# Schedule of the Summer School

## Monday, June 19

10:00 - 10:35 |
Registration & Welcome coffee |

10:35 - 10:45 |
Opening remarks |

10:45 - 12:00 |
Andreas Thom: Algebraic vs. topological K-theory (Lecture 1) |

12:00 - 13:20 |
Lunch break |

13:20 - 14:35 |
Charles Weibel: K-theory of algebraic varieties (Lecture 1) |

14:45 - 16:00 |
Andreas Thom: Algebraic vs. topological K-theory (Lecture 2) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:45 |
Arthur Bartels: K-theory of group rings (Lecture 1) |

afterwards |
Reception |

## Tuesday, June 20

09:00 - 10:15 |
Charles Weibel: K-theory of algebraic varieties (Lecture 2) |

10:15 - 10:45 |
Group photo and coffee break |

10:45 - 12:00 |
Arthur Bartels: K-theory of group rings (Lecture 2) |

12:00 - 14:45 |
Lunch break |

14:45 - 16:00 |
Andreas Thom: Algebraic vs. topological K-theory (Lecture 3) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:45 |
Moritz Kerz: Algebraic K-theory and descent for blow-ups (Lecture 1) |

19:30 - |
Problem session |

## Wednesday, June 21

09:00 - 10:00 |
Charles Weibel: K-theory of algebraic varieties (Lecture 3) |

10:00 - 10:30 |
Coffee break |

10:30 - 11:30 |
Moritz Kerz: Algebraic K-theory and descent for blow-ups (Lecture 2) |

12:00 - 13:00 |
Marc Levine: Refined enumerative geometry (Lecture 1) |

13:00 - |
Lunch break, free time |

16:00 - 16:30 |
Tea and cake |

19:00 - |
Conference dinner at the Restaurant "Brauhaus Bönnsch" (Sterntorbrücke 4) |

## Thursday, June 22

09:00 - 10:15 |
Marc Levine: Refined enumerative geometry (Lecture 2) |

10:15 - 10:45 |
Coffee break |

10:45 - 12:00 |
Arthur Bartels: K-theory of group rings (Lecture 3) |

12:00 - 14:45 |
Lunch break |

14:45 - 16:00 |
Moritz Kerz: Algebraic K-theory and descent for blow-ups (Lecture 3) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:45 |
Marc Levine: Refined enumerative geometry (Lecture 3) |

19:30 - |
Problem session |

## Friday, June 23

09:00 - 10:00 |
Charles Weibel: K-theory of algebraic varieties (Lecture 4) |

10:00 - 10:30 |
Coffee break |

10:30 - 11:30 |
Marc Levine: Refined enumerative geometry (Lecture 4) |

12:00 - 13:00 |
Arthur Bartels: K-theory of group rings (Lecture 4) |

13:00 - |
Lunch break, end of summer school |

# Abstracts

## Arthur Bartels: K-theory of group rings

The Farrell-Jones Conjecture predicts that the K-theory of group rings RG can be computed in terms of K-theory of group rings RV where V varies over the collection of virtually cyclic subgroups of G.

In these talks I will discuss this conjecture, its formulation and methods of proof. Controlled algebra provides a bridge between the algebraic/homotopy theoretic formulation of the conjecture and the often very geometric proofs of instances of the conjecture. I plan to discuss this bridge in some detail.

Recorded Talks:

## Moritz Kerz: Algebraic K-theory and descent for blow-ups

It is well-known that special cases of descent along blow-ups for algebraic K-theory play an important role for calculating K-groups. For example Thomason proved descent for blow-ups in regular centers. However in general descent for K-theory of blow-ups does not hold. M. Morrow suggested a potential solution to this problem by considering pro-K-groups of all infinitesimal thickenings. In my lectures I will explain why this suggested form of pro-descent holds for all blow-ups of noetherian schemes and why this implies Weibel's conjecture on the vanishing of negative K-groups. This is joint work with F. Strunk and G. Tamme.

Recorded Talks:

## Marc Levine: Refined enumerative geometry

Lecture 1: Milnor-Witt sheaves, motivic homotopy theory and Chow-Witt groups

We review the Hoplins-Morel construction of the Milnor-Witt ring of a field, its extension to the unramified Milnor-Witt sheaves and its relation with the classical Grothendieck-Witt ring of quadratic forms. We discuss Morel's theorem identifying the 0th stable homotopy sheaf of the motivic sphere spectrum with the Milnor-Witt sheaves. We introduce the twisted Chow-Witt ring of a smooth scheme as the cohomology of the twisted Milnor-Witt sheaves and describe its functorial properties.

