(X) as concordance classes of families of supersymmetric 2-dimensional Euclidean field theories parametrized by X. Evidence for the conjecture comes from an analogous description of K(X) in terms of 1-dimensional field theories, and our result that the partition function of a supersymmetric 2-dimensional Euclidean field theory is a modular form. The latter is the main focus of the talk.

This talk will describe recent progress, in joint work with M. Ching, on developing standard tools of the homotopy theory of spaces in this new algebraic-topological context of structured ring spectra, with a special emphasis on recovering algebraic and topological structures from associated homology objects.