Geometry/Topology Seminar
Fall 2015
Thursdays (and sometimes Tuesdays) 3-4pm, in
Eckhart 308
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- Tuesday October 13 at 3-4pm in Eck 308
- Igor Belegradek, Georgia Tech
- Spaces of nonnegatively curved metrics
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Abstract: I will explain how to determine
homeomorphism type of the space of complete nonnegatively
curved metrics on the plane and the 2-sphere in the smooth
and in the Holder topologies. If time allows I will also
survey what is known about the space of complete
nonnegatively curved metrics on higher dimensional
manifolds.
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- Tuesday October 20 at 4:30pm (pretalk at 3:00pm) in Eck 203
- Alexander Kupers, Stanford
- En cells and homological stability (joint seminar with algebraic topology)
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Abstract: When studying objects with additional
algebraic structure, e.g. algebras over an operad, it can be
helpful to consider cell decompositions adapted to these
algebraic structures. I will talk about joint work with
Jeremy Miller on the relationship between En-cells and
homological stability. Using this theory, we prove a
local-to-global principle for homological stability, as well
as give a new perspective on homological stability for
various spaces including symmetric products and spaces of
holomorphic maps.
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- Thursday October 22 at 3-4pm in Eck 308
- Andrew Sanders, UIC
- Complex deformations of Anosov representations
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Abstract: An Anosov representation of a hyperbolic
surface group is a homomorphism from the surface group into
a semi-simple Lie group which satisfies a certain dynamical
property: from this property one deduces that Anosov
representations are discrete, faithful and the set of all
Anosov representations is an open subset of the space of all
homomorphisms. In recent years, Guichard-Weinhard produced
examples of co-compact domains of discontinuity for Anosov
representations, which lie in various homogeneous spaces,
thus giving an answer to the question of whether or not
Anosov representations appear as monodromies of locally
homogeneous geometric structures on manifolds. In this talk,
which comprises joint work with David Dumas, I will discuss
some of the complex analytic features of these locally
homogeneous geometric manifolds in the case the relevant
homogeneous space is a generalized flag variety. In
particular, we will give sufficient conditions to compute
the space of all infinitesimal deformations of the complex
manifold underlying these manifolds. Time permitting, we
will discuss the problem of deforming a pair (M,Z) where M
is a holomorphic locally homogeneous manifold and Z is a
complex sub-manifold and indicate an application to the
study of Anosov representations.
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- Tuesday November 03 at 4:30 PM (pre-talk at 3:00) in Eck 203
- Ben Knudsen, Northwestern
- Rational homology of configuration spaces via factorization homology
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Abstract: The study of configuration spaces is
particularly tractable over a field of characteristic zero,
and there has been great success over the years in producing
complexes simple enough for explicit computations, formulas
for Betti numbers, and descriptive results. I will discuss
recent work identifying the rational homology of the
configuration spaces of an arbitrary manifold with the
homology of a Lie algebra constructed from its cohomology.
The aforementioned results follow immediately from this
identification, albeit with hypotheses removed; in
particular, one obtains a new, elementary proof of
homological stability for configuration spaces.
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- Tuesday November 03 at 3-4PM in Eck 206
- Tullia Dymarz, Univ. Wisconsin
- Non-rectifiable Delone sets in amenable groups
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Abstract: In 1998 Burago-Kleiner and McMullen
constructed the first examples of coarsely dense and
uniformly discrete subsets of Rn that are not
biLipschitz equivalent to the standard lattice
Zn. We will show how to find such sets inside
certain other solvable Lie groups. The techniques involve
combining ideas from Burago-Kleiner with quasi-isometric
rigidity results from geometric group theory.
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- Tuesday November 17 at 3-4PM in Eck 206
- John Wiltshire-Gordon, University of Michigan
- Algebraic invariants of configuration space via representation theory of finite sets
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Abstract: The space of n-tuples of distinct points
in a smooth manifold M is called the nth configuration space
of M. As n grows, what happens to configuration space? This
attractive question continues to receive plenty of
attention. Recently, Church-Ellenberg-Farb obtained strong
results on the eventual behavior of the cohomology of
configuration space using the representation theory of
finite sets. I will use recent advances in this theory to
prove a theorem about configuration space when M admits a
nowhere-vanishing vector field. Finally, I will use
Goodwillie calculus to prove a similar result for
configurations of smoothly embedded circles if M has
almost-complex structure. This talk is based on joint work
with Jordan Ellenberg.
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- Tuesday December 8 at 2-4PM in Eck 202
- Curt McMullen, Harvard University
- Cubic curves and Teichmuller theory
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Abstract: TBA
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- Tuesday February 9 at 3-4PM in Eck 206
- Craig Westerland, University of Minnesota
- TBA
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Abstract: TBA
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