Spring, 2019
- Speaker Thomas Lam, University of Michigan
- Speaker Florencia Orosz Hunziker, Yale University
- Title Fusion rules for the Virasoro algebra of central charge 25
- Time/place 2/1/2019, Friday, 12:00 in Hill 705
- Abstract In 1990 Feigin and Fuchs established a correspondence between the Verma modules for the Virasoro algebras of dual central charges $c$ and $26- c$. In later work, the irreducible quotient module $L(c,0)$ was proved to be a vertex operator algebra called the Virsasoro VOA of central charge $c$. In this talk we will discuss an extension of the Feigin-Fuchs correspondence to the vertex algebra setting for the case $c=1$ and $c=25$. We will prove the the fusion rules for the non-Verma irreducible $L(25,0)$-modules coincide the fusion rules for the non-Verma irreducible $L(1,0)$-modules.
- Speaker Robert Laugwitz, Rutgers University
- Speaker Yi-Zhi Huang, Rutgers University
- Speaker Nicola Tarasca, Rutgers University
- Title Vertex algebras and moduli spaces of curves
- Time/place 2/22/2019, Friday, 12:00 in Hill 705
- Abstract This talk will focus on geometric realizations of vertex algebras. The Virasoro uniformization provides an incarnation of the Virasoro algebra in the tangent space of the Hodge line bundle on moduli of algebraic curves with marked points and local coordinates. This allows to assign to certain representations of the Virasoro algebra a sheaf on moduli of curves together with a projective connection.
After reviewing some facts on curves and their moduli spaces, I will discuss the sheaves on moduli of stable curves obtained from coinvariants of modules over conformal vertex algebras, and identify their logarithmic projective connection. This is joint work with Chiara Damiolini and Angela Gibney.
- Speaker Mark Skandera, Lehigh University
- Title Total nonnegativity and induced sign characters of the Hecke algebra
- Time/place 3/1/2019, Friday, 12:00 in Hill 705
- Abstract Gantmacher's study of totally nonnegative (TNN) matrices in
the 1930's
eventually found applications in many areas of mathematics.
Descending from his work are problems
concerning TNN polynomials, those polynomial functions of n^2
variables which take nonnegative
values on TNN matrices. Closely related to TNN polynomials are
functions in the Hecke algebra
trace space whose evaluations at certain Hecke algebra elements yield
polynomials in N[q]. In all
cases, it would be desirable to combinatorially interpret the
resulting nonnegative numbers.
In 2017, Kaliszewski, Lambright, and the presenter found the first
cancellation-free combinatorial formula
for the evaluation of all elements of a basis of V at all elements of
a basis of the Hecke algebra.
We will discuss a recent improvement upon this result which also
advances our understanding of TNN
polynomials. This is joint work with Adam Clearwater.
- Speaker Donald Richards, Pennsylvania State
University
- Title Integral Transform Methods in
Goodness-of-Fit Testing for the Wishart Distributions
- Time/place 3/8/2019, Friday, 12:00 in Hill 705
- Abstract This talk is based on joint work with Elena Hadjicosta.
In recent years, random data consisting of positive definite (symmetric) matrices have appeared in several areas of applied research, e.g., diffusion tensor imaging, wireless communication systems, synthetic aperture radar, and volatility models in finance. Given a random sample of such matrices, we wish to test whether the data are drawn from a given statistical population. In this talk, we apply the Hankel transform of matrix argument to develop goodness-of-fit tests for the Wishart distributions. The asymptotic distribution of the test statistic is derived in terms of the integrated square of a Gaussian random field, and an explicit formula is obtained for the corresponding covariance operator. The eigenfunctions of the covariance operator are determined explicitly, and the eigenvalues are shown to satisfy certain interlacing properties.
Throughout this work, the Bessel functions of matrix argument of Herz (1955) and the zonal polynomials of James (1964) play a crucial role. Also, our results raise the possibility of extending to the Wishart distributions many statistical properties of the classical, one-dimensional gamma distributions.
- Speaker Daniel Nakano, University of Georgia
- Title On BBW Parabolics for Simple Classical
Lie Superalgebras
- Time/place 3/29/2019, Friday, 12:00 in Hill 705
- Abstract Let g be a (simple) classical Lie superalgebra over the complex numbers.
