## Lie Group/Quantum Mathematics SeminarOrganizers Lisa Carbone, Yi-Zhi
Huang, Jim
Lepowsky and Siddhartha Sahi.
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 webmaster@math.rutgers.edu. For the Lie Group/Quantum Mathematics seminar in previous semesters, see this page. For talks in the Quantum Mathematics Seminar from Spring, 1998 to Fall, 2007, see this page. 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 for the Algebra Seminar. For all the seminars and colloquia in the department, see the Seminars and Colloquia page.
## Spring, 2018**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 Urbana-Champaign**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 lattice VOA. 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**Title**Genus Two Generalization of A_1 spherical Double Affine Hecke Algebra**Time/place**3/30/2018, Friday, 12:00 in Hill 705**Abstract**Spherical Double Affine Hecke Algebra can be viewed as a noncommutative (q,t)-deformation of the SL(N,C) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus SL(2,Z) acts by atomorphisms of DAHA. In my talk I will define a genus two analogue of A_1 spherical DAHA and show that the Mapping Class Group of a closed genus two surface acts by automorphisms of such algebra. I will then show that for special values of parameters q,t satisfying q^n t^2=1 for some nonnegative integer n this algebra admits finite dimensional representations. I will conclude with discussion of potential applications to TQFT and knot theory.Based on arXiv:1704.02947, joint with Sh. Shakirov.
**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.
**Speaker**Robert Laugwitz, Rutgers University**Title**A categorification of cyclotomic integers at non-prime orders**Time/place**4/27/2018, Friday, 12:00 in Hill 705**Abstract**I am reporting on recent joint work with You Qi (Caltech). Using the framework of Hopfological algebra, developed by Khovanov-Qi, and used by Elias-Qi to categorify versions of quantum \mathfrak{sl}_2 at a prime root of unity, we construct a monoidal triangulated category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers. The construction removes the restriction that the order has to be a prime number as required in the original work of Khovanov.
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