# Schedule

## Monday, August 17

10:00 - 11:00 |
Registration & Welcome coffee |

11:00 - 12:00 |
Andreas B. Thom: Topological methods to solve equations over groups |

12:00 - 14:30 |
Lunch break |

14:30 - 15:30 |
Thomas Schick: Classification of positive scalar curvature metric: Index theory versus geometry |

15:30 - 16:00 |
Coffee break |

16:00 - 17:00 |
Jesper Grodal: Burnside rings in topology |

## Tuesday, August 18

9:30 - 10:30 |
Günter M. Ziegler: Equivariant methods in discrete geometry: Problems and progress |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Martin R. Bridson: Profinite recognition of groups and 3-manifolds |

12:00 - 14:30 |
Lunch break |

14:30 - 15:30 |
Sylvain E. Cappell: Cobordism of L^{2} acyclic manifolds and an L^{2} structure sequence |

15:30 - 16:00 |
Coffee break |

16:00 - 17:00 |
Mark Behrens: A Lie algebra model for unstable v_{n} periodic homotopy. |

18:00 - |
Dinner at Universitätsclub Bonn (Konviktstr. 9; map) |

## Wednesday, August 19

9:30 - 10:30 |
Thomas Church: Applications of representation stability |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Dan Freed: Diagonalization with singularities and complex volumes of hyperbolic 3-manifolds |

12:00 - |
Lunch break, free afternoon |

## Thursday, August 20

9:30 - 10:30 |
Søren Galatius: Tautological rings for high-dimensional manifolds |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Daniel T. Wise: Nonpositive immersions and counting cycles |

12:00 - 14:30 |
Lunch break |

14:30 - 15:30 |
Thomas Farrell: Bundles with extra geometric structure |

15:30 - 16:00 |
Coffee break |

16:00 - 17:00 |
Jacob Lurie: Rotation invariance in algebraic K-theory |

## Friday, August 21

9:30 - 10:30 |
Birgit Richter: Higher THH – towards integral results |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Lars Hesselholt: Algebraic K-theory of orders |

12:00 - 14:30 |
Lunch break |

14:30 - 15:30 |
Ib Madsen: Automorphisms of manifolds and graph homology |

15:30 - 16:00 |
Coffee / end of conference |

# Abstracts

## Mark Behrens: A Lie algebra model for unstable v_{n} periodic homotopy

Quillen-Sullivan rational homotopy theory encodes rational homotopy groups in a differential graded Lie algebra. I will discuss a generalization of this to unstable v_{n} periodic homotopy. This is joint work with Charles Rezk.

## Martin R. Bridson: Profinite recognition of groups and 3-manifolds

I shall begin by discussing the history of the following problem: to what extent is a residually-finite group determined by its set of finite quotients (equivalently, its profinite completion)? I shall explain how ideas developed in connection with non-positive curvature have led to significant progress on this question in recent years, including the construction of infinite families of finitely presented subgroups of SL(n,Z) all of which have the same finite quotients.

Fundamental groups of orbifolds of dimension at most 3 enjoy a greater degree of profinite rigidity than arbitrary groups. I shall present positive and negative results in this context, and outline how Reid, Wilton and I proved that the fundamental groups of punctured torus bundles can be distinguished from each other and from other 3-manifold groups by means of their profinite completions.

## Sylvain E. Cappell: Cobordism of L^{2} acyclic manifolds and an L^{2} structure sequence

This talk will develop the cobordism theory of L^{2} acyclic manifolds and present new calculations of free and torsion components. A structure sequence for L^{2} related (but not necessarily acyclic) manifolds will also be developed for certain fundamental groups. This is joint work with Jim Davis and Shmuel Weinberger.

## Thomas Church: Applications of representation stability

I will give a gentle survey of the theory of representation stability, viewed through the lens of its applications in topology and elsewhere. These applications include: homological stability for configuration spaces of manifolds; stable and unstable homology of arithmetic lattices; Hecke eigenclasses in stable mod-p cohomology and excision in p-completed K-theory (Calegari-Emerton); uniform generators for congruence subgroups and "congruence" subgroups; and distributional stability for random squarefree polynomials over finite fields. Based on joint work with Jordan Ellenberg, Benson Farb, Andrew Putman, and Rohit Nagpal.

## Thomas Farrell: Bundles with extra geometric structure

This talk is about smooth fiber bundles whose concrete fibers are each equipped with a Riemannian metric whose sectional curvatures are constrained to lie in a fixed interval I of real numbers. Prominent amoung these will be I = (-∞,0), [0,+∞) and (¼,1].

This is a report on joint work with several people including Pedro Ontaneda, Andrey Gogolev, Igor Belegradek, Vitali Kapovitch, Zhou Gang and Dan Knopf.

## Dan Freed: Diagonalization with singularities and complex volumes of hyperbolic 3-manifolds

The volume of a hyperbolic 3-manifold is the imaginary part of the Chern-Simons invariant of the associated flat connection. In joint work in progress with Andy Neitzke we use a 3-dimensional spectral network to achieve a branched diagonalization of such flat connections, and produce formulas for the complex Chern-Simons invariant in terms of dilogarithms. The argument uses ideas from topological field theory.

