Past Algebra Seminar

17 October 2017
Aslak Buan

We give a short reminder about central results of classical tilting theory, 
including the Brenner-Butler tilting theorem, and
homological properties of tilted and quasi-tilted algebras. We then discuss 
2-term silting complexes and endomorphism algebras of such objects,
and in particular show that some of these classical results have very natural 
generalizations in this setting.
(joint work with Yu Zhou)

16 May 2017
Maxim Nazarov

Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at N → ∞ . These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.

3 May 2017
Mikhail Ershov

Let $A$ denote either the automorphism group of the free group of rank $n$ or the mapping class group of an orientable surface of genus $n$ with at most 1 boundary component, and let $G$ be either the subgroup of IA-automorphisms or the Torelli subgroup of $A$, respectively. I will discuss various finiteness properties of subgroups containing $G_N$, the $N$-th term of the lower central series of $G$, for sufficiently small $N$. In particular, I will explain why
(1) If $n \geq 4N-1$, then any subgroup of G containing $G_N$ (e.g. the $N$-th term of the Johnson filtration) is finitely generated
(2) If $n \geq 8N-3$, then any finite index subgroup of $A$ containing $G_N$ has finite abelianization.
The talk will be based on a joint work with Sue He and a joint work with Tom Church and Andrew Putman

2 May 2017
Beth Romano

Let G be a split reductive group over a finite extension k of Q_p. Reeder and Yu have given a new construction of supercuspidal representations of G(k) using geometric invariant theory. Their construction is uniform for all p but requires as input stable vectors in certain representations coming from Moy-Prasad filtrations. In joint work, Jessica Fintzen and I have classified the representations of this kind which contain stable vectors; as a corollary, the construction of Reeder-Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G(k). 

7 March 2017
Niamh Farrell

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Morita Frobenius numbers were introduced by Kessar in 2004 in the context of Donovan’s Conjecture in block theory. I will present the latest results of a project in which we aim to calculate the Morita Frobenius numbers of the blocks of quasi-simple finite groups. I will also discuss the importance of a recent result of Bonnafe-Dat-Rouquier for our methods, and explain the relationship between Morita Frobenius numbers and Donovan’s Conjecture. 

28 February 2017

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is:  what are the minimal models within a birational class?  It is not even clear a priori what the correct definition is of a minimal model in this context.

We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.
This is joint work in progress with Dan Rogalski and Toby Stafford.


24 January 2017

Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an
Euler-Poincare formula for the r-depth Bernstein projector. We establish an Euler-Poincare formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P.  This work is joint with Dan Barbasch and Dan Ciubotaru.

17 January 2017

The local Langlands correspondence for classical groups gives a natural finite-to-one map between certain representations of p-adic classical groups and certain self-dual representations of the absolute Weil group of a p-adic field (and more). On both sides of the correspondence, the description of the representations involves a ``wild part'' of more arithmetic nature and a ``tame part'' of more geometric nature, and the notion of endo-parameter (due to Bushnell--Henniart for general linear groups) is designed to describe the ``wild part'' of the Langlands correspondence. I will explain what this means and the connection with representations of affine Hecke algebras. This is joint work with Blondel--Henniart, with Lust, and with Kurinczuk--Skodlerack.