Geometry/Topology Seminar
Spring 2023
Thursdays 3:30-4:30pm, in
Ryerson 358
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- Thursday March 30 at 3:30-4:30pm in Ry 358
- Jenya Sapir, Binghamton
- Geometry of geodesic currents
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Abstract: The space of projective, filling currents
PFC(S) contains many structures relating to a closed, genus
g surface S. For example, it contains the set of all closed
curves on S, as well as an embedded copy of Teichmuller
space, and many other spaces of metrics on S. We will
discuss a structure theorem that compares each filling
current with a suitably chosen point in Teichmuller space.
We will then use this structure theorem to explore the
geometry of PFC(S) under an extension of the Thurston
metric.
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- Thursday April 6 at 3:30-4:30pm in Ry 358
- Jonathan Johnson, Oklahoma State
- Bi-orderability of link groups
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Abstract: In the past couple of decades, group
orderability has played a surprising role in the study of
3-manifolds. A group is bi-orderable when there is a total
ordering of its elements invariant under group
multiplication. In this talk, we will discuss some recent
developments in the bi-orderability of link exteriors and
explore connections to the Alexander polynomials of links
and the topology of the branched cyclic covers of the link.
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- Thursday April 20 at 3:30-4:30pm in Ry 358
- Paul Apisa, Wisconsin
- Hurwitz Spaces, Hecke Actions, and Orbit Closures in Moduli Space
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Abstract: The moduli space of Riemann surfaces is a
space whose points correspond to the ways to endow a surface
with a hyperbolic metric or, equivalently, complex
structure. Geodesic flow on moduli space can be used to
generate an action of GL(2, R) on its cotangent bundle.
While work of Eskin, Mirzakhani, Mohammadi, and Filip
implies that GL(2, R) orbit closures are varieties, the
question of which ones occur is wide open. Aside from two
well-understood constructions (taking loci of branched
covers and subloci of rank two orbit closures) there are
only 3 known families of orbit closures: the Bouw-Moller
curves, the Eskin-McMullen-Mukamel-Wright (EMMW) examples,
and 2 sporadic examples. Building on ideas of
Delecroix-Rueth-Wright, I will describe work showing that
the Bouw-Moller and EMMW examples can be constructed using
just the representation theory of finite groups. The main
idea is to connect these examples to Hurwitz spaces of
G-regular covers of the sphere for an appropriate finite
group G. In the end, I will describe a construction that
inputs a finite group G and a set of generators satisfying a
combinatorial condition and outputs a GL(2, R) orbit closure
in moduli space. No background on dynamics, Hurwitz spaces,
or Riemann surfaces will be assumed.
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- Thursday April 27 at 4:30-5:30pm in Ry 358
- Tarik Aougab, Haverford College
- Currents with corners
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Abstract: We introduce the notion of a geodesic
current with corners, a generalization of a geodesic current
in which there are singularities (the “corners”)
at which invariance under the geodesic flow can be violated.
Recall that the set of closed geodesics is, in the
appropriate sense, dense in the space of geodesic currents;
the motivation behind currents with corners is to construct
a space in which graphs on S play the role of closed curves.
Another fruitful perspective is that geodesic currents
reside “at infinity” in the space of currents
with corners, in the sense that their (non-existent) corners
have been pushed out to infinity. As an application, we
count (weighted) triangulations in a mapping class group
orbit with respect to (weighted) length, and we obtain
asymptotics that parallel results of Mirzakhani,
Erlandsson-Souto, and Rafi-Souto for curves. This represents
joint work with Jayadev Athreya.
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- Thursday May 4 at 3:30-4:30pm in Ry 358
- Jean-Pierre Mutanguha, Princeton/IAS
- Canonical forms for free group automorphisms
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Abstract: The Nielsen-Thurston theory of surface
homeomorphisms can be thought of as a surface analogue to
the Jordan canonical form. I will discuss my progress in
developing a similar canonical form for free group
automorphisms. (Un)Fortunately, free group automorphisms can
have arbitrarily complicated behaviour. This is a
significant barrier to translating arguments that worked for
surfaces into the free group setting; nevertheless, the
overall ideas/strategies do translate!
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- Thursday May 11 at 3:30-4:30pm in Ry 358
- Hannah Alpert, Auburn
- Combining Papasoglu's trick with simplicial volume
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Abstract: In 1983 Gromov proved the systolic
inequality: if M is a closed, essential
n-dimensional Riemannian manifold where every
loop of length 2 is null-homotopic, then the
volume of M is at least a constant depending only
on n. He also proved a version that depends on
the simplicial volume of M, saying that if the
simplicial volume is large, then the lower bound on volume
becomes proportional to the simplicial volume divided by the
n-th power of its logarithm. Nabutovsky showed in
2019 that Papasoglu's method of area-minimizing separating
sets recovers the systolic inequality and improves its
dependence on n. We introduce simplicial volume
to the proof, recovering the statement that the volume is at
least proportional to the square root of the simplicial
volume.
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- Thursday May 18 at 3:30-4:30pm in Ry 358
- Greg Chambers, Rice
- On the relative isoperimetric problem for the cube
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Abstract: It is conjectured that the minimizers for the relative
isoperimetric problem in the cube are spheres located at corners, tubes
located along edges, or slabs parallel to faces. Constraining ourselves
to sets whose unit normals (where defined) are in the coordinate
directions, we prove an analogous conjecture. We will also describe
some history of the problem as well as some interpretations of this new
result. This talk is based on joint work with Lawrence Mouillé.
Due to the high number of requests, we are no longer accepting speakers via self-invitations.
For questions, contact