Algebra & Combinatorics Seminars
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Previous Seminars - 2018 to 2019

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

Wednesday the 13th of February at 1pm in Lecture Theatre D

Justin Lanier
Georgia Tech
Normal generators for mapping class groups are abundant

Under what conditions do a group element and all of its conjugates form a generating set for the ambient group? Such an element is called a normal generator. We first survey some results about normal generators in a variety of groups. After introducing mapping class groups of surfaces, we provide a number of simple criteria that ensure that a mapping class group element is a normal generator. We then apply these criteria to show that almost every periodic element is a normal generator whenever genus is at least 3. We also show that every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. Our pseudo-Anosov results answer a question of Darren Long from 1986. This is joint work with Dan Margalit.

Wednesday the 6th of February at 1pm in Lecture Theatre D

Matt McDevitt
University of St Andrews
Consecutive patterns in permutations

We consider permutations ordered by the consecutive pattern involvement ordering, and describe a recent result on infinite anti-chains

Wednesday the 6th of February at 1:30pm in Lecture Theatre D

Nayab Khalid
University of St Andrews
A presentation for some other rearrangement groups

In previous talks, I have defined rearrangement groups of fractals (Belk, Forrest 2016), constructed Thompson's group F as a rearrangement group, and subsequently developed an infinite presentation for it which stems directly from the geometric structure of the topological space. In this slightly ambitious talk, I will present the final bit of research of my PhD: a framework for similar infinite presentations for other rearrangement groups.

Wednesday the 30th of January at 1pm in Lecture Theatre D

Peter Cameron
University of St Andrews
The Hall--Paige conjecture and an application

In 1955, Marshall Hall Jr. and Lowell Paige conjectured that the Cayley table of a finite group \(G\) has an orthogonal mate if and only if the Sylow 2-subgroups of \(G\) are trivial or non-cyclic. This was proved in 2009 by the combined efforts of Stuart Wilcox, Anthony Evans, and John Bray, but the final step (involving the sporadic group \(J_4\)) was never published.

Last year, an application emerged, a proof that primitive permutation groups of simple diagonal type with more than two factors in the socle are non-synchronizing. A paper has appeared on the arXiv including both a description of Bray's argument for \(J_4\) and a proof of the application (and a little bit more).

I will discuss these matters, starting from a minimal amount of background.

Wednesday the 31st of October at 2.00pm in Lecture Theatre D

Sanming Zhou
University of Melbourne
The vertex-isoperimetric number of the incidence graphs of unitals and finite projective spaces

I will talk about some recent results on the vertex-isoperimetric number of the incidence graph of unitals and the point-hyperplane incidence graph of \(PG(n, q)\), where a unital is a \(2\)-\((n^3 + 1, n+1, 1)\) design for some integer \(n \ge 2\).

Joint work with Andrew Elvey-Price, Alice M. W. Hui and Muhammad Adib Surani.

Wednesday the 24th of October at 2.00pm in Lecture Theatre D

Wolfram Bentz
University of Hull
Congruences on the product of transformation monoid

Congruences for transformation monoids, were first described in 1952, when Mal’cev determined the congruences of the monoid \(\mathcal{T}_n\) of all full transformations on a finite set \(X_n=\{1, \dots,n\}\). Since then, congruences have been characterized in various other monoids of (partial) transformations on \(X_n\), such as the symmetric inverse monoid \(\mathcal{I}_n\) of all injective partial transformations, or the monoid \(\mathcal{PT}_n\) of all partial transformations.

Although these results are about 60 years old, none of them have previously been generalized to products of two such monoids. Our work closes this gap by describing all congruences of \(\mathcal{T}_m \times \mathcal{T}_n\). As it turns out, the congruence structure of the factors is still visible in the congruences of the product, but the variations introduced by having an extra component adds a high level of technical complexity which accounts for the difficulty in achieving this result.

In addition to presenting the congruences of \(\mathcal{T}_m \times \mathcal{T}_n\), we will also address generalizations to products of \(\mathcal{PT}_n\), \(\mathcal{I}_n\), and matrix monoids, as well as generalizations to products with more than 2 factors.

This is a joint work with {\sc Jo\~{a}o Ara\'{u}jo} and {\sc Gracinda M.S. Gomes} (Lisbon).