Geometry/Topology Seminar
Winter 2023
Thursdays 3:30-4:30pm, in
Ryerson 358
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- Thursday January 5 at 3:30PM-4:30PM in Ryerson 358
- Elia Portnoy, MIT
- A Generalized Isoperimetric Inequality via Thick Embeddings of Graphs
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Abstract: We prove a generalized isoperimetric
inequality for a domain diffeomorphic to a sphere that
replaces filling volume with k-dilation. Suppose
U is an open set in
Rn diffeomorphic to a Euclidean
n-ball. We show that in dimensions at least 4
there is a map from a standard Euclidean ball of radius
about vol(d U)1/(n-1)) to
U, with degree 1 on the boundary, and
(n-1)-dilation bounded by some constant only
depending on n. We also give an example in dimension 3 of an
open set where no such map with small
(n-1)-dilation can be found.
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- Thursday January 12 at 3:30PM-4:30PM in Ryerson 358
- Robert Young, NYU
- Metric differentiation and embeddings of the Heisenberg group
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Abstract: The Heisenberg group is the simplest
example of a noncommutative nilpotent Lie group. In this
talk, we will explore how that noncommutativity affects
geometry and analysis in the Heisenberg group. We will
describe why good embeddings of H must be
bumpy at many scales, how to study embeddings into
L1 by studying surfaces in
H, and how to construct a metric space
which embeds into L1 and
L4 but not in
L2. This talk is joint work with Assaf
Naor.
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- Thursday January 19 at 3:30PM-4:30PM in Ryerson 358
- Kejia Zhu, UIC
- Relatively geometric actions of complex hyperbolic lattices on CAT(0) cube complexes
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Abstract: We prove that for n ≥ 2, a
non-uniform lattice in PU(n,1) does not admit a relatively
geometric action on a CAT(0) cube complex, in the sense of
Einstein and Groves. As a consequence, we prove that if
K is a non-uniform lattice in a non-compact
semisimple Lie group G that admits a relatively
geometric action on a CAT(0) cube complex, then G
is isomorphic to SO(n,1). This work is joint with Corey
Bregman.
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- Thursday January 26 at 3:30PM-4:30PM in Ryerson 358
- Aaron Messerla, UIC
- Quasi-isometries of relatively hyperbolic groups with an elementary hierarchy
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Abstract: Sela introduced limit groups in his work
on the Tarski problem, and showed that each limit group has
a cyclic hierarchy. We prove that a class of relatively
hyperbolic groups, equipped with a hierarchy similar to the
one for limit groups, is closed under quasi-isometry.
Additionally, these groups share some of the algebraic
properties of limit groups. In this talk I plan to present
motivation for and introduce the class of groups studied, as
well as present some of the results for this class.
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- Thursday February 2 at 3:30PM-4:30PM in Ryerson 358
- Daniel Groves, UIC
- Drilling hyperbolic groups
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Abstract: The notion of "filling" of groups has been
very fruitful over recent years. Motivated by questions
around the Cannon Conjecture, I will explain how to take a
(residually finite) hyperbolic group with a two-sphere
boundary and "drill" it to produce a relatively hyperbolic
group with two-sphere boundary. This allows us to relate the
Cannon Conjecture to the relatively hyperbolic version. This
is joint work with Peter Haïssinsky, Jason Manning,
Damian Osajda, Alessandro Sisto, and Genevieve Walsh.
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- Thursday February 9 at 3:30PM-4:30PM in Ryerson 358
- Grigori Avramidi, MPI Bonn
- Homology growth, hyperbolization, and fibering
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Abstract: In this talk I will describe how a
classical conjecture of Hopf about the sign of the Euler
characteristic of non-positively curved manifolds leads to
the construction of odd, Gromov hyperbolic manifolds that do
not virtually fiber over a circle. Joint work with Boris
Okun and Kevin Schreve.
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- Thursday February 16 at 3:30PM-4:30PM in Ryerson 358
- Jonathan Zung, MIT
- Anosov flows and the pair of pants differential
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Abstract: In this talk, I'll explain how to
construct a chain complex associated to a transitive Anosov
flow on a 3-manifold. This story parallels the construction
of embedded contact homology for Reeb flows. I'll show
computations in simple examples and suggest some
conjectures.
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- Thursday February 23 at 3:30PM-4:30PM in Ryerson 358
- Andy Putman, Notre Dame
- The stable cohomology of the moduli space of curves with level structures
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Abstract: I will give an introduction to the
cohomology of the moduli space of curves, and then discuss a
recent theorem of mine saying that passing to certain finite
covers does not change its stable rational cohomology.
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- Thursday March 2 at 3:30PM-4:30PM in Ryerson 358
- Daniel Litt, Toronto
- Mapping class group dynamics via Hodge theory
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Abstract: Let Σ be an orientable
surface. Then the mapping class group of Σ
acts on the set of conjugacy classes of
r-dimensional complex representations of the
fundamental group of Σ. The question of
classifying finite orbits of this action goes back to work
of Painlevé, Fuchs, and Schlesinger in the beginning
of the 20th century. We show that if the genus of
Σ is at least r2,
then any representation with finite orbit has finite image,
answering questions of Kisin and Whang and resolving
conjectures of Esnault-Kerz and Budur-Wang. The proof relies
on non-abelian Hodge theory and input from the Langlands
program. This is joint work with Aaron Landesman.
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- Thursday March 9 at 3:30PM-4:30PM in Eckhart 207A
- James Farre, Heidelberg
- Tight maps to the circle and horocycle orbit closures in Z-covers
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Abstract: The horocycle flow on hyperbolic manifolds
has attracted considerable attention in the last century. In
the 30's, Hedlund proved that horocycle orbits are dense in
closed hyperbolic surfaces. Horocycle orbit closures have
been classified for geometrically finite surfaces, as well.
Our goal is to understand the topology and dynamics of
horocycle orbits in geometrically infinite hyperbolic
manifolds, where our understanding is extremely limited. In
this talk, I will discuss joint work with Or Landesberg and
Yair Minsky, where we give the first complete classification
of orbit closures for a class of Z-covers of closed
surfaces. I will explain why a complete classification may
be difficult to obtain. Our analysis is rooted in a
seemingly unrelated geometric optimization problem: finding
a best Lipschitz map to the circle. We then relate the
topology of horocycle orbit closures with the the dynamics
of the minimizing lamination (!) of maximal stretch.
Due to the high number of requests, we are no longer accepting speakers via self-invitations.
For questions, contact