# School of Mathematics

## Pure Colloquium

All welcome!

Colloquia will be followed by a reception in the School of Maths common room.

Forthcoming colloquia

To be announced.

Past colloquia
Monday 15th May, 2017
MALLs 1&2, 4:00 PM
Mirna Dzamonja (UEA)
Some connections between discrete and continuous mathematics
Recent years have seen the developments by Lovasz and others of graphons, in which the theory of finite graphs has met the theory of measurable functions from [0,1]^2 to [0,1]. Techniques developed in this area allow a transfer of results about graphs to results about graphons and vice versa. A posteriori, by the results of Elek and Szegedy, one can see that in fact graphons are an instance of constructing an ultraproduct measure on an ultraproduct of finite structures, so endowed with a counting measure. A much more general context including this one was actually studied by Loeb in the 1970s, who constructed ultraproduct measures on general measure spaces. We are particularly interested in the context of the ultraproduct of Boolean algebras, which allow us to give a graphon-like version of Boolean algebras. We connect this with our work from 2014 where we modelled the universality spectrum of Banach spaces under isomorphisms using an ultraproduct of Boolean algebras.

Monday 20th March, 2017
MALLS 1&2, 4:00 PM
Ken Brown (Glasgow)
Azumaya loci and discriminant ideals
I will give a very simple-minded account of some aspects of the representation theory of affine associative algebras with big centres, some old results, some new ones. My aim will be to define and motivate all unfamiliar concepts used in the talk.

Monday 6th February, 2017
MALLs 1&2, 4:00 PM
Alexander Strohmaier (Leeds)
Hahn meromorphic functions and their application to scattering theory
I will review some basic theorems in stationary scattering theory of the Schrödinger operator and explain the importance of the meromorphic Fredholm theorem in this context. I will then introduce a new ring of functions that allows to analyse the singularity of the resolvent of the Schrödinger operator near the bottom of the continuous spectrum. The talk will be kept at an elementary level and will not require expertise in scattering theory. (based on joint work with J. Müller)

Monday 5th December, 2016
Roger Stevens LT11, 4:00 PM
Fran Burstall (Bath)
Harmonic Gauss maps
I shall give an overview for non-experts of the modern theory of harmonic maps and how it applies to questions of classical (and sometimes unfashionable) differential geometry via an appropriate notion of Gauss map.

Tuesday 8th November, 2016
MALLs 1&2, 3:15 PM
Karin Erdmann (Oxford)
Periodicity for selfinjective algebras and representations
Selfinjective algebras, or Frobenius algebras, include group algebras of finite groups, and finite-dimensional Hopf algebras, and many others. We will review homological properties: growth of periodic resolutions, and periodicity, and present some open problems. We describe new results on algebras of period four defined via surface triangulations.

Monday 24th October, 2016
Roger Stevens LT10, 4:00 PM
Gerasim Kokarev (Leeds)
Minimal surfaces and eigenvalue problems
I will tell about eigenvalue problems for the Laplace-Beltrami operator on Riemannian manifolds, and will describe a number of relationships between them and the minimal surface theory. In particular, I will discuss a few conjectures and open questions in the area, and a number of related results. If time permits, I will also tell about the extremal eigenvalue problems.

Monday 23rd May, 2016
Roger Stevens LT16, 4:00 PM
Jonathan Pila (Oxford)
A tale about e^z and j(z)
I will describe a diophantine conjecture due to Zilber arising out of the model theory of the exponential function and transcendental number theory, and its relations with classical conjectures. I will then describe how the picture generalises to more exotic settings, involving in particular the elliptic modular function.

