## Algebra, Combinatorics, Measure

Organizers: Gabor Elek, Łukasz Grabowski, Tony Nixon

Contact: Ł.G.

Seminar is on **Mondays**, usually at 1PM in B35.

This is a research seminar with both internal and external speakers. Topics covered might include, but are by no means limited to:

limits of combinatorial and algebraic structures (e.g. sofic groups, graph limits, Borel and measurable combinatorics)

discrete geometry (in particular graph rigidity)

group rings of infinite groups, $l^2$-invariants, group actions, measured group theory

interactions between algebra/combinatorics and operator algebras, geometry, probability, etc.

We have a mailing list: acmseminar@lists.lancs.ac.uk. In order to participate either ask ŁG to sign you up, or send a *plain text* email to majordomo@lists.lancs.ac.uk, containing only the two words `subscribe acmseminar`

in the body (you can leave the subject empty).

**Note for external speakers looking at available slots**: feel free to choose any empty slot or any slot with a Lancaster speaker (local speakers are easy reaccommodate)

### 2017-18

### Abstracts

Joint work with Gabor Elek. We show that if A_1,A_2,..., A_n are either unitary or self-adjoint matrices that almost commute with respect to the rank metric, then one can find commuting matrices B_1,B_2,..., B_n that are close to the matrices A_i in the rank metric. In other words, we solve a variant of the Halmos problem, where the "closeness" of the matrices is measured by the rank of their difference. Our proofs are surprisingly algebraic in flavour - the effective Nullstellenesatz plays the most crucial role, and we also use the Macaulay theorem on growth in graded algebras.

In Ramsey theory, Schur's theorem states that however one colours the positive integers with finitely many colours, there always exists a solution to the equation x+y = z in which each variable receives the same colour. Rado completely characterised which linear equations possess this property and which do not. We discuss analogues of these results for certain non-linear Diophantine equations.

Cremona group Cr(n) is the group of birational self maps of the projective n-space. I will give a gentle introduction to the topic and discuss various results on subgroups of the Cremona group and their conjugacy classes. A motivating question for this investigation is due to Serre: Can all finite groups be embedded in Cr(3)?

A semilattice is a poset in which every pair of elements has a greatest lower bound. Dually, one can realize every semilattice as a set system, on some carrier set, which is closed under taking binary unions. One possible measure of complexity of a semilattice is its breadth, which may be finite or infinite. In recent work on character (in)stability for weighted semilattices, it turns out that every semilattice of infinite breadth always supports some "bad" weight. The key to proving this is a pair of structural theorems for semilattices of infinite breadth.

In my talk I will present some basic examples to illustrate the concept of breadth, and then present the two structural theorems, which both use the perspective of union-closed set systems. Time permitting, I will try to indicate how these are used to construct weights with suitable "bad" properties. This is joint work with M. Ghandehari (Delaware) and H. Le Pham (Wellington).

The classical Robinson-Schensted correspondence is an algorithm that describes a bijection between elements of the symmetric group and pairs of standard Young tableaux. This algorithm is combinatorially rich and has many applications to the representation theory of the symmetric group and the general linear group. In 1988, Steinberg discovered a geometric setting for this algorithm coming from the nilpotent cone of the adjoint representation for the general linear group. In this talk, I will describe an exotic Robinson-Schensted algorithm for the Weyl group of type C coming from Kato's exotic Springer correspondence for the symplectic group.

For a finitely generated group G I will introduce the concept of its equationally compact actions and subgroups. As it turns out, certain subgroups satisfy this property in a trivial way. The main part of the talk is going to concern the construction of graphs corresponding to subgroups of the free product of three copies of the cyclic group of order 2. Using this I will present an example of that group's non-trivial equationally compact subgroup. Furthermore, it allows us to answer the question of R. Rajani and M. Prest about the existence of an equationally compact action of a countable group on a certain space X such that the stabiliser of each point x in X is not equationally compact. Joint work with Gabor Elek.

Formation control and network localization are two important research topics in the broad area of multi-agent cooperative control and estimation. Here an agent may represent an autonomous vehicle or a node in a sensor network. Formation control aims to steer a group agents starting from a given initial configuration to converge to a desired geometric pattern. Network localization aims to estimate the position of each agent in a network by using each agent’s local measurements of their nearest neighbors. In recent years, rigidity theories have been playing important roles in the two research topics. In addition to the classic distance rigidity theory described based on inter-agent distances, novel rigidity theories such as bearing rigidity described based on inter-agent bearings has also been developed and applied. This talk will give a review of the recent advances of the application of rigidity theories in multi-agent systems and discuss potential future research directions.