Forthcoming events in this series
Baumslag-Solitar groups are very simple groups which are not Hopfian (they are isomorphic to proper quotients). I will discuss these groups, as well as their obvious generalizations, with emphasis on their automorphisms and their generating sets
We study non-commutative analogues of Serre-ï¬~Abrations in topology. We shall present several examples of such ï¬~Abrations and give applications for the computation of the K-theory of certain C*-algebras. (Joint work with Ryszard Nest and Herve Oyono-Oyono.)
$PD$-complexes model the homotopy theory of manifolds.
In dimension 3, the unique factorization theorem holds to the extent that a $PD_3$-complex is a connected sum if and only if its fundamental group is a free product, and the indecomposables are aspherical or have virtually free fundamental group [Tura'ev,Crisp]. However in contrast to the 3-manifold case the group of an indecomposable may have infinitely many ends (i.e., not be virtually cyclic). We shall sketch the construction of one such example, and outline some recent work using only group theory that imposes strong restrictions on any other such examples.
We classify 2-dimensional polygonal complexes that are simply connected, platonic (in the sense that they admit a flag-transitive group of symmetries) and simple (in the sense that each vertex link is a complete graph). These are a natural generalization of the 2-skeleta of simple polytopes.
Our classification is complete except for some existence questions for complexes made from squares and pentagons.
(Joint with Tadeusz Januszkiewicz, Raciel Valle and Roger Vogeler.)
A polygonal complex $X$ is Platonic if its automorphism group $G$ acts transitively on the flags (vertex, edge, face) in $X$. Compact examples include the boundaries of Platonic solids. Noncompact examples $X$ with nonpositive curvature (in an appropriate sense) and three polygons meeting at each edge were classified by \'Swi\c{a}tkowski, who also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete. For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete. A Fuchsian building is a two-dimensional also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete. For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete. A Fuchsian building is a two-dimensional hyperbolic building. We study lattices in automorphism groups of Platonic complexes and Fuchsian buildings. Using similar methods for both cases, we construct uniform and nonuniform lattices in $G=Aut(X)$. We also show that for some $X$ the set of covolumes of lattices in $G$ is nondiscrete, and that $G$ admits lattices which are not finitely generated. In fact our results apply to the larger class of Davis complexes, which includes examples in dimension > 2.
It is a classical result that the Alexander polynomial of a fibered knot has to be monic. But in general the converse does not hold, i.e. the Alexander polynomial does not detect fibered knots. We will show that the collection of all twisted Alexander polynomials (which are a natural generalization of the ordinary Alexander polynomial) detect fibered 3-manifolds.
As a corollary it follows that given a 3-manifold N the product S1 x N is symplectic if and only if N is fibered.
A graph in R^n is a closed subset that locally looks like R (edges) or like a wedge of half intervals (vertices). I will describe a topology on the space of all such graphs and determine its homotopy type. This is one step in determining the homology of Aut(F_n), the automorphism group of a free group, in the limit where n goes to infinity.
A 2-dimensional Topological Quantum Field Theory (TQFT) is a symmetric monoidal functor from the category of 2-dimensional cobordisms to the category of vector spaces. A classic result states that 2d TQFTs are classified by commutative Frobenius algebras. I show how to extend this result to open-closed TQFTs using a class of 2-manifolds with corners, how to use the Moore-Segal relations in order to find a canonical form and a complete set of invariants for our cobordisms and how to classify open-closed TQFTs algebraically. Open-closed TQFTs can be used to find algebraic counterparts of Bar-Natan's topological extension of Khovanov homology from links to tangles and in order to get hold of the braided monoidal 2-category that governs this aspect of Khovanov homology. I also sketch what open-closed TQFTs reveal about the categorical ladder of combinatorial manifold invariants according to Crane and Frenkel.
references:
1] A. D. Lauda, H. Pfeiffer:
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras,
Topology Appl. 155, No. 7 (2008) 623-666, arXiv:math/0510664
2] A. D. Lauda, H. Pfeiffer: State sum construction of two-dimensional open-closed Topological Quantum Field Theories,
J. Knot Th. Ramif. 16, No. 9 (2007) 1121-1163,arXiv:math/0602047
3] A. D. Lauda, H. Pfeiffer: Open-closed TQFTs extend Khovanov homology from links to tangles, J. Knot Th. Ramif., in press, arXiv:math/0606331.
I will give a characterization of connected Lie groups admitting a quasi-isometric embedding into a CAT(0) metric space. The proof relies on the study of the geometry of their asymptotic cones
If $G$ is a group and $g$ an element of the derived subgroup $[G,G]$, the commutator length of $g$ is the least positive integer $n$ such that $g$ can be written as a product of $n$ commutators. The commutator width of $G$ is the maximum of the commutator lengths of elements of $[G,G]$. Until 1991, to my knowledge, it has not been known whether there exist simple groups of commutator width greater than $1$. The same question for finite simple groups still remains unsolved. In 1992, Jean Barge and Étienne Ghys showed that the commutator width of certain simple groups of diffeomorphisms is infinite. However, those groups are not finitely generated. Finitely generated infinite simple groups of infinite commutator width can be constructed using "small cancellations." Additionally, finitely generated infinite boundedly simple groups of arbitrarily large (but necessarily finite) commutator width can be constructed in a similar way.
In recent joint work with Maarten Solleveld we could give a complete classification of the set the irreducible discrete series characters of affine Hecke algebras (including the non simply-laced cases). The results can be formulated in terms of the K-theory of the Schwartz completion of the Hecke algebra. We discuss these results and some related conjectures on formal dimensions and on elliptic characters.
The Borel conjecture asserts that aspherical manifolds are topologically rigid, i.e., every homotopy equivalence between such manifolds is homotopic to a homeomorphism. This conjecture is strongly related to the Farrell-Jones conjectures in algebraic K- and L-theory. We will give an introduction to these conjectures and discuss the proof of the Borel conjecture for high-dimensional aspherical manifolds with word-hyperbolic fundamental groups.
Taking the intersection form of a 4n-manifold defines a functor from a category of cobordisms to a symmetric monoidal category of quadratic forms. I will present the theory of the Maslov index and some higher-categorical constructions as variations on this theme.