Pure Maths Seminar Archive - Abstracts 2013
"Diophantine approximaton with almost-prime rationals"
Alexander Gorodnik (Bristol)
January 14th 2013
We discuss the problem of Diophantine approximation with almost-prime numerators/denominators on semisimple groups. Using a suitable ergodic theorem, we establish a quantitative Diophantine approximation result in this setting. This this a joint work with S. Kadyrov.
"Pattern avoidance in permutations"
Vladimir Dotsenko (Dublin)
January 21st 2013
A permutation w is said to contain another permutation w' as a (consecutive) pattern if there is a subword of w that is order-isomorphic to w'; otherwise w is said to avoid w'. Enumeration of permutations avoiding a given set of patterns is a challenging combinatorial question. I shall explain an approach to enumeration questions of that sort which is motivated by homological algebra for shuffle algebras (graded vector spaces which are monoids with respect to a certain unconventional monoidal category structure). The talk is supposed to be rather elementary, though an algebraic mind would help to unravel motivations behind it.
"Derived autoequivalences and braid relations"
Joseph Grant (Leeds)
January 28th 2013
I will give a short introduction to the derived category of an algebra, which is made of complexes of representations, and describe certain symmetries of these categories called spherical twists in the case where the algebra is symmetric. These symmetries can be generalized by a construction involving periodic algebras, which I will explain. The spherical twists satisfy braid group relations, and I will discuss how periodic twists are related to this braid group action.
"p-modular representations of p-adic reductive groups of rank 1"
Ramla Abdellatif (Lyon)
February 4th 2013
Let p be a prime number. The goal of this talk is to explain how we built a classification of the representations of G(F) over an algebraically closed field of characteristic p, where G is a connected reductive group which is defined, quasi-split and of relative rank 1 over a finite extension F of Qp. We will mainly focus on the case G=SL2 and F=Qp: the results we got in this setting are indeed as complete as can be expected, and it is moreover enough to give a full overview of the methods that are used in the general context, without getting annoyed by technical problems.
"Donovan's conjecture"
Charles Eaton (Manchester)
February 11th 2013
A fundamental problem in modular representation theory is whether, having bounded a natural invariant known as the defect of a block, there are only a finite number of blocks of finite groups up to Morita equivalence. Donovan's conjecture is that the answer is yes, and its resolution would profoundly influence the way we view the subject. I will give the background necessary to state the conjecture and attempt to give some insight into it, before describing some recent work using the classification of finite simple groups.
"Euler systems for Rankin-Selberg convolutions of modular forms"
David Loeffler (Warwick)
February 18th 2013
An Euler system is a certain compatible family of classes in the cohomology of a Galois representation, which play a key role in relating arithmetical properties of the representation to values of the associated L-function. Only a few examples of such systems have been constructed to date, although they are conjectured to exist in quite general settings. I will describe a construction of an Euler system for the tensor product of the Galois representations of two modular forms. This is joint work with Antonio Lei and Sarah Zerbes.
"Forms in many variables and p-adic solubility"
Julia Brandes (Bristol)
February 25th 2013
It is a recurrent theme in number theory that the number of integer solutions of an equation is related to the number of solutions over the local field fields R and Qp. We will present a geometric argument that allows us to use the information about p-adic solubility more efficiently than hitherto possible. This leads to improved bounds on the number of variables required in order to solve systems of cubic forms in the integers.
"On maximal subgroups of free idempotent generated semigroups"
Robert Gray (UEA)
March 4th 2013
The set of idempotents of an arbitrary semigroup has the structure of a so called biordered set (or regular biordered set in the case of von Neumann regular semigroups). These structures were studied in detail in work of Nambooripad (1979) and Easdown (1985). There is a notion of a free idempotent generated semigroup on a biordered set and it was conjectured by McElwee in 2002 that the maximal subgroups of such a semigroup are all free (in fact, this had been conjectured since the early 1980s). The first counterexample to this conjecture was given by Brittenham, Margolis and Meakin (2009), where it was shown that the free abelian group of rank 2 is a maximal subgroup of the free idempotent generated semigroup arising from a certain 72-element semigroup. In this talk I will present some recent developments in our understanding of free idempotent generated semigroups and their maximal subgroups.
"2-by-2 matrices as stepping-stones"
Roger Plymen (Manchester)
March 11th 2013
I would like to show how, using the special linear group SL(2) as a stepping-stone, one can go, fairly quickly, from SL(2,C) to the Dirac equation, and from SL(2,Qp) to Langlands parameters.
"Higher Eisenstein congruences"
Tobias Berger (Sheffield)
March 18th 2013
I will discuss a recent result on congruences of Eisenstein series and cuspforms modulo prime powers. This is joint work with Kris Klosin and Kenneth Kramer (CUNY). Time permitting I will also describe its application in proving the modularity of residually reducible Galois representations.
"Probabilistic Galois Theory"
Rainer Dietmann (Royal Holloway)
April 29th 2013
Van der Waerden has shown that `almost' all monic integer polynomials of degree n have the full symmetric group Sn as Galois group. The strongest quantitative form of this statement known so far is due to Gallagher, who made use of the Large Sieve. In this talk we want to explain how one can use recent advances on bounding the number of integral points on curves and surfaces instead of the Large Sieve to go beyond Gallagher's result.
"Long reals"
David Aspero (UEA)
May 7th 2013
The real numbers are constructed from the set of natural numbers in a natural way. I will show some of the things that happen when we replace, in that construction, the set of natural numbers with a higher ordinal. This is joint work in progress with Kostas Tsaprounis.