Monday 9th May, 2016
EC Stoner 9.301 (Physics Research Deck), 4:00 PM
John Barrett (Nottingham)
Non-commutative geometry and physics
The talk will focus on the development of non-commutative geometry, particularly due to the work of Alain Connes, through the investigation of examples from physics. The original examples were from diverse fields such as Penrose tilings, foliations and the non-commutative torus. More recently there has been a lot of work on the non-commutative structure of particle physics. I will add my own perspective on the programme I'm currently working on to understand whether non-commutativity has any role in the description of spacetime (at high energies). An example of the mathematical question this suggests is to investigate the non-commutative approximation of (the usual) commutative manifolds. Generally I will try to talk about the subject and keep technical details to a minimum.

Tuesday 8th March, 2016
MALL1, 4:00 PM
Vitaliy Kurlin (Durham)
Topological Computer Vision
Topological Computer Vision is a new research area within Topological Data Analysis on the interface between algebraic topology and computational geometry. The flagship method of persistent homology quantifies topological structures hidden in unorganized data across all scales. The talk will review recent applications to Computer Vision including auto-completion of contours, parameterless skeletonization and superpixel segmentation of images. The last work is joint with Microsoft Research Cambridge and is funded by the EPSRC Impact Acceleration Account through a Knowledge Transfer Secondment.

Thursday 4th December, 2014
MALL, 4:00 PM
Nicola Gambino (University of Leeds)
From type theory and homotopy theory to Univalent Foundations of Mathematics
Voevodsky's Univalent Foundations of Mathematics programme is an ambitious, long-term, project that seeks to develop a new approach to the foundations of mathematics on the basis of the recent connections between type theory and homotopy theory. This programme aims also at facilitating the use of computer systems for the verification of mathematical proofs. The talk will consist of two parts. In the first part, I will give an introduction to Voevodsky's Univalent Foundations of Mathematics programme, without assuming prior familiarity with type theory or homotopy theory. In the second part, I will explain how the type-theoretic counterpart of the topological notion of contractibility can be used to characterise certain free constructions by means of a universal property that does not involve infinitely many coherence conditions.

Thursday 16th October, 2014
EC Stoner SR 8.90, 4:00 PM
Immanuel Halupczok (University of Leeds)
Transfer between \Q_p and \F_p((t)) and applications to representation theory
The field \Q_p of p-adic numbers and the field \F_p((t)) of formal power series over the finite field with p elements behave very similarly for big p. The Ax-Kochen-Ershov Transfer Principle makes this precise: For a large class of statements \phi, one has that for all sufficiently big p, \phi is true in \Q_p iff it is true in \F_p((t)). This has applications to representation theory of reductive groups over such fields. Most of the talk will consist of a gentle introduction to this Transfer Principle and to new variants of it. At the end, I will show how this can be applied to a conjecture from representation theory about Harish-Chandra characters. (This was done in cooperation with Cluckers and Gordon.)

Wednesday 22nd January, 2014
RSLT 16, 2:00 PM
Azat Gaynutdinov (Hamburg & Paris; Humboldt Fellow)
From the Virasoro algebra to Temperley-Lieb algebras
Deformation of the infinite dimensional Lie algebra Virasoro resembles the well-known deformation of classical simple Lie algebras to quantum algebras. Such a deformation of the Virasoro algebra appeared to be useful in many applications to off-critical statistical models and massive integrable field theories. In the talk, I present my recent results on a surprising realization of this algebra in finite tensor products of the natural representations of the quantum group for sl(2). More formally, it turns out that the deformed Virasoro at N-th root of unity has representations realized by the Temperley-Lieb algebras with N generators. Such finite-dimensional representations are of the cyclic type (similar to those for quantum groups at roots of unity) and were never observed before. I will also discuss a limit when N goes to infinity. [Tea/Coffee from 13:40]

Thursday 7th November, 2013
Roger Stevens Lecture Theatre 11, 4:00 PM
Derek Harland (University of Leeds)
Topological Energy Bounds
Let C be a set of continuous maps between two manifolds, and let E:C->R be a non-negative real "energy function". A topological energy bound is a lower bound on E which depends only on topological invariants of the map. Topological energy bounds have important consequences in both physics and geometry. I will review some of these, including some very recently discovered bounds, and present some open problems.

