Spring, 2010
- Speaker Zajj
Daugherty, University of Wisconsin, Madison
- Title Degenerate two-boundary centralizer algebras
- Time/place 1/22/2010, Friday, 11:45 am in Hill 425
- Abstract Degenerate two-boundary centralizer algebras
arise as algebras of commuting operators for a Lie algebra action
on a tensor space. These algebras generalize the group algebra of
the symmetric group, and graded Hecke and Brauer algebras, and
have similarly elegant combinatorial structure. In this talk, we
will explore centralizers of the actions of .nite dimensional
complex reductive Lie algebras on tensor spaces of the form $M
\otimes N \otimes V^{\otimes k}$. As an example, we will explore
in detail the combinatorics of special cases corresponding to
$\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ and explain how this might
be applied to the study of the combinatorial representation theory
of graded Hecke algebras of type C.
- Speaker Anton
Zeitlin, Yale University
- Title From Lian-Zuckerman Algebras to the Algebraic
Structure of Classical Field Equations
- Time/place 1/29/2010, Friday, 11:45 am in Hill 425
- Abstract We define a quasiclassical limit of the
Lian-Zuckerman homotopy BV
algebra (LZ_{qc} algebra) on the subcomplex, corresponding to
light modes, i.e. the elements of zero conformal weight, of the semi-
infinite (BRST) cohomology complex of the Virasoro algebra associated
to Vertex Operator Algebra (VOA) with a formal parameter.
We show that LZ_{qc} algebra is actually a certain "double" of a
Courant Algebroid associated with the corresponding VOA.
We also discuss certain deformations related to abelian vertex
subalgebras
of the A_{\infty} subalgebra of LZ_{qc} algebra. In particular
case of
VOA associated with beta-gamma systems it will be shown that the
deformations of this kind lead to the A_{\infty} algebra of the
Yang-Mills equations.
- Speaker Miranda Cheng,
Harvard University
- Title Black Holes, Wall-Crossing and Generalized
Kac-Moody Algebras
- Time/place 2/5/2010, Friday, 11:45 am in Hill 425
- Abstract The relation between generalised Kac-Moody
algebras, automorphic forms and hyperbolic reflection groups has
led to many fascinating results in mathematics. In my talk I will
report on results regarding the role of these objects in the
physics of black holes. I will also describe how, through string
theory, these results have lead to interesting conjectures in
geometry.
- Speaker David Jordan, MIT and Eric Larson, Harvard
- Title Fusion categories of dimension pq^2
- Time/place 2/19/2010, Friday, 11:45 am in Hill 425
- Abstract Fusion categories are certain ``small'' tensor
categories, which generalize categories of representations of
finite groups. After introducing fusion categories and their
basic properties, we outline a theory of extensions and their
obstructions due to Etingof, Nikshych, and Ostrik, which is
informed by methods in homotopy theory. Fusion categories have a
notion of dimension, and have been completely classified in
dimensions p^k, pq, and pqr, for distinct primes p, q, r. Using
the theory of extensions, we completely classify fusion categories
of dimension pq^2, producing many new examples which are not
``group-theoretical" in the sense of Ostrik.
- Speaker Tony Giaquinto, Loyola University of Chicago
(canceled)
- Title Graphs, Frobenius functionals, and the classical
Yang-Baxter equation
- Time/place 3/12/2010, Friday, 11:45 am in Hill 425
(canceled)
- Abstract A Lie algebra is Frobenius if it admits a
linear functional F such that the Kirillov form F([x,y]) is
non-degenerate. If g is the m-th maximal parabolic subalgebra
P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a
"cyclic" functional F on P(n,m) and prove it is non-degenerate
using properties of certain graphs associated to F. These graphs
also provide in some cases readily computable associated solutions
of the classical Yang-Baxter equation. We also examine the seaweed
Lie algebras of Dergachev and Kirillov from our perspective.
- Speaker Zhiwei Yun, IAS
- Title A topological shadow of Mirror Symmetry for dual
Hitchin fibrations
- Note Joint Lie Group/Quantum Mathematics/Geometry,
Symmetry,
and Physics Seminar
- Time/place 3/26/2010, Friday, 11:45 am in Hill 425
- Abstract For a reductive group G and its Langlands dual
G^, their
Hitchin fibrations are dual torus fibrations. Homological Mirror
Symmetry and Geometric Langlands duality predicts an equivalence
between categories constructed from the dual fibrations, respecting
certain natural symmetries. Being less ambitious, we will establish
the shadow of this prediction on the level of cohomology, relating
"global Springer actions" on (parabolic) Hitchin fibers of G and
"Chern class actions" on those of G^.
- Speaker Antun Milas, University at Albany (SUNY)
- Title W-algebra extensions of minimal models
- Time/place 4/2/2010, Friday, 11:45 am in Hill 425
- Abstract I will explain structure and representations
of a family of
W-algebras obtained as extension of the Virasoro vertex algebras of
minimal central charge, and in particular the c=0 algebra (joint work
with
D. Adamovic). These W-algebras are of considerable interest
in logarithmic CFT and critical percolation. In addition, their
representation theory is expected to be identical to that of certain
finite dimensional quantum groups (via Kazhdan-Lusztig
correspondence).
Time permitting, I will also describe how considerations of W-algebras
lead to new constant term and q-series identities.
- Speaker
Xiao-Gang Wen, MIT
- Title Symmetric polynomial of infinite variables and
quantum Hall states
- Note Joint Lie Group/Quantum
Mathematics/Mathematical
Physics Seminar
- Time/place 4/9/2010, Friday, 11:45 am in Hill 425
- Abstract I will discuss the close relation between
symmetric polynomial of infinite variables,
vertex algebra, and quantum Hall states. I will present some attempts
to classify
symmetric polynomial of infinite variables.
- Speaker Aaron Lauda, Columbia and MSRI
- Title A diagrammatic categorification of quantum groups
- Time/place 4/30/2010, Friday, 11:45 am in Hill 425
- Abstract I'll explain joint work with Mikhail Khovanov
on a categorification of one-half of the quantum universal
enveloping algebra associated to a Kac-Moody algebra. This
categorification is obtained from the graded representation
category of certain graded algebra that can be defined using a
graphical calculus. Certain finite-dimensional quotients of these
graded algebras give categorifications of irreducible
representations of the quantum enveloping algebra. Extensions of
these rings are used in Webster's recent work categorifying
Reshetikhin-Turaev tangle invariants.
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