Lie Group/Quantum Mathematics Seminar
Organizers Lisa Carbone, Yi-Zhi
Lepowsky and Siddhartha Sahi.
Time Friday, 12:00 am to 1:00 pm.
Place Hill 705.
Starting from Spring, 2008, the
Lie Group Seminar and Quantum Mathematics Seminar
have merged together to a single seminar called the
Lie Group/Quantum Mathematics
Seminar. This seminar also has a page
Lie Groups Quantum Mathematics Seminar,
maintained by email@example.com.
For the Lie Group/Quantum Mathematics seminar in previous
semesters, see this
For talks in the Quantum Mathematics Seminar from Spring, 1998 to
Fall, 2007, see
For the last few years, the Quantum Mathematics Seminar
shared the time and place with the Algebra Seminar.
For talks in both the Algebra and
Quantum Mathematics Seminars in the last few semesters, see the page
Seminar. For all the seminars and colloquia in the department, see
the Seminars and Colloquia page.
- Speaker Vladimir Retakh, Rutgers University
- Title Noncommutative Catalan numbers
- Time/place 1/26/2018, Friday, 12:00 in Hill 705
- Abstract We introduce and study noncommutative Catalan numbers C_n which belong to the free Laurent polynomial algebra in n generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices H_n and introduce accompanying noncommutative binomial coefficients. Joint work with A. Berenstein (Oregon). See arXiv:1708.03316
- Speaker Rekha Biswal, Université Laval
- Title Demazure flags: connections to algebraic
combinatorics and number theory
- Time/place 2/23/2018, Friday, 12:00 in Hill 705
- Abstract In this talk, we will briefly review the basic theory of Demazure modules, which are modules over the standard maximal parabolic subalgebra of an affine Lie algebra. The discussion will be followed by some connections that I have discovered (with collaborators) in my own research between algebraic combinatorics and other areas of mathematics such as representation theory and number theory. For instance, we show that the graded multiplicities of higher level Demazure modules in Demazure flags can be expressed in terms of Dyck paths. The generating series for those graded multiplicities give rise to interesting connections with Ramanujan's mock theta functions. I will describe some results and further questions in this direction.
- Speaker Emily Cliff, University of Illinois
- Title Modules over factorization spaces, and moduli spaces of parabolic G-bundles
- Time/place 2/16/2018, Friday, 12:00 in Hill 705
- Abstract Beilinson and Drinfeld introduced the notion of factorization algebras, a geometric incarnation of the notion of a vertex algebra. An advantage of working with factorization algebras is that they admit non-linear analogues, called factorization spaces, which can be viewed as both generalizations of and ways to produce examples of factorization algebras from algebraic geometry. The resulting factorization algebras can then be studied via the geometry of the spaces from which they arise.
Just as vertex algebras admit interesting categories of representations, so too do factorization algebras and factorization spaces. In this talk we will review the definitions of a factorization algebra and factorization space before introducing the notion of a module over a factorization space. As an example and an application we will construct moduli spaces of principal G-bundles with parabolic structures, and discuss how they can be linearized to recover modules over the factorization algebra corresponding to the affine Lie algebra associated to a reductive group G. The spaces we construct can be considered to form a "modular functor valued in spaces", which we linearize at different levels to recover the well-known WZW modular functors (valued in vector spaces).
- Speaker Sven Moeller, Rutgers University
- Title Cyclic Orbifolds of the Leech lattice vertex operator algebra
- Time/place 3/9/2018, Friday, 12:00 in Hill 705
- Abstract Recent results have established that
the weight-one space of a (suitably regular) holomorphic VOA
of central charge 24 is one of 71 Lie algebras (Schellekens' list)
and all of these cases have now been constructed in a joint
effort by many authors. The main tool for constructing these
VOAs is the orbifold construction, which we established for
arbitrary cyclic groups of automorphisms.
In analogy to the construction of all the Niemeier lattices
from the Leech lattice (via its deep holes) we conjecture
that all 71 cases on Schellekens' list can be obtained in a
uniform way as cyclic orbifold constructions from the Leech
In an ongoing effort we have constructed 63 cases so far. If
all 71 cases could be constructed from the Leech lattice VOA
in this way, this would greatly help to gain a more conceptual
understanding of Schellekens' list. Moreover, this provides
evidence for the effectiveness of cyclic orbifolding. (We do
believe however that the orbifold theory can and should be
extended to more general, in particular non-abelian, groups.)
(This is work in progress joint with Nils Scheithauer)
- Speaker Semeon Artamonov, Rutgers University
- Speaker Joshua Sussan, CUNY Medgar Evers
- Title Braid group actions from zigzag algebras
- Time/place 4/6/2018, Friday, 12:00 in Hill 705
- Abstract There exists a braid group action on the homotopy category of modules for a zigzag algebra associated to a linear quiver. We will explain several variations of this well known fact arising from various areas of categorical representation theory.
- Speaker Eric Schippers, University of Manitoba
- Title Sewing and boundary value problems
in complex function theory and conformal field theory
- Time/place 4/13/2018, Friday, 12:00 in Hill 705
- Abstract Some classical ideas in complex
function theory were rediscovered, from another point of
view, in two-dimensional conformal field theory. On the
other hand, conformal field theory suggests new ways to
look at these old objects, and new theorems about them.
In this talk I will give examples, with particular
attention to sewing, boundary value problems on Riemann
surfaces, and generalizations of the classical period maps.