Thursday 23rd May, 2013
MALL, 4:00 PM
Sanju Velani (University of York)
Metric Diophantine approximation: the Lebesgue and Hausdorff theories
There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We discuss these theorems and show that Lebesgue statement implies the general Hausdorff statement. The key is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup sets to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

Thursday 16th February, 2012
RSLT 12, 4:15 PM
Peter Schuster (University of Leeds)
Proofs by Induction

Thursday 8th December, 2011
RS LT07, 4:00 PM
Stephen Donkin (University of York)
Schur algebras old and new

Thursday 13th October, 2011
MALL 1, 4:00 PM
Simon Willerton (University of Sheffield)
Magnitude of metric spaces
Tom Leinster defined the notion of the magnitude' of a finite metric space, which can be thought of as something like the effective number of points' in the metric space. Leinster and I extended the notion to infinite metric spaces such as subsets of Euclidean spaces --- intervals, circles, Cantor sets, etc. I will try to give you some idea what this has to do with ecological biodiversity measures, geometric measure theory, Euler characteristics of categories and penguins.

Thursday 5th May, 2011
MALL, 4:00 PM
Professor Dirk van Dalen (Utrecht University)
L.E.J. Brouwer. A mathematician between introspection and agitation
Young Brouwer was equally attracted to mathematics and philosophy. His definite choice for mathematics was made writing his PhD thesis, even though the thesis was strongly colored by philosophy. As an intermezzo Brouwer published a series of lectures "Life, Art, and Mysticism", which foreshadowed some of his foundational views. After the proclamation of a constructive program he turned to the second topic of his thesis - topology. Within two years he was the leading topologist, with strong ties to Göttingen. The isolation of the First World War caused Brouwer to return to the foundations. He laid down the principles of his mature intuitionism, which was developed to a viable mathematical discipline. From 1918 onward his publications kept the mathematical world in suspense. At the same time he started, parallel to Hardy's efforts, to campaign for a return to a fully international mathematical community. This, sadly, gave him a reputation of being pro-German. His life offered a fair number of actions of a social/political nature, much to the astonishment and irritation of the public. We will go into Brouwer's intuitionistic program and some of his extra-academic activities.

Thursday 17th March, 2011
MALL, 4:00 PM
Dr. June Barrow-Green (Open University)
A Mathematical War? [Co-sponsored by the Centre for the History and Philosophy of Science]
‘This is a Mathematical War’ declared Sir George Greenhill, erstwhile Professor of Mathematics at the Royal Military Academy, in January 1915. Two years later his words were echoed at the front by a young British soldier who found himself fighting in a ‘war of guns and mathematics’. But were these accurate descriptions or isolated observations? In my talk, I shall consider the contributions of British mathematicians to the war effort as well as discussing the effect of the war on the mathematical community.

Thursday 10th February, 2011
MALL, 4:00 PM
Dr Keith Carne (King’s College, Cambridge)
Two-Generator Discrete Groups
I will consider groups of M\"obius transformations generated by two elements and consider the geometry of the quotient space. This involves hyperbolic geometry in 3 dimensions and classical geometry in representing the generators as compositions of involutions.

Thursday 9th December, 2010
MALL, 4:00 PM
Andrew Hubery (University of Leeds)
Geometry of Quiver Varieties
We will introduce quiver varieties, which are geometric objects describing various matrix problems. We will discuss various results relating geometry and representation theory, in particular looking at some of the arithmetic of these varieties.

Thursday 14th October, 2010
MALL, 4:00 PM
Dr. Diane Maclagan (University of Warwick)
Tropical Geometry
Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. This "tropicalization" procedure turns algebraic varieties (solutions to polynomial equations) into polyhedral complexes, which are combinatorial objects. A surprisingly large amount of information about the variety is still present in tropical "combinatorial shadow". In this talk I will introduce tropical varieties, and indicate some of their applications, both inside and outside algebraic geometry.