Abstract: This will be a gentle introduction to the local Langlands correspondence for SL(n), with emphasis on SL(2).
The talks are all happening at University of Southampton (in person), on Highfield Campus (SO17 1BJ). For the building and the room, see the individual postings. Email the organiser ( s (dot) nishikawa (at) soton.ac.uk ) if you have questions.
More Pure Maths talks at Southampton: we run Pure Maths Colloquia, Topology Seminar, and Pure Postgraduate Seminar (link soon).
Abstract: This will be a gentle introduction to the local Langlands correspondence for SL(n), with emphasis on SL(2).
Abstract: In this talk, I will give a gentle introduction to root systems and Weyl groups with a focus on the E7 case. Following this, I will go over the types of actions we may see from centralisers in the E7 Weyl group on a torus which will be illustrated with an example.
Abstract: Nonlinear optimization deals with optimizing objective functions subject to some nonlinear constraints. Such problems arise in various fields such as engineering, economics, finance, physics, machine learning, and many others. The study of this problem in finite Euclidean space is known as nonlinear programming (though not that simple) and has attracted much attention. However, its study in more generalized spaces involves some knowledge of operator theory and poses significant challenges. For instance, problems such as frictional membrane bending in mechanics can be studied as optimization problems in Sobolev spaces, while modeling of airway systems in human lungs can be studied as nonlinear optimization in Hadamard or CAT(0) spaces. These examples justify the importance of delving into nonlinear optimization in Hilbert and CAT(0) spaces. This talk will focus on the techniques and theoretical framework for solving nonlinear optimization in real Hilbert and CAT(0) spaces. This will include essential concepts from operator theory, fixed point theory, and algorithm designs. We will also look at some applications of this concept in inverse problems, particularly image processing.
Abstract: Polyhedral products are natural subspaces of the Cartesian product of spaces, which have a diverse range of applications across mathematics. One particular special case is the moment-angle complex. Work of various authors has identified families of simplicial complexes for which the loop space of their corresponding moment-angle complex is homotopy equivalent to a product of spheres and loops on spheres. In this talk, I will survey the current progress in this direction, and then expand the family of simplicial complexes for which such a decomposition is known - namely simplicial complexes which are the k-skeleton of a flag complex.
Abstract: A subset of a group is called a tile if the group can be covered by a collection of disjoint translates of the subset. A longstanding question asks whether each finite subset can be extended to a finite file in every group. Until recently, little progress was made on this question and relatively few examples were known to satisfy this property. A new method of tiling in the hyperbolic groups was developed by Akhmedov last year, which we extend to the much broader setting of acylindrically hyperbolic groups. This is a joint work with Joe MacManus.
Abstract: It is well known that group action changes the homological, differential, and topological properties of the space. The aim of this talk is to discuss the action of some finite discrete groups on some non-commutative differential and algebraic spaces. We shall first discuss and compare the (co)homological properties of non-commutative and quantum torus then observe how the quotient spaces resulting from the actions of discrete subgroups of $SL(2, \mathbb Z)$ behave. We shall discuss the flip actions on the non-commutative sphere from homological perspective.
Abstract: I will describe some of my previous work on supersymmetric Quantum Field Theory (QFT), and some of my current interests in functorial QFT, and attempt to tie these things together by discussing the concept of a defect in QFT and that of a fully local QFT.
Abstract: To every group action on a compact space, one can associate a natural C*-algebra which is generated by a unitary representation of the group and by multiplication operators on the compact space. It is natural to ask how much information this construction remembers about the original group action. For actions of torsion-free hyperbolic groups, the associated C*-algebras are so-called Kirchberg algebras and so their isomorphism type is completely determined by K-theory. We calculate the K-theory of the associated C*-algebras for fundamental groups of certain hyperbolic 3-manifolds and prove that it only depends on the first homology of the manifold. In particular, this gives an infinite supply of hyperbolic 3-manifold groups all of whose associated C*-algebras are isomorphic. This is joint work in progress with Johannes Ebert and Shirly Geffen.
Abstract: We introduce a new class of groups with Thompson-like group properties. I will introduce these groups and give some motivation. For example, in the surface case, the asymptotic mapping class group contains every mapping class group of finite type surface with boundary. I will explain how these groups act on a CAT(0) cube complex which allows us to show that they are of type F_infinity. This is joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Xaolei Wu with an appendix by Oscar Randal-Williams.