Ottawa-Carleton Joint Algebra Seminar

Let G be the group of rational points of a connected reductive group over a p-adic field. Under very mild hypotheses on p, we have a complete, explicit classification of the irreducible complex representations of G, with many desirable properties—the classification is in terms of what Bushnell and Kutzko call typical representations. It's expected that this classification is, up to minor renormalization, the only one with such nice properties, but this is only currently known in a few special cases. I'll explain joint work with Monica Nevins which reduces the general case to some explicit questions about the geometry of the Bruhat–Tits building of G.

In this talk we'll give an introduction to multi-point Seshadri constants for ample line bundles on smooth projective varieties, and discuss their relationship with the nef cone. We'll also go over the structure of the nef cone as described by a theorem of Mori, and show some explicit calculations for blow-ups of \(\mathbb{P}^1 \times \mathbb{P}^1\) at an even number of points greater than eight.

There has been lots of interest, motivated by a conjecture from Adler and Prasad (2006), regarding the restriction of representations of p-adic groups to subgroups containing the derived subgroup. Naturally, the question comes down to determining under what conditions this type of restriction might be multiplicity free. In this talk, we will be focusing on a special class of representations, namely the regular depth-zero supercuspidal representations. The construction of such representations requires Deligne-Luzstig cuspidal representations of certain finite groups of Lie type. We will see that the condition for a multiplicity free restriction depends on the properties of these particular cuspidal representations.

Our talk deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. The notion of cohomological invariants was formalized by Serre in the 90's. It enables to study via Galois cohomology the geometry of linear algebraic groups or forms of algebraic stuctures (such as central simple algebras with involution).

We intend to introduce the general ideas of the theory and to present a generalization of a result by Blinstein and Merkurjev on degree 2 invariants with coefficients Q/Z(1), that is invariants taking values in the Brauer group. More precisely our result gives a description of these invariants for every smooth and connected linear groups, in particular for non reductive groups over an imperfect field (as pseudo-reductive or unipotent groups for instance).

To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. The definition of these algebras is motivated by constructions in Heisenberg categorification. They can be viewed both as symmetric algebra deformations of affine Hecke algebras of type A and as quantum deformations of affine wreath algebras. Special cases include affine Yokonuma–Hecke algebras, and quantum deformations of affine zigzag algebras. This is joint work with Daniele Rosso.

How can we understand the spaces embedded into a fixed projective space? By design, Hilbert schemes provide the geometric answer to this question. After surveying some features of these natural parameter spaces, we aim to determine when a Hilbert scheme is smooth.

Let \(\mathbb{F}_p\) denote the finite field of order \(p\) and \(\mathbb{F}\) its algebraic closure. Classifying the \(\mathbb{F}\)-representations of \(\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}\) leads to a simply stated geometric problem involving \(\mathbb{F}_p\)-planes in \(\mathbb{F}\). Solving this leads in turn to an infinite family of polynomials in \(\mathbb{F}[t]\). These polynomials have a number of surprising algebraic and combinatorial properties and satisfy a recursion relation related to that studied by D.H. Lehmer in his thesis. This presentation will be accessible to graduate students and senior undergraduates. This is joint work with H. E. A. Campbell.

TBA

Toda Brackets are a kind of higher homotopy operation, originally introduced by Toda to study unstable homotopy groups of spheres. They can be viewed as successive obstructions to rectifying a certain kind of homotopy commutative diagram. Baues, Blanc and their collaborators gave a more general definition of Toda brackets (and other higher homotopy operations) in pointed simplicial (model) categories.

In this talk will explain how to generalize their work further, defining Toda brackets (and an unstable analogue) in two different models of \((\infty, 1)\)-categories: simplicially enriched categories and quasi-categories. We will also explain how to construct Bousfield-Kan type spectral sequences in quasi-categories satisfying certain assumptions, and explain how the differentials in these spectral sequences can be described in terms of higher homotopy operations.

This is in progress work with David Blanc.

TBA