# Seminars in geometry/topology

## Fall 2013

## Monday November 18, 13:15 - 14:15, room 734, Sentralbygg 2

Marius Thaule: *Higher n-angulations from local algebras*

**Abstract:** The most common examples of triangulated categories are algebraic and topological triangulated categories. Algebraic triangulated categories are categories which are equivalent (as triangulated categories) to the stable category of a Frobenius category, i.e. an exact category with enough injectives and enough projectives in which injectives and projectives coincide. Stable homotopy theory gives rise to topological triangulated categories. Muro, Schwede and Strickland discovered in 2007 triangulated categories which are neither algebraic nor topological. Motivated by these triangulated categories, we provide new examples of n-angulated categories. In particular, for a commutative local ring with principal maximal ideal squaring to zero, the category of finitely generated free modules is n-angulated for every n at least three.

This is joint work with Petter Bergh.

## Monday November 11, 13:15 - 14:15, room 734, Sentralbygg 2

Alexander Schmeding: *Differentiable mappings on products and the exponential law*

**Abstract:** We consider differentiable mappings on products with different degrees of differentiability in the two factors. A basic example for these mappings is the evaluation map, i.e. the map which evaluates differentiable maps (on finite dimensional spaces) in points from their domain. The main result presented in the talk is a version of the so called "Exponential Law"(*) for differentiable functions with (in-)finite degree of differentiability. The Exponential Law is an advanced tool for application in infinite dimensional Lie theory. Unfortunately, for smooth maps on arbitrary infinite-dimensional spaces, the Exponential law fails . However, in [1] a sufficient topological condition was derived which yields a version of the Exponential Law obtaining an isomorphism of topological vector spaces between the function spaces involved. This version of the theorem generalizes the Exponential Laws obtained in [2] and [3].

This project is joint work with H. Alzaareer.

- Alzaareer, H., Schmeding, A.: Differentiable mappings on products with different degrees of differentiability in the two factors, arXiv:1208.6510 [math.FA]
- Biller, H.: The exponential law for smooth functions, 2002, manuscript TU Darmstadt
- Glockner, H.: Lie groups over non-discrete topological fields, arXiv:math/0408008 [math.GR]

(*) A very basic (and well known) version of the Exponential law for continuous maps states:

Let \(X,Y,Z\) be topological spaces. Each continuous map \(f \colon X \times Y \to Z\) induces a unique continuous map \(g \colon Y \to C(X,Z)\), \(y \mapsto f(*,y)\). If \(X\) is locally compact and Hausdorff then this correspondence induces a homeomorphism between \(C(X \times Y,Z)\) and \(C(Y,C(X,Z))\) (if we endow both spaces with the compact-open topology).

Notice that this yields a version of cartesian closedness for a certain category of topological spaces (cf. [4]).

## Tuesday November 5, 13:15 - 14:15, room 656, Sentralbygg 2

Rune Haugseng (Max Planck Institute for Mathematics): *Enriched (infinity,n)-categories*

**Abstract:** The enrichment construction I described yesterday can be iterated to give a theory of "enriched (infinity,*n*)-categories". If we start with sets or spaces this gives theories of *n*-categories and (infinity,*n*)-categories, respectively, with very nice formal properties. I will explain how the latter are related to Barwick's iterated Segal spaces. I will also describe a non-iterative construction of enriched (infinity,*n*)-categories connected to Rezk's
θ_{n}-spaces.

## Monday November 4, 13:15 - 14:15, room 734, Sentralbygg 2

Rune Haugseng (Max Planck Institute for Mathematics): *Enriched infinity-categories*

**Abstract:** Categories enriched in symmetric monoidal categories such as spectra turn up in various places in algebraic topology. Unfortunately these can be difficult to work with in a homotopically meaningful way, which suggests that for many purposes it would be better to work with less rigid structures, where composition is only associative up to coherent homotopy. In this talk I will introduce a general theory of such weak or homotopy-coherent enrichment, built using a non-symmetric variant of Lurie's infinity-operads. I will describe how the correct homotopy theory of these enriched infinity-categories can be constructed as a localization of a homotopy theory defined using
infinity-operads (this is joint work with David Gepner), and also discuss some comparison results.

## Monday October 28, 13:15 - 14:15, room 734, Sentralbygg 2

Richard Williamson: *A glimpse of homotopy type theory*

**Abstract:** There is a deep link between category theory, type theory, logic, and programming languages. In particular, locally cartesian closed categories give a semantics for extensional
Martin-Löf type theory.

Recently, it has begun to be understood that abstract homotopy theories give a semantics for intensional Martin-Löf type theory. This had led to the idea of a foundations for mathematics in which homotopy types are the basic building blocks.

In this talk I will discuss a few aspects of this story, and outline briefly a few ways in which it touches upon work I have been doing over the last few years.

## Seminar series by David Ayala (University of Southern California) October 17 - October 21

David Ayala (University of Southern California): *Manifold invariants as algebraic data*

**Abstract**: Homology theories, sheaves, and topological field theories are all concepts that give rise to manifold invariants. Each of these invariants satisfies a local-to-global property, such as excision, thereby making these invariants computable, at least in principle. This series of talks will investigate manifold invariants in a primitive form and see that they tend to be characterized by the combinatorics of algebras and categories. As we will see, consequences of these investigations include a refined statement of Poincaré duality, a host of examples of higher categories, functorial constructions of manifold invariants, and the construction of various trace maps.

