# Seminars in geometry/topology

## August 27, 13:15 - 14:15, room 734, Sentralbygg 2

**Mark Powell (Durham/UQAM): The 4-dimensional sphere embedding theorem**

**Abstract:** I will describe the classification by Freedman of simply connected 4-dimensional topological manifolds, with emphasis on use of the sphere embedding theorem in its proof. Arunima Ray, Peter Teichner and I found that a technical improvement to Freedman's main embedding theorem is necessary in order to apply the theorem to 4-manifold classification, and I will endeavour to explain this new ingredient.

## September 17, 13:15 - 14:15, room 734, Sentralbygg 2

**Paul Ziegler (Oxford): The fundamental lemma**

**Abstract:** I will start with an introduction to the fundamental lemma and the role it plays in the Langslands program. Then I will talk about the proof of this result due to Ngô as well as a recent new proof due to Groechenig, Wyss and myself.

## September 24, 13:15 - 14:15, room 734, Sentralbygg 2

**Claudia Scheimbauer (NTNU): Understanding the Waldhausen construction via 2-Segal spaces**

**Abstract:** I will first recall the Waldhausen construction for exact categories (for example vector spaces), which is a beautiful approach to K-theory. Then we will look at the structure of its output, which is in itself interesting: it is a simplicial space which satisfies certain "higher Segal" conditions. These in turn come from the combinatorics of triangulations of a polygon and describe associativity of a certain compositional structure. I will explain a generalization of the Waldhausen construction, allowing for vastly more general inputs. The main result, which is joint work with with Bergner, Osorno, Ozornova, and Rovelli, is that every 2-Segal space arises from this construction in a suitable way.

## October 1, 13:15 - 14:15, room 734, Sentralbygg 2

**Severin Bunk (Hamburg): The Smooth Functorial Field Theory of B-Fields and D-Branes**

**Abstract:** A functorial field theory (FFT) on a manifold \(M\) is a symmetric monoidal functor out of a bordism category whose objects and bordisms are decorated with smooth maps to \(M\). For example, the parallel transport on a vector bundle with connection on \(M\) gives rise to a smooth 1-dimensional FFT on \(M\).

Higher dimensional FFTs on \(M\) will need higher geometric objects as input. The categorification of a line bundle with connection is called a bundle gerbe; these are geometric objects whose sections are twisted vector bundles, and whose field strength is a 3-form. In a collaboration with Konrad Waldorf we showed how bundle gerbes and D-branes on \(M\) give rise to smooth 2-dimensional open-closed FFTs on \(M\). In this talk, I will briefly introduce bundle gerbes and their 2-categorical structure, and explain the FFT construction as well as the transgression techniques that go into it.

## October 8, 13:15 - 14:15, room 734, Sentralbygg 2

**Sabrina Pauli (Oslo): \(\mathbb{A}^1\)-contractible varieties**

**Abstract:** The affine line \(\mathbb{A}^1\) is the only smooth, complex \(\mathbb{A}^1\)-contractible curve and it is conjectured that \(\mathbb{A}^2\) is the only smooth complex, \(\mathbb{A}^1\)-contractible surface. However, there exist nontrivial examples of smooth, complex, \(\mathbb{A}^1\)-contractible varieties in dimension three and higher, one being the Koras-Russell cubic \(\{ x^2 y = z^2 + t^3 + x \} \subset \mathbb{A}^4\).

In my talk I will give a short survey on \(\mathbb{A}^1\)-contractible varieties and explain how one can construct some of them using affine modifications.

## October 15, 13:15 - 14:15, room 734, Sentralbygg 2

**Nima Moshayedi (Zurich): Introduction to Perturbative Quantum Gauge Theories**

**Abstract:** I will give an introduction to the mathematical methods of quantization and talk about the concept of a perturbative quantum field theory. Especially, I will give a short introduction on supergeometry and introduce the notion of a so-called gauge fixing in the guise of Batalin-Vilkovisky. If time permits, I will talk about some of these notions in the case of manifolds with boundaries. Moreover, I will briefly explain the connection to TQFT‘s.

## October 22, 15:00 - 16:00, room 656, Sentralbygg 2

**Håkon Kolderup (Oslo): Cohomological correspondences and motivic Eilenberg-Mac Lane spectra**

**Abstract:** Since Suslin and Voevodsky’s introduction of finite correspondences, several alternate correspondence categories have been constructed in order to provide different linear approximations to the motivic stable homotopy category. In joint work with Andrei Druzhinin, we provide an axiomatic approach to a class of correspondence categories that are defined by an underlying cohomology theory. For such cohomological correspondence categories, one can prove strict homotopy invariance and cancellation properties. This results in a well behaved associated derived category of motives, each of which gives rise to a motivic Eilenberg-Mac Lane spectrum.

## October 29, 13:15 - 14:15, room 734, Sentralbygg 2

**Barbara Giunti (Pavia): Classification of filtered chain complexes**

**Abstract:** Persistent homology has proven to be an efficient tool to extract topological information from finite metric spaces. However, homology is a drastic simplification and in certain situations might remove too much information. Therefore, we consider filtered chain complexes: we define and list all and only possible indecomposables of filtered chain complexes, and we provide a structure theorem for enumerating them. In this seminar, I will also provide a brief introduction to persistent homology, to better enlight the difference with our approach, and present an algorithm to compute the decomposition. Finally, I will describe a possible generalisation for a more general type of complexes.

## November 5, 13:15 - 14:15, room 734, Sentralbygg 2

**Jonas Irgens Kylling (Oslo): Hermitian K-theory of rings of integers and the slice spectral sequence**

**Abstract:** A powerful computational tool in topology is the Postnikov filtration, which gives rise to the Atiyah-Hirzebruch spectral sequence computing homotopy groups in terms of cohomology groups. In motivic homotopy theory there is a variant of this known as the slice filtration, which gives a spectral sequence computing motivic homotopy groups in terms of motivic cohomology groups. We will carry out a computation of Hermitian K-theory of rings of integers using the slice filtration. As corollaries we get a variant of Milnor's conjecture on quadratic forms for rings of integers, and an analogy to the Lichtenbaum conjectures on special values of zeta-functions for K-theory. This is joint work with Oliver Röndigs and Paul Arne Østvær.

## November 12, 13:15 - 14:15, room 734, Sentralbygg 2

**Rachael Boyd (NTNU): Low dimensional homology of Coxeter groups**

**Abstract:** Coxeter groups were introduced by Tits in the 1960s as abstractions of reflection groups. They appear in different areas of mathematics such as Lie theory, combinatorics, and geometric group theory. Any Coxeter group can be realised as the reflection group of a contractible complex, called the Davis complex. This talk focuses on a computation of the first three integral homology groups of an arbitrary Coxeter group using an isotropy spectral sequence argument: the answer can be phrased purely in terms of the original Coxeter diagram. I will give a gentle introduction to Coxeter groups and the Davis complex before outlining the proof.

## November 19, 13:15 - 14:15, room 734, Sentralbygg 2

**Elden Elmanto (Copenhagen/Harvard): Stable Profinite Homotopy Theory (after Quick and Lurie)**

**Abstract:** We discuss a model-independent approach to stable profinite homotopy theory of Quick. An amusing feature of this construction is the idea of a ``co-spectrum" which had been considered by Lima when he first invented the notion of a spectrum. As an application, we revisit the construction of etale K-theory in the sense of Dwyer and Friedlander and provide an \(\ell\)-adic analog to semitopological K-theory. This is joint work with David Carchedi.