In this talk I will discuss the developments over the past 10 years that
have led to effective methods to systematically study these algebras via the
construction of detecting subalgebras. The detecting subalgebras play an important role
in the theory. They completely detect the cohomology relative to the even
subalgebra, and provide a structural interpretation of combinatorial invariants
that were defined by Kac and Wakimoto.
Recently, the speaker with his collaborators have constructed parabolic subalgebras, b,
(like Borel subalgebras) where the detecting subalgebras can be viewed as the Levi component.
By comparing the cohomology of g and b, we discovered an important relationship with
the Poincare series of an ambient complex reflection group via the Bott-Borel-Weil theorem.
Furthermore, these Poincare series describe the higher sheaf cohomology groups of the
trivial line bundle over G/B where g=Lie G and b=Lie B.
At the end of the talk, applications will be given to verifying the conjecture due to Boe, Kujawa
and Nakano on the realization of support varieties for g.
This talk represents joint work with D. Grantcharov, N. Grantcharov and J. Wu.
- Speaker Nitu Kitchloo, Johns Hopkins University
- Title Stability for Kac-Moody Groups
- Time/place 4/5/2019, Friday, 12:00 in Hill 705
- Abstract In the class of Kac-Moody groups,
one can extend all the exceptional families of compact Lie groups yielding infinite families (E_n, F_n, G_n), as well as other infinite families. These families often pass through the Affine or loop-groups of compact Lie groups. We will show that these exceptional families stabilize in a homotopical sense and that the (co)homology of their classifying spaces behave well (in a precise sense) for all but a finite set of primes that is determined by the family and not the individual groups in the family. I will also speculate on the underlying structure that might be present in these families.
- Speaker Thomas Creutzig, University of Alberta
- Title Glueing Vertex Algebras
- Time/place 4/12/2019, Friday, 12:00 in Hill 705
- Abstract The aim of this talk is to explain why certain types of extensions of vertex algebras are possible if and only if equivalences of associated braided tensor categories hold.
This result is due to joint work with Shashank Kanade and Robert McRae and it will be motivated via its many nice consequences as e.g. rigidity and fusion rules of categories of W-algebra modules, constructing vertex algebras for S-duality and many more.
- Speaker Lisa Carbone, Rutgers University
University
- Title Imaginary reflections and automorphisms of the monster Lie algebra
- Time/place 4/19/2019, Friday, 12:00 in Hill 705
- Abstract We describe a `hidden symmetry' of the root lattice of the monster Lie algebra m, namely a reflection with respect to an imaginary simple root, that preserves the bilinear form and root multiplicities. This symmetry appears only after the infinite root lattice is specialized to the unique even 2-dimensional unimodular Lorentzian lattice II_{1,1}. When composed with reflection in the unique real simple root, this symmetry extends to the Cartan involution on m. We describe an uncountable family of involutions on m and their associated eigenspaces, known as Cartan pairs, and we examine these involutions as elements of Aut(m).
- Speaker Juan Villarreal, Virginia Commonwealth
University
- Title The Witten Genus and sheaves of
vertex algebras
- Time/place 4/26/2019, Friday, 12:00 in Hill 705
- Abstract In this talk I want to introduce the Witten genus. We can think on the Witten genus as a loop generalization of the Atiyah Singer index theorem for Dirac operators. However, the idea to formalize a Dirac operator on loop spaces has been always a little subtle. Different approaches have been done, and always it has been considered that a vertex algebraic description of this construction must be possible. I want to show a construction using some sheaves of vertex algebras.
- Speaker Apoorva Khare, Indian Institute of Science,
and
Analysis & Probability Research Group
- Title Schur polynomials, entrywise positivity preservers, and weak majorization
- Time/place 7/18/2019, Thursday, 12:00 in Hill 705
- Abstract Which functions preserve positive semidefiniteness (psd) when applied entrywise to the entries of psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, who classified the positivity preservers of matrices of all dimensions. The study of positivity preservers in fixed dimension is harder, and a complete characterization remains elusive to date. In fact it was not known if there exists any analytic preserver with negative coefficients.
We prove such an existence result, and in fact a characterization, for classes of polynomials and power series. Central to the proof are novel determinantal identities involving Schur polynomials. If time permits, we will mention an application to a novel characterization of weak majorization, via Schur polynomials and the Harish-Chandra--Itzykson--Zuber integral. This parallels a conjecture of Cuttler--Greene--Skandera. (Joint with Alexander Belton, Dominique Guillot, and Mihai Putinar; and with Terence Tao.)
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