## Spring 2013

Seminars take place on Monday afternoons 14:00-15:00. Everyone is welcome.

### Talks

- January 14 (ARTS 2.02),
**Alexander Gorodnik (Bristol)**, Diophantine approximation with almost-prime rationals. - January 21 (SCI 3.05),
**Vladimir Dotsenko (Dublin**), Pattern avoidance in permutations. - January 28 (ARTS 2.02),
**Joseph Grant (Leeds)**, Derived autoequivalences and braid relations. - February 4 (SCI 3.05),
**Ramla Abdellatif (Lyon),**p-modular groups of p-adic reductive groups of rank 1. - February 11 (ARTS 2.02),
**Charles Eaton (Manchester)**, Donovan's conjecture. - February 18 (ARTS 2.02),
**David Löffler (Warwick)**, Euler systems for Rankin--Selberg convolutions of modular forms. - February 25 (ARTS 2.02),
**Julia Brandes (Bristol)**, Forms in many variables and p-adic solubility. - March 4 (ARTS 2.02),
**Robert Gray (UEA)**, On Maximal Subgroups of Free Idempotent Generated Semigroups. - March 11 (ARTS 2.02),
**Roger Plymen (Manchester)**, 2 by 2 matrices as stepping-stones. - March 18 (ARTS 2.02),
**Tobias Berger (Sheffield)**, Higher Eisenstein congruences. - April 29 (SCI 3.05),
**Rainer Dietmann (London)**, Probabilistic Galois Theory. - May 07 (SCI 3.05),
**David Aspero (UEA)**, Long reals.**Note**: This seminar takes place on a**Tuesday**. - May 13 (SCI 3.05),
**Misha Klin (Beersheva)**, Constructive enumeration of Schur rings over group A_5:computer results, theoretical explanations, striking examples. - May 20 (SCI 0.31),
**Bill Barton (Auckland)**, LUMOS--Shedding Light on Undergraduate Mathematics.

### Abstracts

**Alexander Gorodnik:** 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.

**Vladimir Dotsenko: **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.

**Joseph Grant: **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.

**Ramla Abdellatif**: 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 Q_p. We will mainly focus on the case G= SL_2 and F=Q_p: 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.

**Charles Eaton**: 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.

**David Löffler:** 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.

**Julia Brandes:** 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 Q_p. 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.

**Robert Gray: **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.

**Roger Plymen: **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,Q_p) to Langlands parameters.

**Tobias Berger: **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.

**David Aspero**: 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.

**Rainer Dietmann: **Van der Waerden has shown that `almost' all monic integer polynomials of degree n have the full symmetric group S_n 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.

**Mikhail Klin: **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 A_5 of order 60, which was fulfilled my Matan Ziv-Av. Special attention will be paid 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.

**Bill Barton: **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.

## Autumn 2012

Unless otherwise stated, seminars take place on Monday afternoons, 14:00-15:00 in Arts 2.01. Everyone is welcome.

### Talks

- October 8
**Chufeng Nien (Tainan)**, Models and their applications in representation theory. - October 15
**Adam Harris (Oxford)**, Categoricity of the j-function. - October 22
**Daniel Loughran (Bristol)**, Rational points of bounded height and the Weil restriction. - October 29, no seminar.
- November 5,
**Grzegorz Plebanek (Wrocław)**, On isomorphisms of Banach spaces of continuous functions. - November 12 two talks in S3.05
- 14:00-15:00
**Robert Jenkins (UEA)**, History of Complex Numbers. - 15:30-16:30
**Chris Smyth (Edinburgh)**, Sequences and congruences related to Ramanujan's Tau Function.

- 14:00-15:00
- November 19, no seminar.
- November 26,
**Stefan Kolb (Newcastle)**, Quantum homogeneous spaces for quantized enveloping algebras. - December 3, (EFRY 1.01)
**Tim Browning (Bristol)**, How frequently does the Hasse principle fail? - December 10, 15:00-16:00 (S3.05),
**Colin Bushnell (King's College London)**, Questions of effectiveness in the local Langlands correspondence.

### Abstracts

**Chufeng Nien**: Models and their applications in representation theory

In this talk, we will introduce some models in representation theory and their applications in obtaining functional equations and defining twisted Gamma factors. If time permits, we can also talk about classification of irreducible cuspidal representations of general linear group over finite fields through the set of twisted Gamma factors.

**Adam Harris: ** Categoricity of the j-function

I will introduce the model theoretic notion of categoricity, and then outline a proof for categoricity of a natural structure involving the modular j-function. It turns out that categoricity in this situation is related to deep geometric and number theoretic results, in particular the Mumford-Tate conjecture, and the theory of complex multiplication. I will give brief introductions to the objects under discussion (e.g. elliptic curves and the j-function) as I go along.

**Daniel Loughran**: Rational points of bounded height and the Weil restriction

If one is interested in studying Diophantine equations over number fields, there is a clever trick due to Weil where one may move the problem from the number field setting to the usual field of rational numbers by performing a "restriction of scalars". In this talk, we consider the problem of how the height of a solution (a measure of the complexity of a solution) changes under this process, and in particular how the number of solutions of bounded height changes.

**Grzegorz Plebanek**: On isomorphisms of Banach spaces of continuous functions

We consider Banach spaces of the form C(K) of real-valued continuous functions on a compact space K. We address the problem what are necessary topological relations between spaces K and L for which C(K) and C(L) are isomorphic.

**Stefan Kolb**: Quantum homogeneous spaces for quantized enveloping algebras

The coordinate ring C[G] of a complex affine algebraic group G is a commutative Hopf algebra. A coideal subalgebra B of C[G] describes an affine homogeneous space for G if and only if C[G] is faithfully flat as a B-module. In the theory of quantum groups one studies various families of non-commutative Hopf algebras. By analogy to the commutative setting, it is natural to define a quantum homogeneous space for a Hopf algebra H to be a coideal subalgebra of H over which H is faithfully flat. An important example of a non-commutative Hopf algebra is the quantized enveloping algebra Uq(g) of a simple, complex Lie algebra g. In this talk, I will report on recent progress in the classification of quantum homogeneous spaces for Uq(g). Classification results will be given in terms of Weyl group combinatrics. The talk is based on joint work with I. Heckenberger.

**Tim Browning (Bristol)**, How frequently does the Hasse principle fail?

Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

You can also view a list of previous years' seminars.