"Constructive enumeration of Schur rings over the group A5: computer results, theoretical explanations, striking examples"
Misha Klin (Beersheva)
May 13th 2013
We will recall main definitions related to a concept of a Schur ring (briefly S-ring), paying special attention to the state of art in enumeration of S-rings over groups of small order. The main part of the lecture is devoted to the results of the computer aided enumeration of S-rings over group A5 of order 60, which was fulfilled my Matan Ziv-Av. Special attention will be payed to so-called non-Schurian S-rings. A number of detected examples of striking S-rings will be discussed in a way, which does not require consideration of computer data and is relative clear to human.
"LUMOS–Shedding Light on Undergraduate Mathematics"
Bill Barton (Auckland)
May 20th 2013
LUMOS is a national research project that seeks to identify, categorise, observe, and report on ALL the desired outcomes for mathematics undergraduate courses. As well as standard content and skills outcomes as measured in tests, assignments and examinations, we will include outcomes desired by lecturers, employers, university graduate profiles, and the students themselves. Mathematical processes and modes, attitudes, advanced thinking will be considered along with broader outcomes of communication, critical thinking, and learning habits. The project includes using three innovative course delivery methods so that we can attempt to differentiate course delivery using the Outcome Profile. Part of the session will provide an opportunity for members of the audience to contribute their own ideas about desired outcomes and how they might be observed.
"On transcendence and generic Painlevé equations"
Joel Nagloo (Leeds)
October 7th 2013
The Painlevé equations are nonlinear 2nd order ODE and come in six families P_I--P_VI, where P_I consists of the single equation y'' = 6y^2 + t, and P_II--P_VI come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications including for example random matrix theory and general relativity.
In this talk we are interested in the following classical transcendence problem: Suppose y'' = f(y, y', t) is one of the generic Painlevé equation from the class II to VI and let y_1, ... , y_n be distinct solutions. Are y_1, y_1' , ... , y_n, y_n' algebraically independent over C(t)?
I explain how one can use model theory, a branch of mathematical logic, to positively answer this question. It is worth mentioning that these results are the culmination of the work started by P. Painlevé (over 100 years ago), the Japanese school and many others on transcendence and the Painlevé equations.
"Nilpotent orbits and representation theory"
Lewis Topley (UEA)
October 14th 2013
A simple algebraic group acts upon its Lie algebra and the subset of
nilpotent elements decomposes into conjugacy classes, which we call
nilpotent orbits. These orbits appear to be subtly connected to the
representation theory of Lie theoretic objects, such as Weyl groups. I
shall begin by sketching some of these connections.
Over the past 12 years the emerging theory of finite W-algebras has offered a new perspective on these connections and allowed for progress to be made in understanding the (ordinary and modular) representation theory of semisimple Lie algebras. I shall describe some of these advances and (hopefully) mention one of my recent results, relating the representation theory of the finite W-algebras to the sheets of orbits in a classical Lie algebra.
"Introduction to 2-representation theory"
Vanessa Miemietz (UEA)
October 28th 2013
I will try to motivate why one would want to study 2-representations of 2-categories (other than to have some fun) and to explain some of the basic concepts and difficulties that one encounters in doing so.
"Permutations and words"
Nik Ruskuc (St. Andrews)
November 4th 2013
In this talk I will explore how concepts from theoretical computer science - automata and languages - can be utilised in a combinatorial context such as the theory of pattern avoidance classes of permutations.
"Algebra with surfaces"
Joseph Chuang (City)
November 11th 2013
Frobenius algebras give rise to topological invariants of surfaces. I will review this idea (two-dimensional topological field theory) and describe joint work with Andrey Lazarev on a similar construction.
"Discrete Riemann mapping and the Poisson boundary"
Agelos Georgakopoulos (Warwick)
November 18th 2013
Answering a question of Benjamini & Schramm, we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which circle arises from a discrete version of Riemann's mapping theorem. I will start by explaining this discrete Riemann map, drawing various analogies between the discrete and continuous case.
The talk will be self contained, assuming no prior knowledge of the mentioned terminology, and will contain many pictures.
"The partition algebra and the Kronecker problem"
Chris Bowman (City)
November 25th 2013
The Kronecker problem asks for a combinatorial understanding of the tensor products of simple modules for the symmetric group. We shall introduce the partition algebra as a natural setting in which to study this problem and discuss new results concerning its representation theory. This is based on joint work with M. De Visscher, O. King, and R. Orellana.
"Fun with partition categories"
Paul Martin (Leeds)
December 2nd 2013
The Brauer category sits inside the partition category - both having elementary set-theoretic constructions. The Temperley-Lieb category sits inside these categories (in at least two different ways), but it's construction has a more geometrical flavour. We will consider geometrically defined extensions of the TL category in the Brauer and partition categories. These constructions are motivated in part by applications in computational physics, but here we will consider them from a representation theory perspective.
"Partial mirror symmetry"
Brent Everitt (York)
December 9th 2013
In this talk (but possibly nowhere else) "mirror symmetry" means the theory of reflection groups. These are groups of linear maps generated by reflections. Partial mirror symmetry is then the theory of "reflection monoids": inverse monoids of partial linear maps generated by partial reflections. This is a subject in its infancy. I'll start at the beginning and try to give a flavour of what has been achieved so far.
"The automorphism group of a (nice) tropical polytope"
Marianne Johnson (Manchester)
December 16th 2013
Tropical linear spaces can look a bit strange. In case you've not come across these objects before, I'll start with the basics and show you some examples. I'll then say a bit about the group of automorphisms of a finitely generated tropical linear space (or tropical polytope for short).