2014–2015 Abstracts
Fall 2014
12 and 19 September 2014
- Nicolas Templier, Cornell University
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- Asymptotics of Whittaker functions and Lagrangian
singularities
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We describe work with F. Brumley on the asymptotics of Whittaker
functions motivated by number theoretic questions. Whittaker
functions arise from generic representations of Lie groups when
imposing invariance properties with respect to a maximal unipotent
subgroup. For ${\bf SL}(2,{\bf R})$ they coincide with the K-Bessel
functions. In general we are interested in ''computing'' them in the
semiclassical limit by methods from the geometry of integrable systems
and representation theory.
27 September 2014
- Wan-Yu Tsai, Cornell University
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- Lift of the trivial representation to a nonlinear cover
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Let $G$ be the real points of a simply laced, simply connected complex
Lie group. We discuss a set of small genuine representations of the
nonlinear double cover of $G$, denoted by $\text{Lift}(C)$, which can
be obtained from the trivial representation of $G$ by a lifting
operator. The representations in $\text{Lift}(C)$ can be
characterized by the following properties: (a) the infinitesimal
character is one fourth of the sum of positive roots; (b) they have
maximal tau-invariant; (c) they have a particular associated variety
$O$. When $G$ is split, we will show that all representations in
$\text{Lift}(C)$ are parametrized by pairs (central character, real
form of $O$) by examples.
10 October 2014
- Allen Knutson, Cornell University
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- $SO(3)$-multiplicities of $SL_3(\mathbb R)$-representations
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Given a topological representation of a noncompact real Lie group like
$SL_3(\mathbb R)$, one classically constructs an algebraic replacement
called a $(\mathfrak g,K)$-module, and from there a geometric
replacement called a $\mathcal D_{G/B}$-module, which is supported on
a $K$-orbit closure on $G/B$ (of which there are finitely many).
Given the audience, I'll recall this story in some detail.
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When the $K$-orbit closure is smooth (and the "infinitesimal
character" is integral), I'll use equivariant localization to compute
the $K$-multiplicities in the representation. This generalizes
Blattner's conjecture.
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Then I'll refine this alternating sum to a combinatorial formula, in
the case of the $SO(3)$-multiplicities in the four types of
$SL(3,\mathbb R)$-irreps.
24 October 2014
- Yao Liu, Cornell University
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- Lacunas of Hyperbolic PDEs and Dunkl operators
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The theory of lacunas of hyperbolic PDEs was initiated by Petrovsky
(1945), and developed further by Leray and Atiyah-Bott-Garding in the
late 70's. I will give a brief overview of the theory, which connects
PDEs with topology and algebraic geometry. Then I'll introduce
Dunkl's "differential-difference" operators related to
finite-dimensional complex semi-simple Lie algebras and show that they
bring out more lacunas than available in the purely differential case.
31 October 2014
- Martina Lanini, Friedrich-Alexander Universität
Erlangen-Nürnberg
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- Finite-dimensional representations of rational Cherednik
algebras
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Given a highest weight irreducible representation of a simple Lie
algebra, the question whether it is finite-dimensional has an
affirmative answer if and only if its highest weight is dominant. In
the case of a lowest weight simple module of a rational Cherednik
algebra for a given parameter, the above, natural, question does not
have an answer in general yet. In this talk, I will discuss joint
work with S.Griffeth, A.Gusenbauer and D.Juteau providing a necessary
condition for finite dimensionality of such modules.
07 November 2014
- Matthias Franz, University of Western Ontario
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- Big polygon spaces and syzygies in equivariant cohomology
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Polygon spaces are configuration spaces of polygons with prescribed
edge lengths. We present a related family of compact orientable
manifolds, called big polygon spaces. They come with a canonical
torus action, whose fixed point set is a polygon space. Big polygon
spaces are particularly interesting because they provide the only
known examples of maximal syzygies in equivariant cohomology. We
therefore review the theory of syzygies in equivariant cohomology and
its relation to the equivariant Poincaré pairing and the "GKM
method". We finally discuss possible relations between the syzygy
order and the dimension of the manifold.
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Part of the talk is based on joint work with Chris Allday and Volker
Puppe.
Spring 2015
30 January 2015
- David Li-Bland, University of California, Berkeley
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- Quantization using colored surfaces
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Quantization - generally speaking, the process of deforming a
`classical’ state space to a `quantum mechanical’ one - is a problem of
fundamental importance.
It becomes even more challenging when one wants to quantize not just a
single space in isolation, but rather to simultaneously quantize a
collection of spaces in a manner which is compatible with some key
structural maps between them. In fact, this is generally not possible.
Nevertheless, in this talk we will describe an approach to this problem
which works for a large class of spaces and structural maps which are
important both mathematically and physically (they generalize moduli spaces
of flat connections). The trick is that these spaces can be understood in
terms of `colored surfaces' (which we will introduce) which are both quite
visual and easy to work with. As one application, we will explain how this
allows one to quantize Lie bialgebras.
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This talk is based on joint work with Pavol Severa.
