Forthcoming events in this series
In this lecture, I will illustrate the cobordism hypothesis by presenting some examples. Exact content to be determined, depending on the interests of the audience.
In this lecture, I will give a more precise statement of the Baez-Dolan cobordism hypothesis, which gives a description of framed bordism (higher) categories by a universal mapping property. I'll also describe some generalizations of the cobordism hypothesis, which take into account the structure of diffeomorphism groups of manifolds and which apply to manifolds which are not necessarily framed.
In this lecture, I'll give an overview of some ideas from higher category theory which are needed to make sense of the Baez-Dolan cobordism hypothesis. If time permits, I'll present Rezk's theory of complete Segal spaces (a model for the theory of higher categories in which most morphisms are assumed to be invertible) and explain how bordism categories can be realized in this framework.
In this lecture, I will review Atiyah's definition of a topological quantum field theory. I'll then sketch the definition of a more elaborate structure, called an "extended topological quantum field theory", and describe a conjecture of Baez and Dolan which gives a classification of these extended theories.
I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations.
This is joint work with Rob Meyerhoff.
The notion of a sutured 3-manifold was introduced by Gabai. It is a powerful tool in 3-dimensional topology. A few years ago, Andras Juhasz defined an invariant of sutured manifolds called sutured Floer homology.
I'll discuss a simpler invariant obtained by taking the Euler characteristic of this theory. This invariant turns out to have many properties in common with the Alexander polynomial. Joint work with Stefan Friedl and Andras Juhasz.
There is a well-known relationship between the theory of formal group schemes and stable homotopy theory, with Ravenel's chromatic filtration and the nilpotence theorem of Hopkins, Devinatz and Smith playing a central role. It is also familiar that one can sometimes get a more geometric understanding of homotopical phenomena by examining how they interact with group actions. In this talk we will explore this interaction from the chromatic point of view.
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.
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.
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.)
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
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).