Our seminar in applied algebra has been running since 1997.

The topics of talks have typically been any mixture of algebra with any other fields: combinatorics, geometry, topology, physics, etc.

We usually have half a dozen to a dozen people in attendance at the seminars, and this usually includes several graduate students and post-docs.
For this reason, we encourage the speakers to devote a portion of their talk to the suggestion of open problems and the direction of research in their area.

• note some info, day and time changes in RED
• dates and speaker in ORANGE are special join seminar AAS-Fields on Friday

• Title: Forms of Conformal Superalgebras
• Date: Monday, March 30, 2009
• Time: 3:00p.m.
• Place: Room 638 North Ross

Conformal superalgebras describe symmetries of superconformal field theories and come equipped with an infinite family of products. They also arise as singular parts of the vertex operator superalgebras associated with some well-known Lie structures (e.g. affine, Virasoro, Neveu-Schwarz).

In joint work with Arturo Pianzola and Victor Kac, we classify forms of conformal superalgebras using a non-abelian Cech-like cohomology set. As the products in scalar extensions are not given by linear extensions of the products in the base ring, the usual descent formalism cannot be applied blindly. As a corollary, we obtain a rigourous proof of the pairwise non-isomorphism of an infinite family of N=4 conformal superalgebras appearing in mathematical physics.

• Title: Twisted vertex operators and Steinberg unitary Lie algebras
• Date: Monday, March 16, 2009
• Time: 3:00p.m.
• Place: Room 638 North Ross

We will use the twisted Heisenberg Lie algebras to construct a family of vertex operators. These will provide representations for certain Steinberg unitary Lie algebras coordinatized by quantum tori. As a by-product we recover some of affine Kac-Moody Lie algebras.

This is a joint work with N.Jing and S.Tan.

• Title: The non-symmetric operad pre-Lie is free
• Date: Monday, March 09, 2009
• Time: 3:00p.m.
• Place: Room 638 North Ross

I will show that the pre-Lie operad is a free non- symmetric operad. I will do some recall on the notions of Operads.

This is joint work with M. Livernet

• Title: Finite-dimensional representation theory of loop algebras
• Date: Monday, March 02, 2009
• Time: 3:00p.m.
• Place: Room 638 North Ross

Let G be a simple complex finite-dimensional Lie algebra, and L=L(G) a Loop algebra corresponding to a diagram automorphism of G. These algebras occur as the main in a construction of the Affine Lie algebras, and their representation theory is subject of continuing interest. Some aspects of the finite-dimensional representation theory of the 'twisted' loop algebras are now well understood. In particular, the universal 'loop-highest weight' Weyl modules have recently been described, has have the blocks of the corresponding (non-semisimple) category.

We will discuss these recent classifications, as provide a more geometric reformulation of the results. We will then discuss possible extensions of this theory to the multiloop generalizations of L, and beyond.

• Title: Non-commutative LLT polynomials
• Date: Monday, February 23, 2009
• Time: 3:00p.m.
• Place: Room 638 North Ross

In 1990, the LLT polynomials were defined as q-analogs of products of Schur functions. In this talk, we define new analogs of LLT polynomials on the non-commutative side. We give an analog of the quotient map, and we also introduce an interpretation of these new polynomials in terms of representations of the Hecke algebra. This non-commutative approach could give us some ideas for solving some problems on the commutative side.

• Title: Non-commutative polynomial invariants of finite groups and word relations
• Date: Monday, February 09, 2009
• Time: 3:30p.m.
• Place: Room 638 North Ross

Let V be a vector space over the complex with basis {x1,x2,...,xn} and G be a finite subgroup of GL(V). Then the tensor algebra T(V) of V over the complex is isomorphic to the polynomials in the non- commutative variables x1, x2, ..., xn with complex coefficients. We consider the graded space of invariants in T(V) with respect to the action of G. More generally, we want to give a combinatorial interpretation for the decomposition of T(V) into irreducible representations. For the symmetric and dihedral groups, we have a link between the representations and a special subalgebra of the group algebra that gives an interpretation of T(V) in terms of words in the associated Cayley graph. In particular, we have an interpretation for the graded dimensions of the invariants in T(V) and we give closed formulas for those graded dimensions. These examples suggest a general method to find the decomposition of T(V). To that end, we would need to find a special set of generators and relations for the group G and then the decomposition of T(V) would be linked to words in the Cayley graph of G associated to those generators.

