Thursdays (and sometimes Tuesdays) 3:40-4:30pm, in
- Thursday January 16 at 3:40-4:30pm in Ry 358
- Michał Marcinkowski, Wrocław University/Univerity of Chicago
- Bounded cohomology of transformation groups
Abstract: Let M be a finite-volume Riemannian
manifold and let \mu be the measure induced by the volume
form. Denote by G the group of all \mu-preserving
homeomorphisms of M isotopic to the identity. It is
well-known that the second bounded cohomology of G is
infinite dimensional due to existence of quasimorphisms on G
(Gambaudo-Ghys, Polterovich). In this talk I will explain
how to construct bounded classes in higher dimensions. As an
application, we will show that under certain conditions on
the fundamental group of M, the third bounded cohomology of
G is infinite dimensional. If time permits, I will discus
how this construction can be used to construct invariants of
foliated fibre bundles. It is a joint work with Michael
Brandenbursky and Martin Nitsche.
- Thursday January 30 at 3:40-4:30pm in Ry 358
- Junho Peter Whang, MIT
- Finiteness of local systems through dynamics and arithmetic
Abstract: A local system on a Riemann surface may be
viewed as a point in a moduli space (which carries mapping
class group dynamics), or as solutions to a differential
equation (which can be studied arithmetically). In this
talk, we present two results that capture intrinsic
properties of a local system from the different viewpoints.
First, on a surface of genus two or higher, a semisimple
special linear rank two local system is finite (resp.
bounded) if and only if its mapping class group orbit is
finite (resp. bounded) in moduli space (joint with Biswas,
Gupta, and Mj). Second, the Grothendieck-Katz p-curvature
conjecture, which ties mod p behaviors of differential
equations to finiteness of monodromy, is true in rank two
for generic Riemann surfaces (joint with Patel and Shankar).
A common ingredient we shall highlight in these works is a
type of local-to-global lemma for rank two local systems on
surfaces, proved using combinatorics of simple loops and
basic algebraic number theory.
- Thursday March 05 at 3:40-4:30pm in Ry 358
- Tam Nguyen Phan, Bonn
- A funny encounter with immersions of spheres
Abstract: It is a common topology homework exercise
to prove that any embedding of a closed, orientable
n-manifold M, such as the n-sphere, into Rn+1, is
separating (i.e. has disconnected complement). However, this
homework problem has never been posed with “embedding”
replaced by “immersion”, probably because it seems at
ﬁrst glance to complicate the problem for no good reason.
But there is actually an interesting reason. This talk is
based on joint work with Michael Freedman.
- Thursday March 12 at 3:40-4:30pm in Ry 358
- Maggie Miller, Princeton University
- Light bulbs in 4-manifolds
Abstract: In 2017, Gabai proved the light bulb
theorem, showing that if R and R' are
2-spheres homotopically embedded in a 4-manifold with a
common dual, then with some condition on 2-torsion in
\pi1(X) one can conclude that
R and R’ are smoothly isotopic.
Schwartz later showed that this 2-torsion condition is
necessary, and Schneiderman and Teichner then obstructed the
isotopy whenever this condition fails. I showed that when
R' does not have a dual, we may still conclude
the spheres are smoothly concordant. I will talk about these
various definitions and theorems as well as joint work with
Michael Klug generalizing the result on concordance to the
situation where R has an immersed dual (and
R’ may have none), which is a common condition
in 4-dimensional topology.
Due to the high number of requests, we are no longer accepting speakers via self-invitations.
For questions, contact