Paul Martin, Mathematics, University of Leeds

Welcome. I am Paul Martin, a professor in the School. I was until relatively recently head of the Department of Pure Mathematics. Here are a few personal thoughts on ...

Some research strengths in the School under heading ``pure maths''

... intended as an ultra-light overview, for not-necessarily-Mathematical visitors to the School.

Firstly some informal defining of terms from the heading above:

Pure maths := not necessarily applied maths (i.e., maths the importance of which extends beyond a single application)
strength := internationally agenda setting activity

Mathematics is the language and toolkit of science and technology. This is a demand-side perspective on Mathematics.
On the supply-side we have the benefits of abstraction; and the capacity for intrinsic beauty.

Pure Mathematics is sometimes broken down, for bookkeeping purposes, into four intrinsic subdisciplines:
algebra; analysis; geometry and number theory.^1,2
Overarching these is a more foundational discipline of logic, consisting roughly of computability; model theory and proof theory.
Leeds Pure Maths is a large research activity in the field, in terms of staff numbers, and we aim to be active in all of these disciplines. In practice, of these disciplines at Leeds only number theory is presently missing as a research strength.
Furthermore, in practice there is no clear division between activities in Pure and Applied mathematics, and the overall connected nature of strong research activity at Leeds across the wide spectrum of mathematics is a particularly striking feature.

Some random examples of strengths are EPSRC funded projects in, or involving:
cluster algebras ; diagram algebras ; ``topological'' quantum groups ; applications of geometry to theoretical physics (e.g. ``type-1.5 superconductors'') ; homotopy-type theory, and indeed all of the abovementioned aspects of logic.

In practice the best way to follow our work is probably through our scholarly and research articles.
These are published in international research journals; are also often found on the arXiv ; and often on our homepages.
Here are some indicative examples:

I Halupczok, Non-archimedean Whitney stratifications. To appear in Proc. Lond. Math. Soc.
M Daws (with R Haydon, Oxford, et al), Shift invariant preduals of l_1(Z). Israel Journal of Mathematics 192:541-585 (2012). ³ pdf
R Marsh (with A Buan, Trondheim), From triangulated categories to module categories via localisation. Trans. Amer. Math. Soc. 365 (2013) 2845-2861.
W Crawley-Boevey (with F Chazal, Saclay, et al), The observable structure of persistence modules. arXiv:1405.5644 [math.AT].
P Martin, The decomposition matrices of the Brauer algebra over the complex field. Trans. Amer. Math. Soc. (2015).
N Gambino (with A Joyal), On operads, bimodules and analytic functors, Memoirs Amer. Math. Soc. to appear.
...more to follow

We also run a large number (typically more than one per day) of open-to-the-public research seminars in the area.
From time to time we run international research conferences and workshops (day long or even week-long sessions with multiple speakers from research groups around the world, covering research themes and agendas set by workers here at Leeds).

Examples of bodies that fund our work (for a variety of reasons of their own) include HEFCE, EPSRC, RS, Leverhulme, IMA, IoP, LMS, EU, USAF, Templeton, ...

This page is about research, but we adhere to the Russell Group philosophy that university teaching and research function symbiotically.

Footnotes:
1. These subdisciplines should perhaps be thought of as regions in a mathematical continuum, with overlapping umbras, penumbras and so on. If the study of (a) sets and relations; (b) functions; (c) shapes; and (d) numbers can be considered as points in this continuum, then "algebra" would be the collection of activities closest to (a), "analysis" closest to (b), and so on.
(Care is needed with such characterisations since, for example, (a) contains (b); but never mind!)
Of course one might prefer to subdivide mathematics according to its applications (or at the other extreme by its axioms), and one would be perfectly entitled to do this.

2. Organisationally, the School includes "research groups" with some similar names to these. This is a different story however, articulating some strategic research aims of the School - which are often cross-disciplinary. For example the AGIS Research Group contains Integrable Systems while the Analysis Research Group does not, although IS is at least as close to (b) as it is to (a,c)!

3. J Partington says of this paper: "A massive (and technically a tour-de-force) paper in one of the leading journals for Analysis. This is combining some of the latest ideas in Banach space theory - in particular, constructions used to solve major open questions - in an operator-theoretic context."

To reiterate, everything on this page is the responsibility of myself, Paul Martin. I wrote it when I was Head of Department, but I am not writing here in that capacity.