9 de dezembro, 2014, 11:00 sala 221
Local Weyl modules for truncated current algebras: recent results and conjectures
Ghislain Fourier
Abstract
I will introduce the truncated current algebra of a simple complex Lie algebra and explain why their local Weyl modules are important finite-dimensional modules. Little is known about these local Weyl modules; so far there is no character formula or dimension formula. I will provide a conjecture on the structure of these modules as tensor products of simple modules. For this I will have to introduce a partial order on partitions of a given dominant weight and explain the relation to Schur positivity results and conjectures of symmetric functions. The conjecture on the tensor product structure has been proved in certain "extremal" cases during the last 15 years, and I will explain several approaches on how to attack this conjecture in general.
2 de dezembro, 2014, 11:00 sala 221
An introduction to geometric representation theory
Rollo Jenkins
25 de novembro, 2014, 11:00 sala 221
Finite dimensional representations of rational Cherednik algebras
Stephen Griffeth
Abstract
We will explain part of our joint work with A. Gusenbauer, D. Juteau, and M. Lanini on the classification of finite dimensional representations for the rational Cherednik algebras of complex reflection groups. Though we have not achieved a complete classification, we obtain necessary conditions for finite dimensionality that seem to be quite close to the truth in practice. The key point is the calculation of the eigenvalues of monodromy for certain systems of differential equations analogous to the Knizhnik-Zamolodchikov equations. For the Cherednik algebra of the symmetric group, our ideas can be simplified quite a bit to give a new, nearly elementary, proof of the classification in this case (originally due to Berest-Etingof-Ginzburg).
18 de novembro, 2014, 11:00 sala 221
Degenerescência da sequência espectral de Hodge–de Rham em característica positiva, II
Nuno Filipe de Andrade Cardoso
11 de novembro, 2014, 11:00 sala 221
Degenerescência da sequência espectral de Hodge–de Rham em característica positiva, I
Nuno Filipe de Andrade Cardoso
4 de novembro, 2014, 11:00 sala 221
Interactions between the representation theory of the symmetric group Sn, the ring of symmetric polynomials An, and the current algebra sl2[t]
Matthew Bennett
28 de outubro, 2014, 11:00 sala 221
Deformations of moduli stacks of vector bundles
Severin Barmeier
21 de outubro, 2014, 11:00 sala 221
Mutations of potentials and B-model Laurent phenomenon
John Alexander Cruz Morales
14 de outubro, 2014, 11:00 sala 221
Givental's mirror theorem
Elizabeth Gasparim
7 de outubro, 2014, 11:00 sala 221
Decomposição celular de variedades flag reais
Jordan Lambert
30 de setembro, 2014, 11:00 sala 221
Symplectic Lefschetz fibrations from a Lie-theoretical viewpoint
Brian Callander
23 de setembro, 2014, 11:00 sala 221
Recent developments of representation theory of nonrelativistic conformal algebras
Naruhiko Aizawa
Abstract
Nonrelativistic conformal algebras are a particular class of non-semisimple Lie algebras.
A member of the class is a finite or an infinite dimensional Lie algebra. The semisimple
part of the finite dimensional algebras is the direct sum of sl2 and sod,
while the Virasoro algebra is the semisimple part of the infinite dimensional algebras.
This class of Lie algebras appears in various kinds of problems in theoretical and mathematical
physics. For instance, one can find them in connection with fluid dynamics, gravity theory, the
AdS/CFT correspondence and vertex operator algebras. This motivates us to study representations
of the nonrelativistic conformal algebras.
In the beginning of this talk, I introduce various nonrelativistic conformal
algebras. Then I pick up some of physical interest and study them in some more detail.
Our first problem is the central extensions of the algebras. The list of possible central
extensions is given. Our second problem is the irreducible representations of highest (lowest)
weight type.
We start with the Verma module and study its irreducibility. This is done by calculating the
Kac determinant and giving an explicit construction of singular vectors. If the Verma module
is reducible, then it will be shown that how to obtain the irreducible modules.
16 de setembro, 2014, 11:00 sala 221
Mirror symmetry in the Hitchin system
Emilio Franco Gomez
9 de setembro, 2014, 11:00 sala 221
Symplectic and Koszul duality, II
Rollo Jenkins
2 de setembro, 2014, 11:00 sala 221
Symplectic and Koszul duality, I
Rollo Jenkins
Abstract
Symplectic duality (in the sense of Braden-Licata-Proudfoot-Webster) is a new construction
that marries together a diverse range of subjects in Lie theory, non-commutative representation theory,
algebraic and symplectic geometry, differential operator theory and theoretical physics.
It is conjectured to give an algebraic formulation of mirror dualities in physics.
I'll give an overview of the construction of geometric and algebraic category O's
with three examples. Commit now for the one-time price of fifty minutes* and get an
introduction to Koszul duality absolutely free!
