As part of the NBFAS meeting at Lancaster University, April 2024, a morning of talks will be held dedicated to the memory of H. Garth Dales.

Garth was a Professor of Mathematics at Lancaster from 2011 until his retirement in 2021. In early 2022 he was diagnosed with motor neurone disease, and passed away on 8th October 2022. He is much missed by colleagues and friends. Garth was a great supporter of NBFAS, and in his honour, a small meeting will be held on the Thursday morning, with research talks from past students of Garth.

- 10:00 - 10:10
- Opening remarks (Yemon Choi)
- 10:15 - 10:40
- Matthew Daws, "Frobenius Algebras" [abstract]
- 10:45 - 11:10
- Joel Feinstein, "Generalised Swiss Cheeses and Bubble Space" [abstract]
- 11:15 - 11:30
- Coffee break
- 11:30 - 11:55
- M. Eugenia Celorrio, "A Walk on Arens Regularity" [abstract]
- 12:00 - 12:25
- Jared White, "Nilpotent left ideals in second dual algebras" [abstract]

I will give a quick introduction to Frobenius Algebras, from a down-to-earth algebraic perspective, and with an explicitly Banach Algebraic viewpoint. I will then discuss some ideas about how to define a Banach Algebraic Frobenius Algebra, concentrating upon how naive ideas don't seem to work. Time allowing, some discussion of Frobenius Extensions will be made.

Swiss cheeses are compact plane sets obtained by deleting from a closed disc some suitable sequence of open discs. They are a valuable source of counterexamples in the theory of uniform algebras and rational approximation. However, in some of the examples in the literature, the open discs (“holes”) may touch or overlap each other or the complement of the original closed disc. Several of my research students and I have looked at how to “improve” these examples to avoid these issues. Many of these results can be extended to far more general settings than just discs in the complex plane. In joint work with my summer undergraduate research bursary students Jonathan Davies and Joshua Reynolds (to be continued with another student this summer), we looked at a variety of settings where many of the same results work, and also at some settings where changes are needed. In particular, Joshua Reynolds and I obtained results on “Bubble Space”, where we consider starting with a sequence of open balls (“bubbles”) in a Banach space, and then using coalescence and transfinite induction to eliminate overlapping and/or touching bubbles. In this talk we discuss some of the results, and future directions for this work.

In 1951, Arens introduced two products in the bidual A'' of a given Banach algebra A. These products are such that A'' is a Banach algebra with respect to both products and A is a closed subalgebra of both these algebras. In this talk, we shall introduce these products and we shall talk about the Arens regularity of weighted semigroup algebras that arise from a family of totally ordered semilattices.

Consider the second dual of l^1(G) with its Arens product. I shall discuss some of Garth’s work on this algebra, before discussing some of my own results. In particular, a question of Garth and Tony Lau asks whether the Jacobson radical of this algebra is always nilpotent of index 2. I shall give examples of groups for which this radical is not nilpotent at all, and discuss what is known about the case where G is the group of integers.

For those who knew Garth, beyond Mathematics, it was clear that hill-walking was a great pleasure. While Garth worked at Lancaster, he lived in Bentham, on the edge of the Yorkshire Dales National Park. On Friday afternoon, an excursion is planned to Ingleton from which we will walk up Ingleborough, about 5 miles round-trip and 1850ft of ascent.

Lancaster hosted a retirement meeting in honour of Garth in August 2021.

Obituaries can be found from the University of Leeds and Lancaster University. An obituary was published by the LMS in the March 2023 edition of the Newsletter.

The Oberwolfach Photo Collection contains some photos of Garth throughout his life.

- NBFAS contacts : Evgenios Kakariadis and Yemon Choi
- Website : Matthew Daws
- Social : Niels Laustsen

**Thursday 4th April 2024**

School of Mathematics and Statistics

Lancaster University