Geometry/Topology Seminar
Fall 2022
Thursdays 3:30-4:30pm, in
Eckhart 207A
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- Thursday October 6 at 3:30PM-4:30PM in Eckhart 207A
- Sahana Vasudevan, UChicago
- Distribution of triangulated surfaces in moduli space
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Abstract: Triangulated surfaces are Riemann surfaces
formed by gluing together equilateral triangles. They are
also the Riemann surfaces defined over the algebraic
numbers. Brooks, Makover, Mirzakhani and many others proved
results about the geometric properties of random large genus
triangulated surfaces, and similar results about the
geometric properties of random large genus hyperbolic
surfaces. These results motivated the question: how are
triangulated surfaces distributed in the moduli space of
Riemann surfaces, quantitatively? I will talk about my work
proving that triangulated surfaces are well distributed in
moduli space as the genus becomes large. In particular, the
number of triangulated surfaces in any Teichmüller unit
ball in moduli space is at most exponential in the number of
triangles, independent of the genus. I will explain some of
the ideas in the proof of these statements.
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- Thursday October 13 at 3:30PM-4:30PM in Eckhart 207A
- Ben Lowe, UChicago
- Beyond Almost Fuchsian Space
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Abstract: Quasi-Fuchsian Manifolds are hyperbolic
3-manifolds homeomorphic to a surface ×
R with well-behaved ends. They provide a
setting where the underlying topology is simple but the
geometry can be very intricate. Uhlenbeck defined a
sub-class of Quasi-Fuchsian manifolds M (called
almost Fuchsian) that contain closed minimal surfaces with
maximum principal curvature less than 1, proved that any
such minimal surface in M is unique, and gave a
way of parametrizing this sub-class in terms of data
attached to this unique minimal surface. I will talk about
joint work with Zeno Huang that goes to the boundary of the
class of Quasi-Fuchsian manifolds covered by Uhlenbeck's
work, as well as difficulties encountered when one tries to
go an epsilon further. In particular, I will describe some
evidence that the results from Uhlenbeck's original paper
are much sharper than one might at first have expected.
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- Thursday October 20 at 3:30PM-4:30PM in Eckhart 207A
- Aaron Landesman, Harvard
- The algebraic geometry of the Putman-Wieland conjecture
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Abstract: Suppose we are given an unramified
covering Σg' \to
Σg of topological surfaces with
g ≥ 2. The Putman-Wieland conjecture predicts
that the action of the mapping class group of
Modg,1 on
H1(Σg') has no
nonzero fixed vectors. This is closely related to Ivanov's
conjecture, predicting that finite index subgroups of the
mapping class group (for g ≥ 3) have finite
abelianization. Based on joint work with Daniel Litt, we
describe how to establish many new cases of the
Putman-Wieland conjecture by studying the derivative of an
associated period map
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- Thursday October 27 at 3:30PM-4:30PM in Eckhart 207A
- Malavika Mukundan, Michigan
- Dynamical approximation of entire functions
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Abstract: Post singularly finite holomorphic
functions are entire functions for which the forward orbit
of the set of critical and asymptotic values is finite.
Motivated by the work of
Bodelón-Devaney-Hayes-Roberts-Goldberg-Hubbard on
approximating exponential functions dynamically by
unicritical polynomials, we ask the following question:
Given a post singularly finite entire function f,
can f be realised as the locally uniform limit of
a sequence of post critically finite polynomials? In joint
work (in progress) with Nikolai Prochorov and Bernhard
Reinke, we show how we may answer this question in the
affirmative.
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- Thursday November 3 at 3:30PM-4:30PM in Eckhart 207A
- Fernando Al Assal, Yale
- Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold
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Abstract: Let M be a closed hyperbolic
3-manifold and let Gr(M) be its 2-plane Grassmann
bundle. We will discuss the following result: the weak-*
limits of the probability area measures on Gr(M)
of pleated or minimal closed connected essential
K-quasifuchsian surfaces as K goes to
1 are all convex combinations of the probability area
measures of the immersed closed totally geodesic surfaces of
M and the probability volume (Haar) measure of
Gr(M).
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- Thursday November 10 at 3:30PM-4:30PM in Eckhart 207A
- Alex Kapiamba, Michigan
- Elephants all the way down: the near-parabolic geometry of the Mandelbrot set
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Abstract: Understanding the geometry of the
Mandelbrot set, which records dynamical information about
every quadratic polynomial, has been a central task in
holomorphic dynamics over the past forty years. Near
parabolic parameters, the structure of the Mandelbrot set is
asymptotically self-similar and resembles a parade of
elephants. Near parabolic parameters on these
"elephants”, the Mandelbrot set is again self-similar
and resembles another parade of elephants. This phenomenon
repeats infinitely, and we see different parades of
elephants at each scale. In this talk, we will explore the
implications of controlling the geometry of these elephants.
In particular, we will partially answer Milnor's conjecture
on the optimality of the Yoccoz inequality, and see
potential connections to the local connectivity of the
Mandelbrot set.
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- Thursday November 17 at 3:30PM-4:30PMM in Zoom (see announcement email for link)
- Nikita Selinger, UAB
- Parameter space of symmetric cubic polynomials
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Abstract: Since the space of all cubic polynomials
is (complex) two-dimensional and thus too difficult to
comprehend, we study a one-dimensional slice of it: the
space of all cubic symmetric polynomials of the form
f(z)=z3+a2 z. Thurston has
built a topological model for the space of quadratic
polynomials f(z)=z2+c by introducing
the notion of quadratic invariant laminations. In the spirit
of Thurston's work, we parametrize the space of cubic
symmetric laminations and create a model for the space of
cubic symmetric polynomials. This is a joint work with
Alexander Blokh, Lex Oversteegen, Vladlen Timorin, and
Sandeep Vejandla.
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- Thursday November 24 at - in -
- Thanksgiving, no seminar
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Abstract: -
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- Thursday December 1 at 3:30PM-4:30PM in Eckhart 207A
- Roman Sauer, KIT
- Higher property T of arithmetic lattices
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Abstract: This is based on joint work with Uri
Bader. We prove that arithmetic lattices in a semisimple Lie
group G satisfy a higher-degree version of property T below
the rank of G. The proof relies on functional analysis and
geometric group theory (polynomiality of higher Dehn
functions of arithmetic lattices below the rank). If time
permits, I will explain some applications to the cohomology
and stability of arithmetic groups (the latter being joint
work with Alex Lubotzky and Shmuel Weinberger).
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- Thursday December 8 at - in -
- End of term, no seminar
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Abstract: -
Due to the high number of requests, we are no longer accepting speakers via self-invitations.
For questions, contact