Past Algebra Seminar

21 May 2013
17:00
Anreas Doering
Abstract
The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from unital C*-algebras to their spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures may lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.
7 May 2013
(All day)
Andreas Doring
Abstract
<p>The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from unital C*-algebras to their spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures may lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.</p>
30 April 2013
17:00
Elizaveta Frenkel
Abstract

In my talk I shall give a small survey on some algorithmic properties of amalgamated products of finite rank 
free groups. In particular, I'm going to concentrate on Membership Problem for this groups. Apart from being algorithmically interesting, amalgams of free groups admit a lot of interpretations. I shall show how to 
characterize this construction from the point of view of geometry and linguistic.  

26 February 2013
17:00
Alessandro Sisto
Abstract

I will discuss similarities and differences between the geometry of
relatively hyperbolic groups and that of mapping class groups.
I will then discuss results about random walks on such groups that can
be proven using their common geometric features, namely the facts that
generic elements of (non-trivial) relatively hyperbolic groups are
hyperbolic, generic elements in mapping class groups are pseudo-Anosovs
and random paths of length $n$ stay $O(\log(n))$-close to geodesics in
(non-trivial) relatively hyperbolic groups and
$O(\sqrt{n}\log(n))$-close to geodesics in mapping class groups.

12 February 2013
17:00
Alex Gorodnik
Abstract

We discuss the problem to what extend a group action determines geometry of the space. 
More precisely, we show that for a large class of actions measurable isomorphisms must preserve 
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

29 January 2013
17:00
Yago Antolin Pichel
Abstract

I will introduce the notion of Kurosh rank for subgroups of 
free products. This rank satisfies the Howson property, i.e. the 
intersection of two subgroups of finite Kurosh rank has finite Kurosh rank.
I will present a version of the Strengthened Hanna Neumann inequality in 
the case of free products of right-orderable groups. Joint work with  A. 
Martino and I. Schwabrow.

15 January 2013
17:00
Peter Kropholler
Abstract

The homological dimension of a group can be computed over any coefficient ring $K$.
It has long been known that if a soluble group has finite homological dimension over $K$
then it has finite Hirsch length and the Hirsch length is an upper bound for the homological
dimension. We conjecture that equality holds: i.e. the homological dimension over $K$ is
equal to the Hirsch length whenever the former is finite. At first glance this conjecture looks
innocent enough. The conjecture is known when $K$ is taken to be the integers or the field
of rational numbers. But there is a gap in the literature regarding finite field coefficients.
We'll take a look at some of the history of this problem and then show how some new near complement
and near supplement theorems for minimax groups can be used to establish the conjecture
in special cases. I will conclude by speculating what may be required to solve the conjecture outright.

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