Lecture 2: Euler classes, Euler characteristics and Riemann-Hurwicz formulas

The Euler class of a vector bundle is defined in the twisted Chow-Witt ring and gives rise to an Euler characteristic for a smooth projective variety over a field k with values in the Grothendieck-Witt ring GW(k). Morel's theorem on the endomorphisms of the motivic sphere spectrum gives a categorically defined Euler characteristic in GW(k) for a larger class of smooth schemes over k. We show how these two classes agree when both are defined and derive a number of consequences, including a quadritc forms version of the classical Riemann-Hurwicz formula.

Lecture 3: Virtual fundamental classes in motivic homotopy theory

Using the formalism of algebraic stacks, Behrend-Fantechi define the intrinsic normal cone, its fundamental class in the Chow group and a virtual fundamental class [Z;φ]^{vir} ∈ CH_{r}(Z) associated to a perfect obstruction theory φ of virtual rank r. Using the six-functor formalism for the motivic stable homotopy category, as developed by Ayoub and Cisinski-Déglise, we define motivic analogs of these constructions (for Z a quasi-projective scheme or G-scheme), which recover the fundamental class and virtual fundamental class of Behrend-Fantechi as a special case. This makes available the "degree" of the motivic virtual fundamental class as an element of GW(k) for a perfect obstruction theory of virtual rank 0 and virtual determinant a square.

Lecture 4: Characteristic classes in Witt-cohomology

Classical enumerative geometry relies heavily on the theory of Chern classes of vector bundles and the splitting principle, which makes possible the computation of the Chern classes of associated bundles (symmetric powers, exterior powers, tensor products, etc.) in terms of the Chern classes of the "input" bundles. This does not seem to be possible in general for the Euler classes described in Lecture 2: a principle obstruction is the lack of classes which correspond to the Chern classes in degree less than the rank of the bundle. However, symplectic bundles and SL_{n}-bundles do admit a good theory of characteristic classes (Borel classes and Pontryagin classes, respectively) when one passes from the Milnor-Witt sheaves to the classical Witt sheaves. We will discuss the construction of the Borel and Pontryagin classes, and the results of A. Ananyevskiy computing the Witt cohomology of B SL_{n}, as well as his SL_{2}-splitting principle. Finally, we discuss a further reduction of an SL_{2}-bundle to an N_{T}-bundle, where N_{T} is the normalizer of the standard torus in SL_{2}, which reduces the computation to the situation reminiscent of the case of real SO(2)-bundles.

References:

[1] A. Ananyevskiy, The special linear version of the projective bundle theorem. Compos. Math. 151 (2015), no. 3, 461-501.

[2] J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles evanescents dans le monde motivique. I+II. Astérisque Nos. 314, 315 (2007).

[3] J. Barge and F. Morel, Groupe de Chow des cycles orientés et classe d'Euler des fibrés vectoriels. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 287-290.

[4] D.C. Cisinski and F. Déglise, Triangulated categories of mixed motives. Preprint 2012, arXiv:0912.2110.

[5] J. Fasel, Groupes de Chow-Witt. Mém. Soc. Math. Fr. (N.S.) No. 113 (2008).

[6] F. Morel, An introduction to A^{1}-homotopy theory. Contemporary developments in algebraic K-theory, 357-441, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.

[7] F. Morel, On the motivic π_{0} of the sphere spectrum. Axiomatic, enriched and motivic homotopy theory, 219-260, NATO Sci. Ser. II Math. Phys. Chem., 131, Kluwer Acad. Publ., Dordrecht, 2004.

[8] F. Morel and V. Voevodsky, A^{1}-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. No. 90 (1999), 45-143 (2001).

Recorded Talks:

## Andreas Thom: Algebraic vs. topological K-theory

I will start out giving a general introduction to K-theory and some basic applications to algebra, geometry and number theory. I will then discuss the comparison of algebraic and topological K-theory for specific rings of functional analytic type. In particular, I will discuss the algebraic K-theory of stable algebras and of rings of continuous functions. If time permits, I will prove a version of the homotopy invariance theorems for algebraic K-theory stabilised by a suitable operator ideal and for negative algebraic K-theory of rings of continuous functions.

## Charles Weibel: K-theory of algebraic varieties

Lecture 1 will present definitions for the Waldhausen K-theory of rings, varieties, additive and exact categories, and dg categories. The "+=wS." theorem for rings and simplicial rings will be mentioned. Basic properties, such as the localization, approximation and additivity properties will be described, and used to construct several localization sequences.

Lecture 2 will continue with localization sequences, using Karoubi filtrations. This includes the construction of negative/non-connective K-theory, and the K-theory of bounded complexes.

Lecture 3 will cover computational issues. This includes Zariski descent, K-cohomology and Bloch's formula, as well as excision for K_{0} and K_{1} of rings. Excision for varieties in low dimensions will also be covered.

Lecture 4 will survey computations for regular rings and smooth varieties. This includes motivic-to-K-theory methods, étale cohomology and regulators.