Forthcoming events in this series
14:15
The space of graphs in Euclidean space.
Abstract
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.
11:00
2-dimensional extended Topological Quantum Field Theories and categorification
Abstract
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.
17:00
Canonical triangulations of quasifuchian convex cores
Abstract
Kleinian groups with an interesting deformation theory. I will show that the convex core of the quotient of hyperbolic 3-space by such a group admits a decomposition into ideal tetrahedra which is canonical in two completely independent senses: one combinatorial, the other geometric. One upshot is a proof of the Bending Lamination Conjecture for such groups.
15:45
On quasi-isometric embeddings of Lie groups into non-positively curved spaces
Abstract
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
14:45
Finitely generated simple groups of infinite commutator width.
Abstract
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.
16:00
Representation theory of affine Hecke algebras and K-theory
Abstract
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.
14:45
Topological rigidity and word-hyperbolic groups
Abstract
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.
14:45
Quadratic forms and cobordisms
Abstract
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.
14:45
Hydra groups
Abstract
I will describe a new family of groups exhibiting wild geometric and computational features in the context of their Conjugacy Problems. These features stem from manifestations of "Hercules versus the hydra battles."
This is joint work with Martin Bridson.
14:45
The arc complex is Gromov hyperbolic
Abstract
The arc complex is a combinatorial moduli space, very similar to the curve complex. Using the techniques of Masur and Minsky, as well as new ideas, I'll sketch the theorem of the title. (Joint work with Howard
Masur.) If time permits, I'll discuss an application to the cusp shapes of fibred hyperbolic three-manifolds. (Joint work with David Futer.)
We are planning to have dinner at Chiang Mai afterwards.
If anyone would like to join us, please can you let me know today, as I plan to make a booking this evening. (Chiang Mai can be very busy even on a Monday.)
14:45
Volumes of knot complements
Abstract
The complement of a knot or link is a 3-manifold which admits a geometric
structure. However, given a diagram of a knot or link, it seems to be a
difficult problem to determine geometric information about the link
complement. The volume is one piece of geometric information. For large
classes of knots and links with complement admitting a hyperbolic
structure, we show the volume of the link complement is bounded by the
number of twist regions of a diagram. We prove this result for a large
collection of knots and links using a theorem that estimates the change in
volume under Dehn filling. This is joint work with Effie Kalfagianni and
David Futer
14:45
Kazhdan and Haagerup properties from the viewpoint of median spaces, applications to the mapping class groups
Abstract
Both Kazhdan and Haagerup properties turn out to be related to actions
of
groups on median spaces and on spaces with measured walls.
These relationships allows to study the connection between Kazhdan
property (T) and the fixed point property
for affine actions on $L^p$ spaces, on one hand.
On the other hand, they allow to discuss conjugacy classes of subgroups
with property (T) in Mapping Class Groups. The latter result
is due to the existence of a natural structure of measured walls
on the asymptotic cone of a Mapping Class Group.
The talk is on joint work with I. Chatterji and F. Haglund
(first part), and J. Behrstock and M. Sapir (second part).
14:45