# Seminars in geometry/topology

## Fall 2018 - Upcoming Talks

## October 22, 13:15 - 14:15, room 734, 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:** TBA.

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

**Jonas Irgens Kylling (Oslo): TBA**

**Abstract:** TBA.

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

**Elden Elmanto (Copenhagen/Harvard): TBA**

**Abstract:** TBA.

## Fall 2018 - Previous Talks

## 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.