4-manifold Discussion Session
dg-categories lecture II, continued (after Toen)
h-cobordism theorem -- end of proof
Stably Cayley groups
A linear algebraic group G over a field k is called a Cayley group if
it admits a Cayley map, i.e., a G-equivariant birational isomorphism
over k between the group variety G and the Lie algebra Lie(G). A
Cayley map can be thought of as a partial algebraic analogue of the
exponential map. A prototypical example is the classical "Cayley
transform" for the special orthogonal group SO_n defined by Arthur
Cayley in 1846. A k-group G is called stably Cayley if the product of
G with a split r-dimensional k-torus is Cayley for some r=0,1,2,....
These notions were introduced in 2006 by Lemire, Popov and Reichstein,
who classified Cayley and stably Cayley simple groups over an
algebraically closed field of characteristic zero. I will explain
their results and also some new results from my recent preprint with
Lemire, Kunyavskii and Reichstein, where we study Cayley and stably
Cayley reductive groups over an arbitrary field k of characteristic
zero.
My view on Mike's designs
The Homotopy Hypothesis
- Prove the best version(s) of the homotopy hypothesis
Relative Categories vs. Quasicategories
- Describe the Barwick-Kan theory of relative categories and its relationship to quasicategories.
- Discuss topic for next year’s seminar
Question and Answer Session
A few steps with Grothendieck derivators
The theory of derivators (going back to Grothendieck, Heller, and others) provides an axiomatic approach to homotopy theory. It adresses the problem that the rather crude passage from model categories to homotopy categories results in a serious loss of information. In the stable context, the typical defects of triangulated categories (non-functoriality of cone construction, lack of homotopy colimits) can be seen as a reminiscent of this fact. The simple but surprisingly powerful idea behind a derivator is that instead one should form homotopy categories of various diagram categories and also keep track of the calculus of homotopy Kan extensions.
In this talk we cover some basics of derivators culminating in a sketch proof that stable derivators provide an enhancement of triangulated categories. Possibly more important than this result itself are the techniques developed along the way as they lay the foundations for further research directions. The aim of this talk is to (hopefully) advertise derivators as a convenient, 'weakly terminal' approach to axiomatic homotopy theory.
Questions about generalized Morse theory
Geometric control and the Design
A Casson handle variant is topologically standard
Exploring Casson handles and their variants, reimbedding
The Dyson-Schwinger equations in the theory of computation
Arithmetic aspects of short random walks
We revisit a classical problem: how far does a random walk travel in a given
number of steps (of length 1, each taken along a uniformly random direction)?
Although such random walks are asymptotically well understood, surprisingly
little is known about the exact distribution of the distance after just a few
steps. For instance, the average distance after two steps is (trivially)
given by 4/pi; but what is the average distance after three steps?
In this talk, we therefore focus on the arithmetic properties of short random
walks and consider both the moments of the distribution of these distances as
well as the corresponding density functions. It turns out that the even
moments have a rich combinatorial structure which we exploit to obtain
analytic information. In particular, we find that in the case of three and
four steps, the density functions can be put in hypergeometric form and may be
parametrized by modular functions. Much less is known for the density in case
of five random steps, but using the modularity of the four-step case we are
able to deduce its exact behaviour near zero.
Time permitting, we will also discuss connections with Mahler measure.
h-cobordism theorem -- end of proof
IMPRS-Seminar on Morse theory
From the classification of Cuntz-Li-$C^*$-algebras to the group cohomology of crystallographic groups
The talk consists of three parts.
1.) Cuntz-Li associated to the ring of algebraic integers of a number
field a $C^*$-algebra.
We will discuss a preprint by Li and Lueck, where a full classification
of these $C^*$-algebras is presented.
2.) There is a Conjecture due to Adem-Ge-Pan-Petroysan about the
cohomology of certain crystallographic groups
with cyclic holonomy. In papers by Langer and Lueck the conjecture has
been disproved in general, but proved
in the case where the action of the holonomy group is free.
3.) We answer the question what the relation between these two on the
first glance unlinked problems is.
Actually, we needed the positive answer to 2.) in order to handle the
classification appearing in 1.)
This is done by an interesting tour from group cohomology, divided power
algebras, characteristic classes
of Charlap-Vasquez, classifying spaces for proper group actions,
equivariant topological $K$-theory of group $C^*$-algebras,
the Atiyah-Segal Completion Theorem for infinite discrete groups, the
Baum-Connes Conjecture, and the classification theorem of Kirchberg.
Casson handles
Segal Categories vs. Quasicategories
- Introduce quasicategories and describe Joyal-Teirney’s comparison with Segal 1-categories.
- Discuss timing of last lectures
English
Deutsch