You missed it?
The 33rd NRW Topology Meeting will be hosted in Essen in early 2020.

# 32nd NRW Topology Meeting

## Heinrich Heine University Düsseldorf

### Schedule

(→ url with better printing support)
 Friday, 8 November 11.00 – 11.50 Tobias Barthel (Copenhagen/Bonn) The affine line in tensor triangular geometry 12.00 – 12.50 Jens Reinhold (Stanford/Münster) Non-smoothable bundles over surfaces Lunch 14.15 – 15.05 Rachael Boyd (Trondheim/Bonn) Embedding spaces of Hopf links 15.15 – 16.05 Oliver Bräunling (Bonn/Freiburg) K-theory of locally compact abelian groups (an equivariant version) Tea/coffee 16.45 – 17.35 Inna Zakharevich (Cornell) The Dehn complex: scissors congruence, K-theory, and regulators 19.00 Dinner Saturday, 9 November 09.30 – 10.20 Konrad Voelkel (Osnabrück) A p-adic Kato-Nakayama functor 10.30 – 11.20 Maria Yakerson (Osnabrück/Regensburg) Modules over algebraic cobordism Tea/coffee 12.00 – 12.50 Viktoriya Ozornova (Bochum) Homotopy theory for 2-categories

All talks will take place in the Felix Klein Lecture Hall (5F in building 25.21).

The dinner will be at Wilma Wunder (Martin-Luther-Platz 27, 40212 Düsseldorf).

### Registration

There is no formal registration, but please send an email to Frau May so that we can estimate the number of participants:

May (at) math.uni-duesseldorf.de

In particular, please indicate whether you will be joining us for dinner on Friday evening, and if so, let us know of any dietary requirements. The restaurant Wilma Wunder is ideally situated between Düsseldorf's Altstadt and Düsseldorf's main station, just a ten minute walk or one or two underground stops away from either. So whether you are looking forward to a relaxed evening in Düsseldorf or planning to spend the night to Saturday elsewhere, there's no reason to miss out.

### Arrival

From Düsseldorf Hbf, take underground line U79 to „Uni-Ost/Botanischer Garten“ (ca. 15 minutes). This is the terminal stop. From there, walk 400m to get to buildings 25.XX. For alternative travel options, see hhu.de.

### Accommodation

The speakers will stay at the HK-Hotel Düsseldorf City (Varnhagenstraße 37, 40225 Düsseldorf). (We had a block of rooms reserved at a fixed rate until October 31.)

### Abstracts

#### Oliver Bräunling: K-theory of locally compact abelian groups (an equivariant version)

The exact category of locally compact abelian (LCA) groups has a natural duality: Pontryagin duality. Deligne attaches to any exact category a universal determinant functor. The graded determinant line is the most popular example. Deligne also interprets this as the truncation of the K-theory spectrum to its stable (0,1)-type. Applying this to LCA groups, one gets an interpretation of the Haar measure.

With some applications to number theory in mind, we will present some thoughts (and theorems) on an equivariant version of this picture. There probably is a precise bridge to the condensed mathematics of Clausen–Scholze, but I don't understand it yet. Audience suggestions welcome!

#### Inna Zakharevich: The Dehn complex: scissors congruence, K-theory, and regulators

Hilbert's third problem asks: do there exist two polyhedra with the same volume which are not scissors congruent? In other words, if $$P^{}_{}$$ and $$Q$$ are polyhedra with the same volume, is it always possible to write $$P = \bigcup^n_{i=1} P_i$$ and $$Q = \bigcup^n_{i=1} Q_i$$ such that the $$P$$'s and $$Q$$'s intersect only on the boundaries and such that $$P_i \cong Q_i$$? In 1901 Dehn answered this question in the negative by constructing a second scissors congruence invariant now called the "Dehn invariant," and showing that a cube and a regular tetrahedron never have equal Dehn invariants, regardless of their volumes.

We can then restate Hilbert's third problem: do the volume and Dehn invariant separate the scissors congruence classes? In 1965 Sydler showed that the answer is yes; in 1968 Jessen showed that this result extends to dimension 4, and in 1982 Dupont and Sah constructed analogs of such results in spherical and hyperbolic geometries. However, the problem remains open past dimension 4.

By iterating Dehn invariants Goncharov constructed a chain complex, and conjectured that the homology of this chain complex is related to certain graded portions of the algebraic K-theory of the complex numbers, with the volume appearing as a regulator. In joint work with Jonathan Campbell, we have constructed a new analysis of this chain complex which illuminates the connection between the Dehn complex and algebraic K-theory, and which opens new routes for extending Dehn's results to higher dimensions. In this talk we will discuss this construction and its connections to both algebraic and Hermitian K-theory, and discuss the new avenues of attack that this presents for the generalized Hilbert's third problem.

#### Viktoriya Ozornova: Homotopy theory for 2-categories

The goal of this long-term project joint with Julie Bergner and Martina Rovelli is the comparison between simplicial and globular approaches to higher categories. In particular, we are working on the connection of 2-categories and $$\Theta_2$$-spaces on the globular side, and weak complicial sets on the other side. In the talk, I will present the results on 2-categories which also shed light onto possible direct comparison with $$\Theta_2$$-spaces.