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"Elliptic curves and the conjecture of Birch--Swinnerton-Dyer", Sarah Zerbes (UCL)
An important problem in number theory is to understand the rational solutions to algebraic equations. One of the first non-trivial examples, cubics in two variables, leads to the theory of so-called elliptic curves. The famous Birch—Swinnerton-Dyer conjecture, one of the Clay Millennium Problems, predicts a relation between the rational points on an elliptic curve and a certain complex-analytic function, the L-function on an elliptic curve. In my talk, I will give an overview of the conjecture and of some new results establishing the conjecture in certain cases.
Located in Academic Departments / Mathematics / Research
"Minimal surfaces and geometric flows" Melanie Rupflin (Oxford)
Abstract: The classical Plateau problems has been one of the most influential problems in the development of modern analysis. Posed initially by Lagrange, it asks whether a closed curve in Euclidean space always spans a surfaces with minimal possible area, a question that was answered positively by Douglas and Rado around 1930. In this talk I want to consider some aspects of the classical Plateau Problem and its generalisations and discuss furthermore how one can "flow" to such minimal surfaces by following a suitably defined gradient flow of the Dirichlet energy, i.e. of the integral of gradient squared.
Located in Academic Departments / Mathematics / Research
"Modulations and potentials for triangulated surfaces" by Jan Geuenich (Bielefeld)
Triangulations of surfaces with marked points and weighted orbifold points encode the mutation combinatorics of "almost all" cluster algebras of finite mutation type. From a representation-theoretic point of view, Jacobian algebras of triangulations play a key role. Daniel Labardini Fragoso defined and investigated these algebras for surfaces without orbifold points. In joint work with him we extend this theory to surfaces that may have weighted orbifold points. The natural set-up for this generalization are modulations for weighted quivers.
Located in Academic Departments / Mathematics / Research
"The additivity problem" Muneerah Saad ALNuwairan (King Faisal University)
Abstract: In our talk, we present the well- known problem in the field of quantum information theory known as “The additivity problem”. It concerns transferring classical information using quantum channels. We will build a mathematical model for the problem, and show that the field of Representation theory provides rich source for solutions. We will also introduce EPOSIC channels, a class of SU(2)-covariant quantum channels that form the extreme points of all SU(2)-irreducibly covariant channels. These channels provide us with a potential solution for the problem.
Located in Academic Departments / Mathematics / Research
"The fundamental crossed module of the complement of a knotted surface in the 4-sphere"Joao Faria Martins (Leeds)
After a review on homotopy 2-types and crossed modules, I will show a method to calculate the homotopy 2-type of the complement of a knotted surface in the 4-sphere given a movie presentation of it. We therefore generalise Wirtinger relations for the fundamental group of a knot complement. [1] Faria Martins J.: The Fundamental Crossed Module of the Complement of a Knotted Surface. Transactions of the American Mathematical Society. 361 (2009), 4593-4630. [2] Faria Martins J., Kauffman L.H.: Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces, Compositio Mathematica. 144:04 (2008) 1046-1080.
Located in Academic Departments / Mathematics / Research
Greg Stevenson (Glasgow) "In search of greener pastures"
There is an instructive and (miraculously) faithful analogy between commutative rings and tensor triangulated categories. One can speak of prime ideals, the Zariski topology, localization, and the resulting spectra in both contexts and these concepts relate to one another in a way that far exceeds what one would naively, and perhaps reasonably, expect. However, there is trouble in paradise: there is no existing analogue of closed subschemes for tensor triangulated categories and, in particular, residue fields are a problematic concept. I'll discuss joint work with Paul Balmer and Henning Krause which is aimed toward elucidating these elusive residue fields.
Located in Academic Departments / Mathematics / Research
Ilke Canakci (Durham) "Infinite rank surface cluster algebras"
Abstract: Cluster algebras were introduced by Fomin and Zelevinsky in the context of Lie theory in 2002, however they received much attention since many applications in diverse areas of mathematics have been discovered including quiver representations, Teichmuller theory, integrable systems and string theory. An important class of cluster algebras, called surface cluster algebras, were introduced by Fomin, Shapiro and Thurston by associating cluster algebra structure to triangulated marked surfaces. I will review the state of the art in research associated to surface cluster algebras and report on joint work with Anna Felikson where we introduce a generalisation of surface cluster algebras to infinite rank by associating cluster algebras to surfaces with finitely many accumulation points of boundary marked points.
Located in Academic Departments / Mathematics
Justin Lynd (Aberdeen) "Fusion systems and classifying spaces"
Abstract: The Martino-Priddy conjecture asserts that two finite groups have equivalent p-completed classifying spaces if and only if their associated fusion systems at the prime p are isomorphic. This was first proved by Bob Oliver in 2004 and 2006. Andrew Chermak generalized this in 2013 by showing that each saturated fusion system over a finite p-group has a unique classifying space attached to it. The proofs of these results depend on the classification of the finite simple groups (CFSG). The focus for this talk will be on the group theoretic aspects of joint work with George Glauberman that helped us remove the dependence of these results on the CFSG. These results concern the question: given a finite group G acting on a finite abelian p-group, when can one find a p-local subgroup H (i.e., a normalizer of a nonidentity p-subgroup) having the same fixed points on the module as does G itself?
Located in Academic Departments / Mathematics / Research
Nathan Broomhead (Plymouth) "Thick subcategories, arc-collections and mutation"
Abstract: I will explain some work, in which I describe the lattices of thick subcategories of discrete derived categories. This is done using certain collections of exceptional and sphere-like objects related to non-crossing configurations of arcs in a geometric model.
Located in Academic Departments / Mathematics
Prof. Fran Burstall (University of Bath)
Located in Academic Departments / Mathematics

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