6 February 2015
- Chris Dodd, University of Toronto
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- Modules over algebraic quantizations and representation theory
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Recently, there has been a great deal of interest in the theory
of modules over algebraic quantizations of so-called symplectic
resolutions. In this talk I'll discuss some recent joint work that open
the door to giving a geometric description to
certain categories of such modules; generalizing classical theorems of
Kashiwara and Bernstein in the case of D-modules on an algebraic variety.
20 February 2015
- Peter Samuelson, University of Toronto
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- The elliptic Hall and HOMFLY skein algebras
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For an abelian category A with some finiteness properties,
one can define an associative algebra, called the Hall algebra of A,
which encodes the structure of extensions in A. If X is an elliptic
curve over a finite field, Burban and Schiffmann gave an explicit
description of the Hall algebra of the category Coh(X) of
coherent sheaves on X. On the other hand, for any surface S, there is
a topologically defined `skein algebra', which is spanned by
links in S modulo the so-called HOMFLY skein relations. In this talk,
we describe joint work with Morton, where we show that the Hall algebra
of an elliptic curve and the HOMFLY skein algebra of the torus are
isomorphic. This can be viewed as a manifestation of Mirror Symmetry
for the torus. We also discuss some consequences for knot theory.
13 March 2015
- Ruth Charney, Brandeis University
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- Subgroups of right-angled Artin groups
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It is well known that right-angled Artin groups (RAAGs) contain a
wide variety of interesting, and sometimes exotic, subgroups. Yet a theorem
of Baudisch from 1981 states that all 2-generator subgroups are either free
or free abelian. I will talk about some recent results of my student
Michael Carr showing that these 2-generator subgoups are always
quasi-isometrically embedded in the RAAG and discuss consequences of this
theorem for cubulating 3-manifold groups.
20 March 2015
- Craig Dodge, Allegheny College
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- Simple modules of centralizer algebras
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Let G be a finite group, H a subgroup of G. The centralizer algebra of H is the subalgebra
of the group algebra kG over a field k that consists of elements commuting with H.
As part of a larger project, Ellers and Murray have been working to uncover information
about the blocks and the simple modules of these centralizer algebras. In this talk,
we will address the problem of classifying the simple modules of centralizer algebras
of symmetric groups S_l in S_n. We will examine a potential solution to the
problem proposed by Ellers and Murray, which was inspired by a classification of James for the
simple kS_n-modules.
10 April 2015
- Joe Gallagher, Cornell University
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- Factorization homology and applications
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Factorization homology is a new technology developed by Lurie and
others - it is a more sensitive notion of a homology theory, specified to
manifolds of a given dimension. While the specialization and the development
require some new language, the result is a collection of invariants which
are able to distinguish properties inaccessible to classical homology
theories. A tactile application comes about by studying the configuration
space of points on a fixed manifold M, which has classical homology groups
detected by current Lie algebras associated to the manifold. We will inspect
the construction of this technology and see what ties it may have to Lie
theory.
17 April 2015
- Jan Möllers, Ohio State University
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- The K-spectrum of symmetry breaking operators
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Symmetry breaking operators are intertwining operators from a representation
of a group G to a representation of a subgroup G',
intertwining for the subgroup. For spherical principal series of G = O(1,n+1)
and G' = O(1,n), these operators have been classified recently by
Kobayashi-Speh in the smooth category. We study symmetry breaking operators
in the category of Harish-Chandra modules, recovering the results by
Kobayashi-Speh in this setting, and thus providing the K-picture of their
operators. We further indicate how the same method can be used in the case
of G = U(1,n+1) and G' = U(1,n) to classify symmetry breaking operators.
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Joint work with Bent Ørsted.
24 April 2015
- Ana Caraiani, Princeton University
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- On the Hodge-Tate period morphism
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In this talk, I will focus on one of the key ingredients in constructing Galois
representations for torsion classes: the Hodge-Tate period morphism. This is a
G(A_f)-equivariant map from a perfectoid Shimura variety into a flag variety which only has
an action of G(Q_p) and can be thought of as a p-adic analogue of the embedding of the upper
half plane into the complex projective line. I will motivate and describe a new, canonical
construction of the Hodge-Tate period morphism and of automorphic vector bundles for Shimura
varieties of Hodge type.
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This is part of ongoing joint work with Peter Scholze.
1 May 2015
- Sema Salur, University of Rochester
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- Deformation theory of calibrated submanifolds
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It is well known that a smooth 2n-dimensional symplectic manifold N is
equipped with a closed, nondegenerate differential 2-form w. An
n-dimensional submanifold L of N is called Lagrangian if the
restriction of w to L is zero. Lagrangian submanifolds and their
deformations have important applications in symplectic geometry and
mathematical physics. In particular, they play a role in establishing the
correspondence between `Calabi-Yau mirror pairs' in string theory via the
Fukaya category. In this talk, we first study the deformations of (special)
Lagrangian submanifolds and then extend the theory to `Lagrangian type'
submanifolds inside G_2 manifolds.
