# Seminars in geometry/topology

## Fall 2012

## Thursday November 22, 10:15 - 12:00, room 656, Sentralbygg 2

Georgios Raptis (University of Osnabrück): *Cobordism categories and the A-theory characteristic*

**Abstract:** The A-theory characteristic of a fibration is a parametrized Euler characteristic which is defined in terms of Waldhausen's algebraic K-theory of spaces. This talk will focus on relating the A-theory characteristic to the cobordism category. A link between the two was proposed recently by Bökstedt and Madsen who defined a map from the cobordism category to A-theory. I will discuss the connection between this map, the A-theory characteristic and the smooth Riemann-Roch theorem of
Dwyer, Weiss and Williams.

## Tuesday November 20, 14:15 - 16:00, room 734, Sentralbygg 2

Georgios Raptis (University of Osnabrück): *Bivariant A-theory*

**Abstract:** This talk will introduce bivariant A-theory and some of its properties. If time permits, the definition of the A-theory characteristic and the smooth Riemann-Roch theorem of Dwyer, Weiss and
Williams will also be discussed.

## Tuesday November 13, 14:15 - 16:00, room 734, Sentralbygg 2

Dimitri Ara (Radboud Universiteit Nijmegen): *On Grothendieck* \(\infty\)*-groupoids and Grothendieck's conjecture*

**Abstract:** In *Pursuing Stacks*, Grothendieck defines a notion of (weak) \(\infty\)-groupoid and constructs a fundamental \(\infty\)-groupoid functor from topological spaces to (GrothendiecK) \(\infty\)-groupoids. He conjectures that these \(\infty\)-groupoids classify homotopy types in a precise way. In this talk, we will explain (a variant on) Grothendieck's definition. We will then present a series of conjectures due to Maltsiniotis (related to Grothendieck's theory of test categories) that lead to Grothendieck's conjecture. We will finally explain what is known about these conjectures.

## Monday November 12, 13:15 - 15:00, room 734, Sentralbygg 2

Dimitri Ara (Radboud Universiteit Nijmegen): *n-quasi-categories and their comparison to*
\(\Theta_n\)*-spaces*

**Abstract:** We introduce a notion of n-quasi-categories as the fibrant objects of a model category structure on the category of presheaves on the category of n-cells \(\Theta_n\) of Joyal. We construct two Quillen equivalences between this model category of n-quasi-categories and the model category of Rezk \(\Theta_n\)-spaces. For n = 1, we recover the two Quillen equivalences between quasi-categories and complete Segal spaces defined by Joyal and Tierney.

## Monday October 29, 13:15 - 15:00, room 734, Sentralbygg 2

Richard Williamson: *The cobordism hypothesis - part II*

**Abstract:** We will discuss the ingredients from higher category theory which are involved in Lurie's formulation of the cobordism hypothesis. I will introduce certain bisimplicial models for (∞,1)-categories due to Rezk, known as complete Segal spaces. We will see that there are different ways to generalise the idea towards gadgets intended as models for (∞,n)-categories. I will explain how cobordisms assemble into an example of an (∞,n)-category in one of these models. Finally, we will discuss higher categories with duals.

## Monday October 22, 13:15 - 15:00, room 734, Sentralbygg 2

Marius Thaule: *The cobordism hypothesis - part I*

**Abstract:** In the mid 1990s Baez and Dolan formulated the cobordism hypothesis. In 2009, Lurie published a detailed sketch of a proof. The cobordism hypothesis seeks to build a bridge between the geometric world of topological quantum field theories and the world of (higher) category theory. In the first talk I will try to motivate the cobordism hypothesis looking at low dimensional examples. I will introduce the version of the cobordism hypothesis that Lurie aims to prove.

## Monday October 8, 13:15 - 15:00, room 734, Sentralbygg 2

Truls Bakkejord Ræder: *Massey products and linking*

**Abstract:** In this talk we present the Massey triple products in dgas, typically the cohomology algebras of spaces. As an example, by an argument of Sullivan, we use Massey products of differential forms to distinguish the Brunnian 3-link from the Borromean rings.

## Monday October 1, 13:15 - 15:00, room 734, Sentralbygg 2

Magnus Bakke Botnan: *Discrete Morse theory*

**Abstract:** All topologists are familiar with Morse theory in the smooth category but not so many are familiar with discrete Morse theory. In this talk I will introduce a Morse theory for simplicial complexes (or, more generally, for CW complexes) and discuss how it shares several beautiful
properties with its smooth counterpart. I will also briefly discuss algebraic discrete Morse theory which is a Morse theory for free chain complexes, i.e. chain complexes consisting of finitely generated modules over a commutative ring with unit.

## Monday September 24, 13:15 - 15:00, room 734, Sentralbygg 2

Gard Spreemann: *Topology and genetic data*

**Abstract:** Persistent homology gives, as I talked about last year, a multiscale way to study the topology of a point cloud. While point clouds certainly are ubiquitous as data in many parts of the natural sciences and engineering, not all data comes as point clouds. An example is that of gene expressions; for a single person such expression data consists of a single point in something like
\(\mathbb{R}^{20000}\). I will briefly review persistent homology, and then mention 3-4 different approaches we are considering for applying persistent homology to datasets like gene expressions.

## Monday September 10, 13:15 - 14:30, room 734, Sentralbygg 2

Wu-Yi Hsiang (University of California, Berkeley): *Geometric Invariant Theory of the Space – solid geometry, sphere packing, and a new simple proof of Kepler's conjecture*

**Abstract:** We shall give a brief overview of such an approach, beginning with a concise summary on the geometric invariant theory of spherical triangles and tetrahedra (i.e. spherical trigonometry and tetranometry). As an outstanding example of application of such a theory, we shall outline a much simplified new proof of Kepler’s conjecture and the least action principle of crystal formation of dense type.

## Wednesday September 5, 14:15 - 15:15, room 734, Sentralbygg 2

Wu-Yi Hsiang (University of California, Berkeley): *On equivariant Riemannian geometry and Ricci flows - part 2*

**Abstract:** We shall use the technique of equivariant Riemannian geometry to analyze the geometry of equivariant Ricci flows, especially those of O(3) Ricci flows of 3-manifolds. The evolution equation of such special cases is much simpler than that of general settings, thus providing a natural testing ground for down-to-earth study of recent celebrated results on the Ricci flows of 3-manifolds by R. Hamilton and G. Perelman. However, such a study leads to the discovery of some puzzling examples of O(3) equivariant Ricci flows which seems to us to seriously contradict their main results.

This is a continuation of the talk given on Monday September 3.

## Monday September 3, 13:15 - 14:15, room 734, Sentralbygg 2

Wu-Yi Hsiang (University of California, Berkeley): *On equivariant Riemannian geometry and Ricci flows - part 1*

**Abstract:** We shall use the technique of equivariant Riemannian geometry to analyze the geometry of equivariant Ricci flows, especially those of O(3) Ricci flows of 3-manifolds. The evolution equation of such special cases is much simpler than that of general settings, thus providing a natural testing ground for down-to-earth study of recent celebrated results on the Ricci flows of 3-manifolds by R. Hamilton and G. Perelman. However, such a study leads to the discovery of some puzzling examples of O(3) equivariant Ricci flows which seems to us to seriously contradict their main results.

This is the first talk in a series of two talks with the first talk aiming to be expository while the second talk will be more technical.