Spring, 2010

Spring, 2010

  • Speaker Zajj Daugherty, University of Wisconsin, Madison
    • Title Degenerate two-boundary centralizer algebras
    • Time/place 1/22/2010, Friday, 11:45 am in Hill 425
    • Abstract Degenerate two-boundary centralizer algebras arise as algebras of commuting operators for a Lie algebra action on a tensor space. These algebras generalize the group algebra of the symmetric group, and graded Hecke and Brauer algebras, and have similarly elegant combinatorial structure. In this talk, we will explore centralizers of the actions of .nite dimensional complex reductive Lie algebras on tensor spaces of the form $M \otimes N \otimes V^{\otimes k}$. As an example, we will explore in detail the combinatorics of special cases corresponding to $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ and explain how this might be applied to the study of the combinatorial representation theory of graded Hecke algebras of type C.

  • Speaker Anton Zeitlin, Yale University
    • Title From Lian-Zuckerman Algebras to the Algebraic Structure of Classical Field Equations
    • Time/place 1/29/2010, Friday, 11:45 am in Hill 425
    • Abstract We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (LZ_{qc} algebra) on the subcomplex, corresponding to light modes, i.e. the elements of zero conformal weight, of the semi- infinite (BRST) cohomology complex of the Virasoro algebra associated to Vertex Operator Algebra (VOA) with a formal parameter. We show that LZ_{qc} algebra is actually a certain "double" of a Courant Algebroid associated with the corresponding VOA. We also discuss certain deformations related to abelian vertex subalgebras of the A_{\infty} subalgebra of LZ_{qc} algebra. In particular case of VOA associated with beta-gamma systems it will be shown that the deformations of this kind lead to the A_{\infty} algebra of the Yang-Mills equations.

  • Speaker Miranda Cheng, Harvard University
    • Title Black Holes, Wall-Crossing and Generalized Kac-Moody Algebras
    • Time/place 2/5/2010, Friday, 11:45 am in Hill 425
    • Abstract The relation between generalised Kac-Moody algebras, automorphic forms and hyperbolic reflection groups has led to many fascinating results in mathematics. In my talk I will report on results regarding the role of these objects in the physics of black holes. I will also describe how, through string theory, these results have lead to interesting conjectures in geometry.

  • Speaker David Jordan, MIT and Eric Larson, Harvard
    • Title Fusion categories of dimension pq^2
    • Time/place 2/19/2010, Friday, 11:45 am in Hill 425
    • Abstract Fusion categories are certain ``small'' tensor categories, which generalize categories of representations of finite groups. After introducing fusion categories and their basic properties, we outline a theory of extensions and their obstructions due to Etingof, Nikshych, and Ostrik, which is informed by methods in homotopy theory. Fusion categories have a notion of dimension, and have been completely classified in dimensions p^k, pq, and pqr, for distinct primes p, q, r. Using the theory of extensions, we completely classify fusion categories of dimension pq^2, producing many new examples which are not ``group-theoretical" in the sense of Ostrik.

  • Speaker Tony Giaquinto, Loyola University of Chicago (canceled)
    • Title Graphs, Frobenius functionals, and the classical Yang-Baxter equation
    • Time/place 3/12/2010, Friday, 11:45 am in Hill 425 (canceled)
    • Abstract A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a "cyclic" functional F on P(n,m) and prove it is non-degenerate using properties of certain graphs associated to F. These graphs also provide in some cases readily computable associated solutions of the classical Yang-Baxter equation. We also examine the seaweed Lie algebras of Dergachev and Kirillov from our perspective.

  • Speaker Zhiwei Yun, IAS
    • Title A topological shadow of Mirror Symmetry for dual Hitchin fibrations
    • Note Joint Lie Group/Quantum Mathematics/Geometry, Symmetry, and Physics Seminar
    • Time/place 3/26/2010, Friday, 11:45 am in Hill 425
    • Abstract For a reductive group G and its Langlands dual G^, their Hitchin fibrations are dual torus fibrations. Homological Mirror Symmetry and Geometric Langlands duality predicts an equivalence between categories constructed from the dual fibrations, respecting certain natural symmetries. Being less ambitious, we will establish the shadow of this prediction on the level of cohomology, relating "global Springer actions" on (parabolic) Hitchin fibers of G and "Chern class actions" on those of G^.

  • Speaker Antun Milas, University at Albany (SUNY)
    • Title W-algebra extensions of minimal models
    • Time/place 4/2/2010, Friday, 11:45 am in Hill 425
    • Abstract I will explain structure and representations of a family of W-algebras obtained as extension of the Virasoro vertex algebras of minimal central charge, and in particular the c=0 algebra (joint work with D. Adamovic). These W-algebras are of considerable interest in logarithmic CFT and critical percolation. In addition, their representation theory is expected to be identical to that of certain finite dimensional quantum groups (via Kazhdan-Lusztig correspondence). Time permitting, I will also describe how considerations of W-algebras lead to new constant term and q-series identities.

  • Speaker Xiao-Gang Wen, MIT
    • Title Symmetric polynomial of infinite variables and quantum Hall states
    • Note Joint Lie Group/Quantum Mathematics/Mathematical Physics Seminar
    • Time/place 4/9/2010, Friday, 11:45 am in Hill 425
    • Abstract I will discuss the close relation between symmetric polynomial of infinite variables, vertex algebra, and quantum Hall states. I will present some attempts to classify symmetric polynomial of infinite variables.

  • Speaker Aaron Lauda, Columbia and MSRI
    • Title A diagrammatic categorification of quantum groups
    • Time/place 4/30/2010, Friday, 11:45 am in Hill 425
    • Abstract I'll explain joint work with Mikhail Khovanov on a categorification of one-half of the quantum universal enveloping algebra associated to a Kac-Moody algebra. This categorification is obtained from the graded representation category of certain graded algebra that can be defined using a graphical calculus. Certain finite-dimensional quotients of these graded algebras give categorifications of irreducible representations of the quantum enveloping algebra. Extensions of these rings are used in Webster's recent work categorifying Reshetikhin-Turaev tangle invariants.