Past Algebra Seminar

11 November 2014
Lukasz Grabowski

 Recently several conjectures about l2-invariants of
CW-complexes have been disproved. At the heart of the counterexamples
is a method of computing the spectral measure of an element of the
complex group ring. We show that the same method can be used to
compute the finite field analog of the l2-Betti numbers, the homology
gradient. As an application we point out that (i) the homology
gradient over any field of characteristic different than 2 can be an
irrational number, and (ii) there exists a CW-complex whose homology
gradients over different fields have infinitely many different values.

28 October 2014
Mike Prest

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them.  But not all.  To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest,  Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

17 June 2014
Prof. Travis Schedler.
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.
10 June 2014
Jon Gonzalez Sanchez
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.
3 June 2014
Tsachik Gelander
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.
27 May 2014
Michael Collins
In 1878, Jordan showed that if $G$ is a finite group of complex $n \times n$ matrices, then $G$ has a normal subgroup whose index in $G$ is bounded by a function of $n$ alone. He showed only existence, and early actual bounds on this index were far from optimal. In 1985, Weisfeiler used the classification of finite simple groups to obtain far better bounds, but his work remained incomplete when he disappeared. About eight years ago, I obtained the optimal bounds, and this work has now been extended to subgroups of all (complex) classical groups. I will discuss this topic at a “colloquium” level – i.e., only a rudimentary knowledge of finite group theory will be assumed.