*price excludes ten minutes of question time.
26 de agosto, 2014, 11:00 sala 221
Lagrangian skeleton and mirror symmetry
Lino Grama
19 de agosto, 2014, 11:00 sala 221
A realization of tilting modules for sl2[t]
Matthew Bennett
12 de agosto, 2014, 11:00 sala 221
Higher symmetries of Laplacian via quantization
Jean-Philippe Michel
Abstract
We first review the seminal results of M. Eastwood on the so-called higher
symmetries of the Laplacian [2]. In particular, in dimension n, they
form an algebra isomorphic to the quotient of the universal enveloping algebra
of o(n+2, C) by the Joseph ideal [3]. We propose a new method to classify
those symmetries [4], relying on the conformally equivariant quantization [1].
This allows to compute the Joseph ideal and provide a geometric interpretation
for it. Moreover, in signature (p, q), we provide links between: higher
symmetries of the Laplacian, the minimal representation of O(p+1, q+1)
on the kernel of the Laplacian and the invariant star-product on the minimal
coadjoint orbit of O(p+1, q+1).
References.
[1] C. Duval, P.B.A. Lecomte & V. Yu. Ovsienko, "Conformally equivariant
quantization: existence and uniqueness", Ann. Inst. Fourier (Grenoble),
49(6): 1999-2029, 1999.
[2] M.G. Eastwood, "Higher symmetries of the Laplacian", Ann. Math.,
161(3): 1645-1665, 2005.
[3] A. Joseph, "The minimal orbit in a simple Lie algebra and its associated
maximal ideal", Ann. Sci. Ecole Norm. Sup., 9(1): 1-29, 1976.
[4] J.-Ph. Michel, "Higher symmetries of Laplacian via quantization",
Ann. Inst. Fourier (to appear).
26 de junho, 2014, 11:00 sala 121
Trendy topics in Lie theory
Adriano Moura
11 de junho, 2014, 11:00 sala 321
Fibrações de Lefschetz em órbitas adjuntas
Elizabeth Gasparim
3 de junho, 2014, 11:00 sala 121
Lie algebras, modular forms and elliptic curves
Reimundo Heiluani
Abstract
Ever since Frobenius it has been noticed that characters of representations
are special functions. Every known family of orthogonal polynomials, Bessel
functions, trigonometric functions, spherical functions and such appear this
way. A new phenomenon started to unveil in the late 70's and 80's when it
was discovered that characters of modules of certain infinite dimensional
Lie algebras and groups where modular invariant. At the time there were
plenty of examples both coming from physics and representation theory, but
no good explanation of this phenomenon. It was in the mid 90's that Zhu
settled this question putting in a rigorous framework some ideas from string
theory: these characters are naturally defined (flat) sections of certain
bundles on the moduli space of elliptic curves. This immediately shows the
modular invariance since a coarse version of this moduli space is simply H
(the upper half plane) divided by SL(2,Z) (the modular group).
In later years many examples have arisen involving super Lie algebras instead
of simply Lie algebras. In all these examples some extended version of
modularity is found. We show that under certain conditions the characters of
modules for these Lie algebras are Jacobi modular forms (that is invariant
for the group SL(2,Z) \ltimes Z²) by naturally constructing them as sections
of some flat bundles on the moduli space of elliptic supercurves. The
reduced part of this moduli space parametrizes an ellptic curve and a line
bundle over it, so it is simply the universal elliptic curve (or rather its
Jacobian), but some very involved subtleties arise when working on families
over supercommutative rings.
This is joint work with Jethro Van Ekeren
29 de maio, 2014, 11:00 sala 121
Mirror symmetry as duality between LG models
Elizabeth Gasparim
22 de maio, 2014, 11:00 sala 121
Classes de Chern de variedades flag
Ailton Ribeiro de Oliveira
15 de maio, 2014, 11:00 sala 121
Grassmannianas de espaços de Hilbert
Jordan Lambert
Jordan Lambert
29 de abril, 2014, 11:00 sala 121
The moduli stack of G-bundles, II
Emilio Franco Gomez
24 de abril, 2014, 11:00 sala 121
The moduli stack of G-bundles, I
Emilio Franco Gomez
Abstract
We will start by recalling the notions of sheaves, G-bundles and families of G-bundles. Introducing the classification problem of G-bundles, we will define the moduli functors on sets and grupoids. This motivates the definition of stacks as sheaves of grupoids on Grothendieck topologies. Later we will see an equivalent notion of stacks as categories fibered on groupoids. Finally (if we still have time) we will describe certain stacks of G-(Higgs-)bundles over elliptic curves and some nice properties of them.
16 de abril, 2014, 10:00 sala 221
Quantum differential operators on the quantum torus
Uma N. Iyer
Abstract
Following the definition of the algebra of quantum differential operators by Lunts and Rosenberg, we present concrete examples of these algebras and their properties.