Spring, 2015
- Speaker Yi-Zhi Huang, Rutgers University
- Title Vertex operator algebras, fractional quantum Hall states
and topological orders
- Time/place 1/30/2015, Friday, 12:00 in Hill 705
- Abstract In Landau's symmetry-breaking theory,
different orders in states of matter correspond to different
symmetries (or more precisely, the breaking of symmetries).
Fractional quantum Hall states
give a new type of orders called
topological orders that cannot be described by symmetries or symmetry-breaking.
In this talk, using the constructions of wavefunctions
for fractional quantum Hall states by Moore and Read as examples, I will propose
that topological orders in $2+1$ dimension can in fact also be viewed as
orders corresponding to "symmetries" that are described not by groups
but instead by vertex operator algebras. In particular,
all the conceptual and powerful mathematical methods and
results in the representation theory of vertex operator algebras
can be applied to the study of
fractional quantum Hall states and topological orders in $2+1$ dimension.
- Speaker Vladimir Retakh, Rutgers University
- Title Noncommutative triangulations
- Time/place 2/6/2015, Friday, 12:00 in Hill 705
- Abstract The celebrated Ptolemy relation plays
an important role in various
studies of triangulated surfaces including hyperbolic geometry,
geometrical applications of cluster algebras and so on. We will
discuss a noncommutative version of the relation which can
be seen as a "categorification" of the classical one.
This leads to new noncommutative invariants of the surfaces and
provides several examples of the noncommutative Laurent phenomenon
answering some questions by Kontsevich.
(Joint work with Arkady Berenstein from University of Oregon)
- Speaker Semeon Artamonov, Rutgers University
- Title Noncommutative Inverse Scattering Method
for the Kontsevich system
- Time/place 2/13/2015, Friday, 12:00 in Hill 705
- Abstract In my talk I will formulate an analog
of Inverse Scattering Method for integrable systems on
noncommutative associative algebras. In particular I will
define Hamilton flows, Casimir elements
and noncommutative analog of the Lax matrix. The noncommutative
Lax element generates infinite family of commuting Hamilton
flows on an associative algebra. The proposed approach to
integrable systems on associative algebras satisfy certain
universal property, in particular it incorporates both
classical and quantum integrable systems as well as provides
a basis for further generalization.
The motivation for definition will be given by explicit
construction of noncommutative analog of
Lax matrix for a system of differential equations on
associative algebra recently proposed by Kontsevich.
First these equations will be presented in the Hamilton
form by defining a bracket of Loday type on the group
algebra of the free group with two generators. To make
the definition more constructive I will utilize (with
certain generalizations) the Van den Bergh approach to
Loday brackets via double Poisson brackets. Finally, it
will be shown that there exists an infinite family of
commuting flows generated by the noncommutative Lax element.
- Speaker Siddhartha Sahi, Rutgers University
- Title Generalized Whittaker functionals for real reductive groups
- Time/place 2/20/2015, Friday, 12:00 in Hill 705
- Abstract Let (\pi,V) be a smooth representation of a
real reductive group G. A generalized Whittaker functional is a linear
functional on V that transforms by a character of a suitable unipotent
subgroup of G. Such functionals play an important role in various applications,
especially in the study of automorphic forms. In this talk I will describe
a number of recent results obtained in this direction.
- Speaker Siddhartha Sahi, Rutgers University
- Title Generalized Whittaker functionals for real reductive groups,
Part II
- Time/place 2/27/2015, Friday, 12:00 in Hill 705
- Abstract Let (\pi,V) be a smooth representation of a
real reductive group G. A generalized Whittaker functional is a linear
functional on V that transforms by a character of a suitable unipotent
subgroup of G. Such functionals play an important role in various applications,
especially in the study of automorphic forms. In this second talk I
will continue to describe
a number of recent results obtained in this direction.
- Speaker Robert McRae, Peking University
- Title On the braided tensor category of standard affine
Lie algebra modules at fixed non-negative integral level
- Time/place 3/6/2015, Friday, 12:00 in Hill 705
- Abstract For a simple Lie algebra g, Huang and Lepowsky
showed that the category of standard modules for the affine Lie
algebra g^ at a fixed non-negative integral level is a braided tensor
category. This category of g^-modules is the category of modules for
a certain simple vertex operator algebra, and Frenkel and Zhu showed
that this category is equivalent to the category of finite-dimensional
modules for a certain quotient of U(g). In this talk, I will explain
this equivalence between the category of fixed-level standard affine
Lie algebra modules and the corresponding subcategory of finite-dimensional
g-modules, and I will describe how the subcategory of finite-dimensional
g-modules inherits braided tensor category structure from the category
of affine Lie algebra modules. The most interesting problem is to describe
the (non-trivial) associativity isomorphisms on triple tensor products of
g-modules; these isomorphisms turn out to be related to "Drinfeld associator"
isomorphisms coming from the KZ equations.
- Speaker Pierre Cartier, IHES
- Title From the exponential mapping to
Witt vectors through jet spaces; a tale of categories
- Time/place 3/6/2015, Friday, 2:30 in Hill 705 (note the special time)
- Abstract
- Speaker Shashank Kanade, Rutgers University
- Title Some Results on the Representation Theory of
Vertex Operator Algebras and Integer Partition Identities
- Time/place 4/9/2015, Thursday, 2:00 in Hill 525 (note the special time)
- Abstract Ph.D. thesis defense.
- Speaker Francesco Fiordalisi, Rutgers University
- Title Logarithmic Intertwining Operator and
Genus-One Correlation Functions
- Time/place 4/9/2015, Thursday, 3:30 in Hill 525 (note the special time)
- Abstract Ph.D. thesis defense.
- Speaker Alex Kontorovich, Rutgers University
- Title On Lusztig's Conjecture
- Time/place 4/10/2015, Friday, 12:00 in Hill 705
- Abstract We will attempt to describe the problem in the title, and some
recent work on the problem by Geordie Williamson.
- Speaker Misha Ershov, University of Virginia
- Title Tarski numbers of groups
- Time/place 5/1/2015, Friday, 12:00 in Hill 705
- Abstract A group G is said to admit a paradoxical decomposition
if it can be represented as a disjoint union of finitely many subsets
A_1,.., A_n and B_1,..., B_m such that for some elements g_1,..., g_n and
h_1,...,h_m of G, the translates g_i A_i cover the entire G, and the same
is true for the translates h_j B_j. Such decompositions are ``responsible''
for the Banach-Tarski paradox and other related phenomena. It is well known
that G admits a paradoxical decomposition if and only if G is non-amenable,
in which case the smallest possible number of pieces in such a decomposition
of G is called the Tarski number of G.
- Speaker Eugene Gorsky, Columbia University
- Title Stable Khovanov-Rozansky homology of torus knots
- Time/place 5/5/2015, Tuesdayday, 12:00 in Hill 705 (note special date)
- Abstract The Jones and HOMFLY-PT polynomials of torus knots were computed by
Rosso and Jones in the beginning of 90's. The computation of their
"categorified" versions (Khovanov and Khovanov-Rozansky homology)
remains an important open problem in knot theory. In the talk, I will
explain some recent results and conjectures describing the stable
limit of these homologies in terms of a certain explicit Koszul
complex. If both parameters of a torus knot tend to infinity, there is
a surprising relation to generalized Gordon and Rogers-Ramanujan
identities.
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