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.

This study is motivated by the class SP of sum-product graphs, introduced by Golumbic and RJ, where the tolerance coupling function is the sum of the two tolerances and and the rank coupling function is the product of the two ranks. Many properties of sum-product graphs remain valid when product is replaced by a general eutactic function --- one that satisfies a certain convexity condition. The class of eutactic functions is quite broad. For example, it includes all symmetric polynomials (in two variables) with positive coefficients.

We show that SP strictly contains co-TT as well as all complete bipartite graphs $K_{2,n}$, and we survey a number of other results. My advisor Victor Klee vowed as graduate student that he would avoid formulae involving binomial coefficients. I vowed that I would never get involved with multidimensional calculus. In the talk I will explain (briefly) why we were both wrong.

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.

It was proved earlier that for sufficiently large $t$ in terms of $s$, every graph with chromatic number $s+t$ has a $K^*_{s,t}$-minor. In particular, with $t_0(s) = \max\{4^{15s^2+s},(240s\log_2{s})^{8s\log_2{s}+1}\}$, the conclusion holds when $t>t_0(s)$. This result confirmed a special case of a conjecture by Woodall and Seymour. We show that the conclusion holds already for much smaller $t$, namely, for $t>C(s\log s)^3$.

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).

Robert Ghrist is the Andrea Mitchell Penn Integrating Knowledge Professor in the Departments of Mathematics and Electrical/Systems Engineering at the University of Pennsylvania.

A reception will be held in AH 314 immediately following the lecture.

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.