• Title: q,t-counting Dyck paths below a staircase
• Date: Monday, February 02, 2009
• Time: 3:00p.m.
• Place: Room 638 North Ross

I will present recent joint work with Jim Haglund and Jennifer Morse about the action of the operator nabla on Hall-Littlewood symmetric functions. The combinatorics of Dyck paths leads us to very surprising symmetric function identities relating Hall-Littlewood symmetric functions indexed by compositions.

• Title: Volumes of matroid polytopes
• Date: Monday, January 26, 2009
• Time: 3:00p.m.
• Place: Fields Institute, Room 210

Given a matroid M we can associate with it its matroid polytope $P_M$ as well as its independent set polytope I_M. In this talk I will show an explicit way to decompose these polytopes as a signed Minkowski sum of simplices. Using this decomposition, which involves a lot of information of M such as its beta invariant, I will give a nice formula to calculate the volumes of P_M and I_M, offering a geometric point of view for beta (M). Finally, we will see analogous results in the case of a nice class of flag matroids, namely the cascading flag matroids. In this case the role of beta (M) will be done by the gamma invariant.
This is joint work with Federico Ardila (San Francisco State University) and Jeffrey Doker (UC Berkeley).

• Title: Imperfect gases and graph invariants
• Date: Monday, November 24, 2008
• Time: 3:00p.m.
• Place: Fields Institute, Room 210

The Mayer and Ree-Hoover theories for the virial expansions in the context of a non-ideal gas reveal certain invariants (weights) associated to graphs. We give a special attention to the case of the hard-core continuum gas in one dimension. We present the method of graph homomorphisms that we apply to compute the Mayer and Ree-Hoover weights of various classes of graphs.

• Title: MV polytopes and components of quiver varieties
• Date: Monday, November 17, 2008
• Time: 3:00p.m.
• Place: Fields Institute, Room 210

A number of interesting bases exist for the upper half of the universal envelopping algebra of a semisimple Lie algebra.
One such basis is Lusztig's semicanonical basis which is indexed by components of quiver varieties.
Another interesting basis is indexed by Mirkovic-Vilonen cycles which lead to the combinatorics of MV polytopes.
In this talk, I will explain a natural bijection between the components of quiver varieties and the MV polytopes.
This is joint work with Pierre Baumann.

• Title: Antichains in the poset of positive roots, Catalan phenomena, and some conjectures of Panyushev
• Date: Monday, November 10, 2008
• Time: 3:00p.m.
• Place: Fields Institute, Room 210

• Title: Hyperoctahedral species
• Date: Monday, November 3, 2008
• Time: 3:00p.m.
• Place: Room 638 North Ross

We introduce a new definition for species of type B, called H-species, analog to the classical species (of type A), but on which we consider the action of the groups Bn of signed permutations.
Classical species are modules of the symmetric group and H-species are modules of the hyperoctahedral group.
We are interested in algebraic structure on these H-species and give examples of Hopf monoids.
Many functors allow us to obtain H-species from species, and one functor is particularly interesting as it sends the regular representation of Sn to the regular representation of Bn.
With this functor and the natural way to get a graded vector space from a species, we give a link between the classical species of set composition and the combinatorial Hopf algebra DQSym.

• Title: Combinatorial Hopf algebras
• Date: Monday, October 27, 2008
• Time: 3:00p.m.
• Place: Room 638 North Ross

Many recent papers are devoted to some infinite dimensional Hopf algebras called collectively "combinatorial Hopf algebras". Among the examples we find the Faa di Bruno algebra, the Connes-Kreimer algebra and the Malvenuto-Reutenauer algebra. We give a precise definition of such an object and we provide a classification. We show that the notion of preLie algebra and of brace algebra play a key role.

Back to main page...

Other archives