## Søren Galatius: Tautological rings for high-dimensional manifolds

To each fiber bundle f: E → B whose fibers are closed oriented manifolds of dimension d and each polynomial p ∈ H*(BSO(d)) there is an associated "tautological class'' κ_{p} ∈ H*(B) defined by fiberwise integration. The set of polynomials in these classes which vanish for all bundles whose fibers are oriented diffeomorphic to M forms an ideal I_{M} ⊂ ℚ[κ_{p}] and the quotient ring R_{M} = ℚ[κ_{p}]/I_{M} is the "tautological ring'' of M. In this talk I will discuss some recent results about the structure and particularly Krull dimension of this ring for various M. This is joint work with Ilya Grigoriev and Oscar Randal-Williams.

## Jesper Grodal: Burnside rings in topology

The Burnside ring of a finite group G enters equivariant stable homotopy theory as the 0th equivariant stable homotopy group. The Segal conjecture, proved in the 80s by Carlsson, identifies the 0th stable cohomotopy group of the classifying space BG as the completion of the Burnside ring at the augmentation ideal.

In my talk I'll present an unstable analog of the Segal conjecture, identifying the group completion of the spaces of maps from BG to ∐_{n}(BΣ_{n})^{+}, for + Quillen's plus construction. The answer turns out to be, in a certain sense, an 'uncompletion' of Carlsson's result, and is related to Burnside rings studied for p-fusion systems.

## Lars Hesselholt: Algebraic K-theory of orders

It is a classical problem in algebraic K-theory to determine the algebraic K-groups of Z-orders in semi-simple Q-algebras, notably, group rings of finite groups. I will first explain how a joint result with Thomas Geisser reduces this to a problem in topological cyclic homology. I will then explain a new result joint with Ayelet Lindenstrauss and Michael Larsen which solves one half of the problem.

## Jacob Lurie: Rotation invariance in algebraic K-theory

For every triangulated category C, there is an associated K-group K_{0}(C). This group is generated by symbols [X], where X is an object of C, with relations [X] = [X'] + [X''] whenever there is a distinguished triangle X' → X → X''.

A simple consequence of these relations is that for every object X, we have [σX] = - [X] in K_{0}(C), so that [σ^{^2}X] = [X]. In this talk, I will discuss a "delooping" of this observation which involves a curious juxtaposition of algebraic and geometric ideas.

## Ib Madsen: Automorphisms of manifolds and graph homology

The lecture will describe the "stable" rational cohomology of the classifying spaces of homotopy automorphisms and block diffeomorphisms of 2d-dimensional "generalized surfaces". The result will be given in terms of Lie algebra cohomology and graph homology.

The lecture represents joint work with Alexander Berglund.

## Birgit Richter: Higher THH – towards integral results

In earlier work we identified higher topological Hochschild homology of rings of integers in number fields with coefficients in the corresponding residue fields. In this talk I will describe how to extend the calculation of higher THH of the integers with coefficients in a prime field to integral coefficients and will describe other related examples. This talk is based on joint work with Bjorn Dundas and Ayelet Lindenstrauss.

## Thomas Schick: Classification of positive scalar curvature metric: Index theory versus geometry

The Atiyah-Singer index theorem and its variants provide powerful tools to understand spin manifolds with invertible Dirac operators (can there be a metric with invertible Dirac operator, how many are there,...).

More or less "by accident" (via the Schroedinger-Weitzenboeck-Lichnerowicz method) this also helps us to understand spin manifolds which admit a Riemannian metric with everywhere positive scalar curvature.

Due to work of Gromov-Lawson and Stolz, in certain situations these two questions (invertibility of Dirac operator - positive scalar curvature) are essentially the same. But we also know that there are differences.

The talk will introduce the main classical constructions and results; and the focus on recent work which display the differences between these two questions.

## Andreas B. Thom: Topological methods to solve equations over groups

We present a large class of groups (no group known to be not in the class) that satisfy the Kervaire-Laudenbach Conjecture about; solvability of non-singular equations over groups. We also show that certain singular equations with coefficients over groups in this class are solvable.

Our method is inspired by seminal work of Gerstenhaber-Rothaus, which was the key to prove the Kervaire-Laudenbach Conjecture for residually finite groups. Exploring the structure of the p-local homotopy type of the projective unitary group, we manage to show that many singular equations with coefficients in unitary groups can be solved in the unitary group. This is joint work with Anton Klyachko.

## Daniel T. Wise: Nonpositive immersions and counting cycles

The "nonpositive immersion" property is a condition on a 2-complex X that generalizes being a surface. When X has this property, its fundamental group appears to have has some very nice properties which I will discuss. I will spend the remainder of the talk outlining a proof that the nonpositive immersion property holds for a 2-complex obtained by attaching a single 2-cell to a graph. This was proven recently with Joseph Helfer and also independently by Lars Louder and Henry Wilton.

## Günter M. Ziegler: Equivariant methods in discrete geometry: Problems and progress

In this lecture, I will discuss three different problems from Discrete Geometry,

- the Topological Tverberg Problem,
- the Colored Tverberg Problem, and
- the Grünbaum Hyperplane Problem.

These problems have many things in common:

- They are easy to state, and may look harmless,
- they have very nice and classical configuration spaces,
- they may be attacked by "Equivariant Obstruction Theory'',
- this solves the problems — but only partially,
- which leads us to ask more questions, look for new tools ...
- and this yields surprising new results.

(Joint work with Pavle Blagojević, Florian Frick, Albert Haase, and Benjamin Matschke)