—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Spring 2019

Wednesdays, 3:45–4:35. IES111

Webpages from previous semesters

Organizers:  Emily Peters, Peter Tingley


January

No talks (please attend special colloquia!
February
6
No talk
13
No talk
20
Organizational meeting
27
Peter Tingley
Quiver varieties and root multiplicities for symmetric Kac-Moody algebras
March
6 No talk (spring break)
13
Nicolas Mayers (Lehigh University)
Calculating the Index of Poset Algebras
20
Marius Radulescu
Vector Bundles on Elliptic Curves
27

April
3
Daniele Rosso (Indiana University Northwest)
10
Aaron Lauve
17
Christopher Drupieski
24
Peter or Tony?

Directions

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)

Parking is available on-campus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.



Abstracts

February 27: Peter tingley. Quiver varieties and root multiplicities for symmetric Kac-Moody algebras

We discuss combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from a realization of the infinity crystal using quiver varieties. The framework is quite general, but we only work out specifics for one special case. We conjecture that our bound is quite tight, and give both computational evidence and heuristic justification for this conjecture, but unfortunately not a proof.



March 13: Nicholas Mayers. Calculating the Index of Poset Algebras

The index of a Lie algebra is an important algebraic invariant which is usually quite difficult to compute. Recently, there has been a push to find closed-form formulas for the index of families of Lie algebras which are combinatorially defined. Much of this work is related to what are known as Seaweed algebras which are subalgebras of sl(n) which can be parametrized by a pair of compositions of n. In this talk I will discuss some recent work related to finding combinatorial formulas for the index of a different family of Lie algebras which are defined via posets.

March 20: Marius Radulescu. Vector Bundles on Elliptic Curves

In his paper "Vector bundles over an elliptic curve," Michael Atiyah classified indecomposable vector bundles of arbitrary rank and degree on elliptic curves. The set of isomorphic classes of indecomposable vector bundles of fixed rank and degree is isomorphic with the Jacobian of the elliptic curve, which in turn is isomorphic with the curve itself. Based on Atiyah's paper, I will provide a detailed description of rank 2, 3 and 4 vector bundles on an elliptic curve.