Algebra & Combinatorics Seminars

## Previous Seminars - 2019 to 2020

Previous seminars from: 2019/20, 2018/19, 2017/18, 2016/17, 2015/16, 2014/15, 2013/14, 2012/13, 2011/12, 2010/11, 2008/09, 2007/08

Wednesday the 20th of November at 1.30pm in Lecture Theatre D

Ruth Hoffman and Ozgur Akgun
University of St Andrews
How can Constraints help Mathematicians and vice versa?

We are looking at writing a grant which combines forces between mathematicians and computer scientists some more. In the past mathematicians have come to us with interesting problems to be modelled and we have helped out. In the process of this we have found that there are many hidden mechanisms that we are not utilising to their fullest (or do not exist).

This talk will cover our experiences and explain what we are looking for. We want to create a discussion that will hopefully lead to an innovative (and successful) grant application.

Wednesday the 20th of November at 1.00pm in Lecture Theatre D

Saul Freedman
University of St Andrews
Non-commuting, non-generating graphs of finite groups

Given a group G, we can construct associated graphs that encode certain relations between the elements of G. A well-known example is the generating graph of G, whose vertices are the nontrivial elements of G, with two vertices joined if the elements form a generating set for G. Breuer, Guralnick and Kantor showed in 2008 that if G is a finite non-abelian simple group, then this graph is connected with diameter 2.

We explore, for various families of finite groups, the connectedness and diameter of a related graph, obtained by taking the complement of the generating graph and then removing edges between elements that commute. In many cases, this is achieved by studying the maximal subgroup structures of the relevant groups.

Wednesday 13 November at 1:00pm in Lecture Theatre D

Rosemary Bailey
University of St Andrews
Some applications of finite group theory in the design of experiments

Group theory is used in (at least) two different ways in the design of experiments. The first is in randomization, the process by which an initial design is turned into the actual layout for the experiment by applying a permutation of the experimental units, chosen at random from a certain group of permutations. Which group? What properties should it have? The second is in design construction. The set of treatments is identified with a finite Abelian group, and the blocks are all translates of one or more initial blocks. The characters of this group form its dual group: they are the eigenvectors of the matrix that we need to consider to see how good the proposed design is.

Wednesday 6 November at 1:00pm in Lecture Theatre D

Mariapia Mosciatiello
On a bound of the number of maximal systems of imprimitivity of a finite transitive permutation group

We will introduce some standard definitions in the context of transitive permutation groups and we will give a very gentle introduction to the concept of crowns of a finite group.

Using these bits of knowledge we will sketch how to prove that there exists a constant $$a$$ such that a transitive permutation group of degree $$n$$ has at most $$an^{3/2}$$ maximal systems of imprimitivity.

When $$G$$ is soluble, generalizing a classic result of Tim Wall, we will see that a much stronger bound holds, that is, a soluble transitive permutation group of degree $$n\ge2$$ has at most $$n-1$$ maximal systems of imprimitivity.

Wednesday 30 October at 1:00pm in Lecture Theatre D

Collin Bleak
University of St Andrews
On maximal subgroups of R. Thompson's group $$V$$

The subgroup structure of the R. Thompson groups $$F$$,$$T$$, and $$V$$ have come under serious scrutiny lately due to to some groundbreaking results in C*-algebras, arising initially from work of Haagerup and Olesen on the amenability of $$F$$. Their work opened a door, and peoples' interest immediately led (by further transformative work of Kalantar and Kennedy, Adrienne Le Boudec, Nicolas Matte-Bon, and others) to a complete transformation of our understanding of C*-simplicity. One key and surprising result here: Matte Bon and Le Boudec provide examples of finitely presented C*-simple groups without free subgroups.

In any case, the work above leads to many interesting questions around the subgroup structures of $$F$$, $$T$$, and $$V$$. In this talk, we give a report on our (with Jim Belk, Martyn Quick, and Rachel Skipper) current work towards classifying the maximal subgroups of $$V$$.

Wednesday 9 October at 1:00pm in Lecture Theatre D

Peter Cameron
University of St Andrews
Growth rates for oligomorphic groups

A permutation group $$G$$ on an infinite set $$\Omega$$ is oligomorphic if the number $$f_n(G)$$ of $$G$$-orbits in its induced action on the $$n$$-element subsets of $$\Omega$$ is finite for all positive integers $$n$$.

Oligomorphic groups are essentially the same as automorphism groups of countably categorical first-order structures.

In the 1980s I started investigating these, and found what appeared to be a detailed structure of gaps in the growth rates. Apart from some very nice results by Dugald Macpherson, not much was done until very recently. But in the last year or two, many of my conjectures have been proved by Justine Falque, Nicolas Thiéry, Pierre Simon and Samuel Braunfeld. The techniques were algebraic (Cohen--Macauley algebras) and model-theoretic (monadic stability).

Wednesday the 2nd of October at 1.00pm in Lecture Theatre D

Louis Theran
University of St Andrews
Unlabelled distance geometry

In the 1930’s Schönberg and Young-Householder classified when $$\binom{n}{2}$$ numbers $$m_{ij}$$ are the pairwise distances among n points $$p_1, …, p_n$$ in a Euclidean space and showed how to find the points from the distances. This question of finding the points becomes more difficult when you take away the association between the $$m_{ij}$$ and the points $$p_i$$, but it was solved by Boutin and Kemper in the mid 2000’s. I’ll talk about some generalisations of Boutin and Kemper’s results obtained jointly with Shlomo Gortler and Dylan Thurston and Ioannis Gkioulekas, Shlomo Gortler, and Todd Zickler.