Algebra & Combinatorics Seminars

## 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 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).