Forthcoming events in this series
17:00
Group actions on rings and the Cech complex.
Abstract
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.
17:00
A closed formula for the Kronecker coefficients.
Abstract
17:00
The width of a group
Abstract
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.
17:00
Superrigidity for mapping class groups?
Abstract
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.
17:00
Rank Gradient of Artin Groups and Relatives
Abstract
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.
17:00
17:00
Artin groups of large type: from geodesics to Baum-Connes
Abstract
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.
1
17:00
'More words on words'
Abstract
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.
17:00
Reflection group presentations arising from cluster algebras
Abstract
17:00
Type theories and algebraic theories.
Abstract
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.
17:00
"Tits alternatives for graph products of groups".
Abstract
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.
17:00
"From Algebra (1895) to Moderne Algebra (1930): Changing Conceptions of a Discipline. A Guided Tour Using the Jahrbuch über die Fortschritte der Mathematik"
17:00
"On the undecidability of profinite triviality"
Abstract
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.
17:00
"Group presentations in which the relators weigh less than the generators"
17:00
Representation Theoretic Patterns in Digital Signal Processing I: Computing the Matched Filter in Linear Time
Abstract
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).
17:00
17:00
"Biaffine geometries, amalgams and group recognition"
Abstract
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.