# Past Algebra Seminar

Dimer models with boundary were introduced in joint work with King and Marsh as a natural

generalisation of dimers. We use these to derive certain infinite dimensional algebras and

consider idempotent subalgebras w.r.t. the boundary.

The dimer models can be embedded in a surface with boundary. In the disk case, the

maximal CM modules over the boundary algebra are a Frobenius category which

categorifies the cluster structure of the Grassmannian.

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)

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.

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

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).