Spring 2013

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

Talks

  • January 14th (ARTS 2.02), Alexander Gorodnik (Bristol), ​Diophantine approximaton with almost-prime rationals.
  • January 21st (SCI 3.05), Vladimir Dotsenko (Dublin), Pattern avoidance in permutations. 
  • January 28th  (ARTS 2.02), Joseph Grant (Leeds), Derived autoequivalences and braid relations.
  • February 4th (SCI 3.05), Ramla Abdellatif (Lyon), TBA.
  • February 11th  (ARTS 2.02), Charles Eaton (Manchester), TBA.
  • February 18th  (ARTS 2.02), David Löffler (Warwick), Euler systems for Rankin--Selberg convolutions of modular forms.
  • February 25th  (ARTS 2.02), Julia Brandes (Bristol), TBA.
  • March 4th  (ARTS 2.02), Robert Gray (UEA), TBA.
  • March 11th  (ARTS 2.02), Mirna Džamonja (UEA), TBA.
  • March 18th  (ARTS 2.02), Tobias Berger (Sheffield), TBA.
  • March 25th (CDA0.01), David Aspero (UEA), TBA.
  • April 29th  (ARTS 2.02), Rainer Dietmann (Royal Holloway), TBA.

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.

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.

Autumn 2012

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

Talks

  • October 8th, Chufeng Nien (Tainan), Models and their applications in representation theory.
  • October 15th, Adam Harris (Oxford), Categoricity of the j-function.
  • October 22nd, Daniel Loughran (Bristol), Rational points of bounded height and the Weil restriction.
  • October 29th, no seminar.
  • November 5th, Grzegorz Plebanek (Wrocław), On isomorphisms of Banach spaces of continuous functions.
  • November 12th, 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.
  • November 19th, no seminar.
  • November 26th, Stefan Kolb (Newcastle), Quantum homogeneous spaces for quantized enveloping algebras.
  • December 3rd, (EFRY 1.01) Tim Browning (Bristol), How frequently does the Hasse principle fail?
  • December 10th, 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 combinatorics. 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.

A list of previous years' seminars can be found here.