Past Algebra Seminar

17 November 2014
17:00
Dawid Kielak
Abstract

We will introduce both the class of right-angled Artin groups (RAAG) and
the Nielsen realisation problem. Then we will discuss some recent progress
towards solving the problem.

• Algebra Seminar
17 June 2014
17:00
Prof. Travis Schedler.
Abstract
The Springer Correspondence relates irreducible representations of the Weyl group of a semisimple complex Lie algebra to the geometry of the cone of nilpotent elements of the Lie algebra. The zeroth Poisson homology of a variety is the quotient of all functions by those spanned by Poisson brackets of functions. I will explain a conjecture with Proudfoot, based on a conjecture of Lusztig, that assigns a grading to the irreducible representations of the Weyl group via the Poisson homology of the nilpotent cone. This conjecture is a kind of symplectic duality between this nilpotent cone and that of the Langlands dual. An analogous statement for hypertoric varieties is a theorem, which relates a hypertoric variety with its Gale dual, and assigns a second grading to its de Rham cohomology, which turns out to coincide with a different grading of Denham using the combinatorial Laplacian.
• Algebra Seminar
10 June 2014
17:00
Jon Gonzalez Sanchez
Abstract
Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G is called p-nilpotent if S has a normal complement N in G, that is, G is the semidirect product between S and N. The notion of p-nilpotency plays an important role in finite group theory. For instance, Thompson's criterion for p-nilpotency leads to the important structural result that finite groups with fixed-point-free automorphisms are nilpotent. By a classical result of Tate one can detect p-nilpotency using mod p cohomology in dimension 1: the group G is p-nilpotent if and only if the restriction map in cohomology from G to S is an isomorphism in dimension 1. In this talk we will discuss cohomological criteria for p-nilpotency by Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the 1960s and 1970s. Finally, we will discuss how one can extend Tate's result to study p-solvable and more general finite groups.
• Algebra Seminar
3 June 2014
17:00
Tsachik Gelander
Abstract
Gromov and Piatetski-Shapiro proved the existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about $v^v$ such manifolds of volume at most $v$, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi- isometry classes of lattices in $SO(n,1)$. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms. A joint work with Arie Levit.
• Algebra Seminar