Past Algebra Seminar

Affinoid enveloping algebras arise as certain p-adic completions of ordinary enveloping algebras, and are closely related to Iwasawa algebras. I will explain how to use Beilinson-Bernstein localisation to compute their (non-commutative) Krull dimension. This is recent joint work with Ian Grojnowski.

Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup corresponding to the nilradical n of b. Extremal projection operators are projection operators onto the subspaces of n-invariants in certain g-modules the action of n on which is locally nilpotent. Zhelobenko also introduced a family of operators which are analogues to extremal projection operators. These operators are called now Zhelobenko operators.
I shall show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence (N, h) -> b given by the restriction of the adjoint action. Simple geometric proofs of  formulas for the ``classical'' counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.

We will first recall the known notion of commensurating actions

and its link to actions on CAT(0) cube complexes. We define a

group to have Property FW if every isometric action on a CAT(0)

cube complex has a fixed point. We conjecture that every

irreducible lattice in a semisimple Lie group of higher rank has

Property FW, and will give some instances beyond the trivial

case of Kazhdan groups.

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.

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.

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.  

I will discuss some recent developments in Schubert calculus and a potential relation to classical combinatorics for symmetric groups and possible extensions to complex reflection groups.

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.

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.

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.

 One of the applications of the study of assymptotics of
homology groups in residually free groups of type FP_m is the calculation
of their analytic betti numbers in dimension up to m.

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.

I'll discuss some results about lattices in totally
disconnected locally compact groups, elaborating on the question:
which classical results for lattices in Lie groups can be extended to
general locally compact groups. For example, in contrast to Borel's
theorem that every simple Lie group admits (many) uniform and
non-uniform lattices, there are totally disconnected simple groups
with no lattices. Another example concerns with the theorem of Mostow
that lattices in connected solvable Lie groups are always uniform.
This theorem cannot be extended for general locally compact groups,
but variants of it hold if one implants sufficient assumptions. At
least 90% of what I intend to say is taken from a paper and an
unpublished preprint written jointly with P.E. Caprace, U. Bader and
S. Mozes. If time allows, I will also discuss some basic properties
and questions regarding Invariant Random Subgroups.

Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of Aut(G) on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice Γ in G is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of G is connected.  I will introduce all necessary notions and try to motivate and explain this result.

Most results about the Cayley graph of a hyperbolic surface group can be replicated in the context of more general hyperbolic groups. In this talk I will discuss two results about such Cayley graphs which I do not know how to replicate in the more general context.

Let G be a permutation group acting on a set Omega. For g in G, let pi(g) denote the partition of Omega given by the orbits of g. The set of all partitions of Omega is naturally ordered by refinement and admits lattice operations of meet and join. My talk concerns the groups G such that the partitions pi(g) for g in G form a sublattice. This condition is highly restrictive, but there are still many interesting examples. These include centralisers in the symmetric group Sym(Omega) and a class of profinite abelian groups which act on each of their orbits as a subgroup of the Prüfer group. I will also describe a classification of the primitive permutation groups of finite degree whose set of orbit partitions is closed under taking joins, but not necessarily meets.

This talk is on joint work with John R. Britnell (Imperial College).

In our 2004 paper, Fritz Grunewald and I constructed the first
pairs of finitely presented, residually finite groups $u: P\to G$
such that $P$ is not isomorphic to $G$ but the map that $u$ induces on
profinite completions is an isomorphism. We were unable to determine if
there might exist finitely presented, residually finite groups $G$ that
with infinitely many non-isomorphic finitely presented subgroups $u_n:
P_n\to G$ such that $u_n$ induces a profinite isomorphism. I shall
discuss how two recent advances in geometric group theory can be used in
combination with classical work on Nielsen equivalence to settle this
question.

 We present a new, more conceptual proof of our result that, when a finite group acts on a polynomial ring, the regularity of the ring of invariants is at most zero, and hence one can write down bounds on the degrees of the generators and relations. This new proof considers the action of the group on the Cech complex and looks at when it splits over the group algebra. It also applies to a more general class of rings than just polynomial ones.

I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|. 
I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.

There is a well-acknowledged analogy between mapping class
groups and lattices in higher rank groups. I will discuss to which
extent does Margulis's superrigidity hold for mapping class groups:
examples, very partial results and questions.

We prove that the rank gradient vanishes for mapping class groups, Aut(Fn) for all n, Out(Fn), n > 2 and any Artin group whose underlying graph is connected. We compute the rank gradient and verify that it is equal to the first L2-Betti number for some classes of Coxeter groups.

I’ll report on my recent work (with co-authors Holt and Ciobanu) on Artin

groups of large type, that is groups with presentations of the form

G = hx1, . . . , xn | xixjxi · · · = xjxixj · · · , 8i 3. (In fact, our results still hold when some, but not all

possible, relations with mij = 2 are allowed.)

Recently, Holt and I characterised the geodesic words in these groups, and

described an effective method to reduce any word to geodesic form. That

proves the groups shortlex automatic and gives an effective (at worst quadratic)

solution to the word problem. Using this characterisation of geodesics, Holt,

Ciobanu and I can derive the rapid decay property for most large type

groups, and hence deduce for most of these that the Baum-Connes conjec-

ture holds; this has various consequence, in particular that the Kadison-

Kaplansky conjecture holds for these groups, i.e. that the group ring CG

contains no non-trivial idempotents.

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In recent years there has been extensive interest in word maps on groups, and various results were obtained, with emphasis on simple groups. We shall focus on some new results on word maps for more general families of finite and infinite groups.

By recent work of Voevodsky and others, type theories are now considered as a candidate

for a homotopical foundations of mathematics. I will explain what are type theories using the language

of (essentially) algebraic theories. This shows that type theories are in the same "family" of algebraic

concepts such as groups and categories. I will also explain what is homotopic in (intensional) type theories.

 Graph products of groups naturally generalize direct and free products and have a rich subgroup structure. Basic examples of graph products are right angled Coxeter and Artin groups. I will discuss various forms of Tits Alternative for subgroups and
their stability under graph products. The talk will be based on a joint work with Yago Antolin Pichel.

In this talk I'll describe recent work with Henry Wilton (UCL) in which

we prove that there does not exist an algorithm that can determine which

finitely presented groups have a non-trivial finite quotient.

In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H=C(Z/p) of complex valued functions on Z/p={0,...,p-1}, the integers modulo a prime number p>>0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form

R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t),

where W(t) in H is a white noise, and τ,ω in ℤ/p, encode the distance from, and velocity of, the object.

Problem (digital radar problem) Extract τ,ω from R and S.

I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p^2⋅log(p) operations. I will then explain how to use techniques from group representation theory to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations. I will demonstrate additional applications to mobile communication, and global positioning system (GPS).

This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley).

A polar space $\Pi$ is a geometry whose elements are the totally isotropic subspaces of a vector space $V$ with respect to either an alternating, Hermitian, or quadratic form. We may form a new geometry $\Gamma$ by removing all elements contained in either a hyperplane $F$ of $\Pi$, or a hyperplane $H$ of the dual $\Pi^*$. This is a \emph{biaffine polar space}.

We will discuss two specific examples, one with automorphism group $q^6:SU_3(q)$ and the other $G_2(q)$. By considering the stabilisers of a maximal flag, we get an amalgam, or "glueing", of groups for each example. However, the two examples have "similar" amalgams - this leads to a group recognition result for their automorphism groups.

I shall discuss recent work in which bounds are obtained, generalising/specialising earlier work for general linear groups