The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar has been running since 1997. The topics of talks have typically been any mixture of algebra with any other field: combinatorics, geometry, topology, physics, etc. Further down this page you will find links to the seminar webpages for previous years. The audience usually consists of 6–12 people, including several graduate students and post-docs. For this reason, speakers are encouraged to devote a portion of their talk to the suggestion of open problems and the directions for research in their area. If you are interested in speaking at the seminar, contact Nantel Bergeron.

You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

Dates are listed in reverse-chronological order. Unless otherwise indicated, all talks will take place at 3:00pm in N638 Ross Building (York University).

Date Speaker Title (click titles for abstract) 04 Oct. 2010

2:30 PM SPECIAL TIMEFlorian Block Computing Node Polynomials for Plane Curves Abstract: Enumeration of plane algebraic curves has a 150-year-old history. A combinatorial approach to this problem, inspired by tropical geometry, was recently suggested by Brugalle, Fomin, and Mikhalkin. I will explain this approach and its applications to computing Gromov-Witten invariants (or Severi degrees) of the complex projective plane, and their various generalizations.

According to Goettsche's conjecture (now a theorem), these invariants are given by polynomials in the degree d of the curves being counted, provided that d is sufficiently large. I will discuss how to compute these "node polynomials," and how large d needs to be.25 Oct. 2010 Luis G Serrano

Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials. Abstract:

We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between k-triangulations of the n-gon and k-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for k-triangulations with rotation to the language of k-flagged tableaux with promotion.

A preprint is available at http://arxiv.org/abs/1009.4690.

29 Nov. 2010 Letitia Banu Betti numbers of a rationally smooth toric variety Consider an irreducible representation of a semisimple algebraic group with $\lambda$ its highest weight and look at the action of the Weyl group W on the rational vector space spanned by the roots. Take the convex hull of the W-orbit of \lambda and obtain the polytope P_{\lambda}=Conv(W.\lambda). We are interested in describing the Betti numbers of the toric variety X(J) associated to the polytope $P_{\lambda}$ when the Weyl group is the n symmetric group and X(J) is a rationally smooth variety which doesn't depend on the highest weight \lambda but on the set of reflections that fix \lambda called J.The main result is a recursion formula for the Betti numbers of X(J) in terms of Eulerian polynomials. The theory of algebraic monoids developed by Renner and Putcha is effectively used in our computations.13 Dec. 2010 Mohammad Hassanzadeh (UNB) Hopf Algebras and Hopf Cyclic Cohomology in Noncommutative Geometry We briefly bring up the role of Hopf algebras in noncommutative geometry. As an example we mention the Hopf Galois extensions. We continue with Hopf cyclic cohomology which was discovered by Connes and Moscovici while studying the noncommutative geometry of foliations. At the end, we talk about some generalizations of Hopf algebras such as $\times$-Hopf algebras.17 Jan. 2011 Vivien Ripoll (LaCIM, UQAM) Noncrossing partition lattice and discriminant of a reflection group When W is a finite reflection group, the noncrossing partition lattice

NCP_W of type W is a very rich combinatorial object, extending

the notion of noncrossing partitions of an n-gon. A formula (only

proved case-by-case for now) expresses the number of multichains of a

given length in NCP_W as a generalized Fuss-Catalan number, depending

on the invariant degrees of W. We explain how to understand some

specifications of this formula geometrically, using an interpretation

of the chains of NCP_W as fibers of a "Lyashko-Looijenga covering",

constructed from the discriminant hypersurface of W. We deduce new

enumeration formulas for certain factorisations of a Coxeter element

of W.24 Jan. 2011 Grégoire Dupont (U. Sherbrooke) A combinatorial approach to cluster characters In the fast-growing theory of cluster algebras, a fruitful approach consists in studying cluster combinatorics through the lens of representation theory of algebras. More precisely, it is known that cluster combinatorics can be encoded in the tilting theory of certain triangulated 2-Calabi-Yau categories. One can even go further in these connections by defining a map, called the \emph{cluster character}, which provides a closed formula for realising distinguished elements in a cluster algebra in terms of objects in the corresponding category. Cluster characters are very efficient as a theoretical tool. However, they are usually hard to compute on concrete examples.

In this talk, I will present several methods for computing explicitly or algorithmically cluster characters in certain contexts, namely those of cluster characters associated to modules over tame hereditary algebras or to string modules over 2-Calabi-Yau tilted algebras. This is based on joint works with Assem and Assem-Schiffler-Smith.

Below you will find links to the seminar webpages for previous years.