Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.
I will prove that the knot Floer homology group
HFK-hat(K, g-1) for a genus g fibered knot K is isomorphic to a
variant of the fixed points Floer homology of an area-preserving
representative of its monodromy. This is a joint work with Gilberto
Spano.
When studying a group, it is natural and often useful to try to cut it up
onto simpler pieces. Sometimes this can be done in an entirely canonical
way analogous to the JSJ decomposition of a 3-manifold, in which the
collection of tori along which the manifold is cut is unique up to isotopy.
It is a theorem of Brian Bowditch that if the group acts nicely on a metric
space with a negative curvature property then a canonical decomposition can
be read directly from the large-scale geometry of that space. In this talk
we shall explore an algorithmic consequence of this relationship between
the large-scale geometry of the group and is algebraic decomposition.
My talk is based on joint work with Claire Debord (Univ. Auvergne).
We will explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory.
In our approach -originating from ideas of Connes, various examples of Lie groupoids
- allow to generalize index problems,
- can be used to construct the index of pseudodifferential operators without using the pseudodifferential calculus,
- give rise to proofs of index theorems,
- can be used to construct the pseudodifferential calculus.
The problem of "unbounded rank expanders" asks
whether we can endow a system of generators with a sequence of
special linear groups whose degrees tend to infinity over quotient rings
of Z such that the resulting Cayley graphs form an expander family.
Kassabov answered this question in the affirmative. Furthermore, the
completely satisfactory solution to this question was given by
Ershov and Jaikin--Zapirain (Invent. Math., 2010); they proved
Kazhdan's property (T) for elementary groups over non-commutative
rings. (T) is equivalent to the fixed point property with respect to
actions on Hilbert spaces by isometries.
We provide a new framework to "upgrade" relative fixed point
properties for small subgroups to the fixed point property for the
whole group. It is inspired by work of Shalom (ICM, 2006). Our
main criterion is stated only in terms of intrinsic group structure
(but *without* employing any form of bounded generation).
This, in particular, supplies a simpler (but not quantitative)
alternative proof of the aforementioned result of Ershov and
Jaikin--Zapirain.
If time permits, we will discuss other applications of our result.