These talks will be accessible to students who are familiar with basics in algebraic topology and category theory, and who have an interest in geometric applications thereof. The first talk will be the most accessible and of broadest interest. The second talk will be the most technical, dwelling on a specific construction. The third talk will state, with some explanation, consequences of the main result of the second talk, which can be taken as a black box.

**Thursday October 17, 13:00 - 14:00, room 734, Sentralbygg 2**

*Manifold homology and Poincaré duality*

**Abstract:** We will consider homology theories that are only defined for manifolds of a specific dimension, and give some examples thereof. We will characterize them in terms of algebraic data, akin to Eilenberg-Steenrod's characterization of ordinary homology theories as their coefficients. We will see that Poincaré duality is valid for such homology theories, as it intertwines with Koszul duality, and point out some consequences which are of independent interest.

**Friday October 18, 10:15 - 11:15, room 734, Sentralbygg 2**

*Higher categories as sheaves on manifolds*

**Abstract:** We will take lesson from the construction of the bordism category to construct a higher category from any sheaf admitting a notion of transversality. We will axiomatize this latter notion, and see that in fact all higher categories arise in this way. The main concept here is a particular formulation of the tangent bundle of a stratified manifold, as well as moduli of stratifications of a manifold.

**Monday October 21, 13:15 - 14:15, room 734, Sentralbygg 2**

*Constructions of manifold invariants*

**Abstract:** We will construct invariants of manifolds as well as embedded submanifolds. Such invariants include Khovanov knot invariants and the Turaev-Viro 2D topological field theories, and generally lend to trace methods. The specifics of this talk will depend on interest.

## Monday October 7, 13:15 - 14:15, room 734, Sentralbygg 2

Truls Bakkejord Ræder: *Stable homotopy groups of spheres*

**Abstract:** In this talk, I will present the problem of computing stable homotopy groups of spectra.
I will then demonstrate one method of doing this, the Adams spectral sequence, on the sphere spectrum.

## Monday September 30, 13:15 - 14:15, room 734, Sentralbygg 2

Gard Spreemann: *Efficiently computing persistent (co)homology of sequences of simplicial
maps*

**Abstract:** The "classical" algorithms for persistent homology treat sequences of simplicial complexes where the maps are inclusions. Our approach to simplifying complexes while computing persistence relies on also being able to collapse simplices, and thus requires general simpicial maps between the complexes (if one is to avoid zigzags).

In this talk, I will present recent algorithms due to Dey, Boissonnat et al. [1,2,3] that track localized bases for homology to facilitate efficient collapses. Warning: May contain traces of implementation details.

- Dey, Fan, Wang: "Computing Topological Persistence for Simplicial Maps". arXiv:1208.5018v3
- Boissonnat, Maria: "The Simplex Tree: an Efficient Data Structure for General Simplicial Complexes". INRIA RR 7993.
- Boissonnat, Dey, Maria: "The Compressed Annotation Matrix: an Efficient Data Structure for Computing Persistent Cohomology". INRIA RR 8195.

## Monday September 23, 13:15 - 14:15, room 734, Sentralbygg 2

Magnus Bakke Botnan: *Capturing topology from finite point samples*

**Abstract:** How do we capture the 'shape' of an unordered sequence of points in high-dimensional Euclidean space? Such data occurs naturally in biology, image analysis, surveys etc. In this talk I show how tools from algebraic topology can help answer this question. It will also be clear that the greatest bottleneck of the standard constructions is that the memory needed grows exponentially in the number of points. As a solution to this problem I will present a new memory-efficient approach which approximates the standard constructions.

## Monday September 16, 13:15 - 14:15, room 734, Sentralbygg 2

Andrew Stacey: *That which we call a manifold …*

**Abstract:** It’s well known that the mapping space of two finite dimensional manifolds can be given the structure of an infinite dimensional manifold modelled on Fréchet spaces (provided the source is compact). However, it is not that the charts on the original manifolds give the charts on the mapping space: it is a little bit more complicated than that. These complications become important when one extends this construction, either to spaces more general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to transport local properties of a space to local properties of its mapping space. As an application, we shall show that applying the mapping construction to a regular map is again regular.

## Monday September 9, 13:15 - 14:15, room 734, Sentralbygg 2

Alexander Schmeding (Universität Paderborn / NTNU): *Diffeomorphisms of orbifolds*

**Abstract:** Orbifolds are a generalization of manifolds. They arise naturally in different areas of mathematics and physics, e.g.:

- Spaces of symplectic reduction are orbifolds,
- Orbifolds may be used to construct a conformal field theory model (cf. [1]),

We consider the diffeomorphism group of non-compact smooth reduced orbifolds. Our main result is the construction of an infinite dimensional Lie-group structure on the diffeomorphism group. Here orbifold morphisms are understood as maps in the sense of [2]. The aim of the talk is to give an introduction to these topics. In particular, we sketch the construction and its main ingredients. Moreover, we shall explain the relation of orbifolds to certain Lie groupoids, called atlas groupoids.

- Dixon, L.J., Harvey, J.A., Vafa, C. and Witten, E.:
*Strings on orbifolds. I*, Nuclear Phys. B 261:4 (1985), 678-686 - Pohl, A.D.:
*Convenient categories of reduced orbifolds*. arXiv:1001.0668v4 [math.GT]

## Monday September 2, 13:15 - 15:00, room 734, Sentralbygg 2

Nathan Perlmutter (University of Oregon): *Classifying spaces of cobordism categories with singularities*