Geometry Seminar
Spring 2021
Thursdays (and sometimes Tuesdays) 3:404:30pm, in
Ryerson 358 (currently on zoom)

 Thursday April 22 at 3:40PM5:00PM in Zoom
 Laurent Bartholdi, University of Göttingen
 Dimension series and homotopy groups of spheres

Abstract: The lower central series of a group
G is defined by 𝛾_{1} = G
and 𝛾_{n} =
[G,𝛾_{n1}]. The "dimension series",
introduced by Magnus, is defined using the integral group
algebra: 𝛿_{n} = {g: g is in the
n^{th} power of the augmentation ideal}.
It has been, for the last 80 years, a fundamental problem of
group theory to relate these two series. One always has
𝛿_{n} ≥ 𝛾_{n}, and a
conjecture by Magnus, with false proofs by Cohn, Losey,
etc., claims that they coincide; but Rips constructed an
example with 𝛿_{4} / 𝛾_{4}
cyclic of order 2. On the positive side, Sjogren showed that
𝛿_{n} / 𝛾_{n} is always a
torsion group, of exponent bounded by a function of
n. Furthermore, it was believed (and falsely
proven by Gupta) that only 2torsion may occur.
In joint work with Roman Mikhailov, we prove however that
every torsion abelian group occurs as a quotient
𝛿_{n} / 𝛾_{n}; this proves
that Sjogren's result is essentially optimal. Even more
interestingly, we show that this problem is intimately
connected to the homotopy groups
𝛑_{n}(S^{m}) of spheres; more
precisely, the quotient 𝛿_{n} /
𝛾_{n} is related to the difference between
homotopy and homology. We may explicitly produce
ptorsion elements starting from the
orderp element in the homotopy group
𝛑_{2p}(S^{2}) due to Serre.

 Thursday April 29 at 4:10PM5:00PM in Zoom
 Joshua Greene, Boston College
 Peg Problems

Abstract: Toeplitz asked in 1911 whether every
Jordan curve in the Euclidean plane contains the vertices of
a square. The problem remains open, but it has given rise to
many interesting variations and partial results. I will
describe some of these and the proof of a result which is
best possible when the curve is smooth: for any four points
on the circle and for any smooth Jordan curve in the
Euclidean plane, there exists an orientationpreserving
similarity which carries the four points onto the curve. The
proof involves symplectic geometry in a surprising way.
Joint work with Andrew Lobb.

 Thursday May 13 at 4:10PM5PM in Zoom
 Kasia Jankiewicz, University of Chicago
 Background on growth rates and hyperbolic group

Abstract: This is a preliminary talk for the talk by
Koji Fujiwara. We will provide background on the growth
rates of groups and hyperbolic groups.

 Thursday May 13 at 5:10PM6PM in Zoom
 Koji Fujiwara, Kyoto University
 The rates of growth in a hyperbolic group

Abstract: I will discuss the set of rates of growth
of a hyperbolic group with respect to all its finite
generating sets. It turns out that the set is wellordered,
and that every real number can be the rate of growth of at
most finitely many generating sets up to automorphism of the
group. This is a joint work with Sela. If time permits, I
will discuss other families of groups.

 Thursday May 20 at 4:10PM5:00PM in Zoom
 Danny Calegari, University of Chicago
 An introduction to slice knots

Abstract: A knot K in the 3sphere is (smoothly)
slice if it bounds a smooth embedded disk in the 4ball;
more generally, the *slice genus* is the least genus of a
smooth embedded surface in the 4ball K bounds. The
signature of a slice knot is zero, and its Alexander
polynomial factors as a product
f(t)f(t^{1}). Detecting whether a knot
is slice is difficult; Conway’s knot was recently shown to
be slice after 50 years by Lisa Piccirillo, using
Rasmussen’s sinvariant. We shall define this invariant,
and how it gives a bound on slice genus. This is an
expository talk.

 Thursday May 27 at 4:10PM5:00PM in Zoom
 Ciprian Manolescu, Stanford University
 Khovanov homology and the search for exotic 4spheres

Abstract: A wellknown strategy to disprove the
smooth 4D Poincare conjecture is to find a knot that bounds
a disk in a homotopy 4ball but not in the standard 4ball.
Freedman, Gompf, Morrison and Walker suggested that
Rasmussen’s invariant from Khovanov homology could be
useful for this purpose. I will describe three recent
results about this strategy: that it fails for Gluck twists
(joint work with Marengon, Sarkar and Willis); that an
analogue works for other 4manifolds (joint work with
Marengon and Piccirillo); and that 0surgery homeomorphisms
provide a large class of potential examples (joint work with
Piccirillo).

 Thursday June 17 at 3:004:00pm in Zoom
 Bruno Martelli, Università di Pisa
 Hyperbolic 5manifolds that fiber over the circle

Abstract: We show that the existence of hyperbolic
manifolds fibering over the circle is not a phenomenon
confined to dimension 3 by exhibiting some examples in
dimension 5. More generally, there are hyperbolic manifolds
with perfect circlevalued Morse functions in all dimensions
n ≤ 5, a fact that leads us naturally to ask
whether this may hold for any n. One consequence of this
result is the existence of hyperbolic groups with
finitetype subgroups that are not hyperbolic. The main tool
is Bestvina  Brady theory applied to some hyperbolic
nmanifolds that decompose very nicely into rightangled
polytopes, enriched with the combinatorial game recently
introduced by Jankiewicz, Norin and Wise. These are joint
works with Battista, Italiano, and Migliorini.
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact