Past Advanced Class

In this talk I want to outline the proofs our of main results, i.e. the localisation theorem and the identification of the homotopy type of Grothendieck-Witt theory in terms of K- and L-theory.
Finally, as a small application I want to present a refinement and extension of certain maps relating certain Madsen-Tillmann spectra and orthogonal/symplectic algebraic K-theory spectra of the integers.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.
 

I will start by briefly reviewing the Tate construction and in particular, the Tate diagonal. Using these I will then illustrate Lurie’s notion of Poincaré categories by considering Poincaré structures on module categories over a ring (spectrum) in detail. In particular, I will describe the somewhat subtle genuine Poincaré structure on the category of perfect complexes of an ordinary ring, which conjecturally links the classical notion of the Grothendieck-Witt spectrum to our derived version. Finally, I will compute its associated L-groups.

 The theory of derivators is an approach to homotopical algebra
that focuses on the existence of homotopy Kan extensions. Homotopy
theories (e.g. model categories) typically give rise to derivators by
considering the homotopy categories of all diagrams categories
simultaneously. A general problem is to understand how faithfully the
derivator actually represents the homotopy theory. In this talk, I will
discuss this problem in connection with algebraic K-theory, and give a
survey of the results around the problem of recovering the K-theory of a
good Waldhausen category from the structure of the associated derivator.

A cactus product is much like a wedge product of pointed spaces, but instead of being uniquely defined there is a moduli space of possible cactus products. I will discuss how this space can be interpreted geometrically and how its combinatorics calculates the homology of the automorphism group of a free product with no free group factors. Then I will reinterpret the moduli space with Outer space in mind: the lobes of the cacti now behave like boundaries and our free products can now include free group factors.

The idea of three-tier conformal field theory (CFT) was first proposed by Greame Segal. It is an extension of the functorial approach to CFT, where one replaces the bordism category of Riemann surfaces by a suitable bordism 2-category, whose objects are points, whose morphism are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. The Baez-Dolan cobordism hypothesis is a meta-mathematical principle. It claims that functorial quantum field theory (i.e. quantum field theory expressed as a functor from some bordism category) becomes simper once "you go all the way down to points", i.e., once you replace the bordism category by a higher category. Three-tier CFT is an example of "going all the way down to points". We will apply the cobordism hypothesis to the case of three-tier CFT, and show how the modular invariance of the partition function can be derived as a consequence of the formalism, even if one only starts with genus-zero data.

Three short talks by the authors of essays on topics related to c3 Algebraic topology: Whitehead's theorem, Cohomology of fibre bundles, Division algebras

The central axis of the famous DNA double helix can become knotted
or linked as a result of numerous biochemical processes, most notably
site-specific recombination. Site-specific recombinases are naturally
occurring enzymes that cleave and reseal DNA molecules in very precise ways.
As a by product of their main purpose, they manipulate cellular DNA in
topologically interesting and non-trivial ways. So if the axis of the DNA
double helix is circular, these cut-and-seal mechanisms can be tracked by
corresponding changes in the knot type of the DNA axis.  In this talk, I'll
explain several topological strategies to investigate these biological
situations. As a concrete example, I will disscuss my recent work, which
predics what types of DNA knots and links can arise from site-specific
recombination on DNA twist knots.

Persistent homology is a relatively new tool to analyse the topology of data sets.

We will give a brief introduction and tutorial as preparation for the third talk in the afternoon.