Spring, 2016
- Speaker Siddhartha Sahi, Rutgers University
- Title Positivity of Shimura operators
- Time/place 1/29/2016, Friday, 12:00 in Hill 124
(note the special room)
- Abstract I will describe some recent progress on an old problem of G. Shimura
("Invariant differential operators on hermitian symmetric spaces." Annals of Mathematics (1990): 237-272.) This is joint work with Genkai Zhang (Gothenburg).
- Speaker Lisa Carbone, Rutgers University
- Title `Arithmetic' constructions of hyperbolic
Kac-Moody groups
- Time/place 2/5/2016, Friday, 12:00 in Hill 705
- Abstract Tits defined Kac-Moody groups over commutative rings, providing
infinite dimensional analogues of the Chevalley-Demazure group schemes. Tits' presentation can be simplified considerably when the Dynkin diagram is hyperbolic and simply laced. In joint work with Daniel Allcock, we have obtained finitely many generators and defining relations for simply laced hyperbolic Kac-Moody groups over Z. We compare this presentation with a representation theoretic construction of Kac-Moody groups over Z. We also present some preliminary results with Frank Wagner about uniqueness of representation theoretic hyperbolic Kac-Moody groups.
- Slides pdf file.
- Archive papers arXiv:1602.02319, arXiv:1512.04623,
arXiv:1409.5918.
- Speaker Yi-Zhi Huang, Rutgers University
- Title Associative algebras for (logarithmic)
twisted modules for a vertex operator algebra
- Time/place 2/12/2016, Friday, 12:00 in Hill 705
- Abstract We construct two associative algebras
from a vertex operator algebra V and a general automorphism g
of V. The first, called g-twisted zero-mode algebra, is a
subquotient of what we call g-twisted universal enveloping
algebra of V. The other is a generalization of the
g-twisted version of Zhu's algebra for g-twisted
modules constructed by Dong-Li-Mason when the order of g is
finite. Our main interest is in the case that g is of
infinite order and does not act on V semisimply. We construct
functors between the categories of modules for these
associative algebras and the category of (logarithmic)
g-twisted V-modules. We conjecture that these two algebras
are isomorphic. This is a joint work with Jinwei Yang.
- Slides pdf file.
- Paper
pdf file
- Speaker Ling Chen, University of Chinese Academy of
Sciences and Rutgers University
- Title On Axiomatic Approaches to Intertwining
Operator Algebras
- Time/place 3/4/2016, Friday, 12:00 in Hill 705
- Abstract We study intertwining operator
algebras which are direct sums of modules (not necessarily
irreducible) for vertex operator algebras equipped with
intertwining operators among these modules satisfying some
basic properties for intertwining operators. In the case that
the intertwining operator algebras involve only irreducible
modules for the vertex operator algebras, a number of results
were given by Huang. We formulate and prove the
generalizations of these results in the general case.
In particular, we construct fusing and braiding isomorphisms
for intertwining operator algebras in the general case and
prove that they satisfy the genus-zero Moore-Seiberg
equations. Moreover, we study the duality properties of
intertwining operator algebras and prove various equivalence
results between axioms and properties. Furthermore, using the
skew-symmetry property and the genus-zero Moore-Seiberg
equations, we prove an S_{3}-symmetry of the Jacobi identity
for intertwining operator algebras.
- Slides pdf file.
- Archive papers arXiv:1503.06428, arXiv:1507.05159.
- Speaker Simon Wood, Australian National University
- Title The rationality of the N=1 minimal model
vertex algebra and its connection to symmetric functions
- Time/place 3/11/2016, Friday, 12:00 in Hill 705
- Abstract I will review the universal N=1
Virasoro vertex algebra, its simple quotients, called the N=1
minimal model vertex algebras, and free field realisations of
these vertex algebras. These free field realisations have
interesting connections to the theory of symmetric functions
which I will use to present new proofs of the rationality of
the N=1 minimal models and of the classification of their
modules.
- Speaker David Radnell, Aalto University
- Speaker Scott Carnahan, University of Tsukuba
- Title A proof of Generalized Moonshine
- Time/place 3/25/2016, Friday, 12:00 in Hill 705
- Abstract In the late 1970s, Monstrous moonshine arose from the discovery of numerical relationships between the representations of the Monster sporadic group and the Fourier expansions of modular functions. This was codified into the Monstrous Moonshine conjecture by Conway and Norton, asserting the existence of a graded representation of the Monster whose characters satisfy nice modularity properties. A candidate representation with vertex operator algebra structure was constructed by Frenkel, Lepowsky, and Meurman, and the conjecture was proved for this representation by Borcherds in 1992. Based on additional massive computations, Norton conjectured a generalization in 1987 that concerns modularity of "higher characters" on pairs of commuting elements in the monster. The underlying vector spaces were quickly interpreted by physicists as twisted sectors of an orbifold conformal field theory, and in the algebraic world are known as twisted modules of the monster vertex operator algebra. I will outline the recent proof of this conjecture, following the Borcherds-Höhn program.
- Speaker Bud Coulson, Rutgers University
- Title An affine Weyl group interpretation of the
"motivated proofs'' of the Gordon-Andrews-Bressoud identities
- Time/place 4/1/2016, Friday, 12:00 in Hill 705
- Abstract A "motivated proof" of the Rogers-Ramanujan identities was given by G. E. Andrews and R. J.
Baxter. This proof was generalized to the odd-moduli
case of Gordon's identities by J. Lepowsky and M. Zhu, and later
to the even-moduli case of the Andrew-Bressoud identities by S.
Kanade, Lepowsky, M.C. Russell and A. Sills. We present a
reinterpretation of these proofs, with new motivation coming
from the affine Weyl group of sl(2)^. (Ph.D. thesis defense.)
- Speaker Pavel Etingof, MIT
- Title Symmetric tensor categories in
characteristic p
- Time/place 4/1/2016, Friday, 2:00 pm in Hill 705
(note the special time)
- Abstract I will review V. Ostrik's
generalization of Deligne's theorem
that symmetric tensor categories of subexponential growth in
characteristic zero are super-Tannakian to fusion categories
in positive
characteristic, which asserts that any such category is the
representation category of an affine group scheme in the
Verlinde category Ver_p, obtained by reduction mod p of the
fusion category of the SU(2) Wess-Zumino-Witten model at level
p-2.
I will also use this result to give a classification of
semisimple
triangular Hopf algebras in characteristic p. Finally, I will
discuss
Deligne's construction of tensor categories in positive
characteristic
of superexponential growth, and discuss the notion of p-adic
dimension
of objects in tensor categories in characteristic p. This is
based on
joint work with N. Harman, V. Ostrik, and S. Venkatesh.
- Speaker Andrew Douglas, CUNY
- Title Subalgebras of semisimple Lie algebras
- Time/place 4/8/2016, Friday, 12:00 in Hill 705
- Abstract While the semisimple subalgebras of semisimple Lie alagebras have received considerable attention, much less is known about their non-semisimple subalgebras. In this talk, I will outline joint work with Joe Repka on the classification of non- semisimple subalgebras of semisimple Lie algebras. Next, I will describe an application of the above work to a problem in physics. All Lie algebras and representations in this talk are finite-dimensional and over the complex numbers.
- Slides pdf file.
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