Mon, 02 Dec 2024
15:30
L5

Building surfaces from equilateral triangles

Lasse Rempe
(Manchester University)
Abstract
In this talk, we consider the following question. Suppose that we glue a (finite or infinite) collection of closed equilateral triangles together in such a way that we obtain an orientable surface. The resulting surface is a Riemann surface; that is, it has a natural conformal structure (a way of measuring angles in tangent space). We ask which Riemann surfaces are *equilaterally triangulable*; i.e., can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface is equilaterally triangulable if and only if it is defined over a number field. These *Belyi surfaces* - and their associated “dessins d’enfants” - have found applications across many fields of mathematics, including mathematical physics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces). The talk should be accessible to a general mathematical audience, including postgraduate students.


 

Tue, 02 Mar 2021
14:15
Virtual

Graded Clifford-Drinfeld algebras

Kieran Calvert
(Manchester University)
Abstract

We combine the notions of graded Clifford algebras and Drinfeld algebras. This gives us a framework to study algebras with a PBW property and underlying vector space $\mathbb{C}[G] \# Cl(V) \otimes S(U) $ for $G$-modules $U$ and $V$. The class of graded Clifford-Drinfeld algebras contains the Hecke-Clifford algebras defined by Nazarov, Khongsap-Wang. We give a new example of a GCD algebra which plays a role in an Arakawa-Suzuki duality involving the Clifford algebra.

Tue, 01 Dec 2020

14:15 - 15:15
Virtual

The deformed Dixmier-Moeglin equivalence for completed enveloping algebras

Adam Jones
(Manchester University)
Abstract

An algebra $R$ is said to satisfy the Dixmier-Moeglin equivalence if a prime ideal $P$ of $R$ is primitive if and only if it is rational, if and only if it is locally closed, and a commonly studied problem in non-commutative algebra is to classify rings satisfying this equivalence, e.g. $U(\mathfrak g)$ for a finite dimensional Lie algebra $\mathfrak g$. We explore methods of generalising this to a $p$-adic setting, where we need to weaken the statement. Specifically, if $\hat R$ is the $p$-adic completion of a $\mathbb Q_p$-algebra $R$, rather than approaching the Dixmier-Moeglin equivalence for $\hat R$ directly, we instead compare the classes of primitive, rational and locally closed prime ideals of $\hat R$ within suitable "deformations". The case we focus on is where $R=U(L)$ for a $\mathbb Z_p$-Lie algebra $L$, and the deformations have the form $\hat U(p^n L)$, and we aim to prove a version of the equivalence in the instance where $L$ is nilpotent.

Thu, 28 Nov 2019

11:30 - 12:30
C4

Actions of groups of finite Morley rank

Alexandre Borovik
(Manchester University)
Abstract

I will be talking of recent results by Ayse Berkman and myself, as well as about a more general program of research in this area.

Mon, 11 Feb 2019

15:45 - 16:45
L3

Small time asymptotics for Brownian motion with singular drift

TUSHENG ZHANG
(Manchester University)
Abstract

We consider Brownian motion with Kato class measure-valued drift.   A small time large deviation principle and a Varadhan type asymptotics for the Brownian motion with singular drift are established. We also study the existence and uniqueness of the associated Dirichlet boundary value problems.

Thu, 24 Jan 2019

14:00 - 15:00
L4

Bespoke stochastic Galerkin approximation of nearly incompressible elasticity

Prof David Silvester
(Manchester University)
Abstract

We discuss the key role that bespoke linear algebra plays in modelling PDEs with random coefficients using stochastic Galerkin approximation methods. As a specific example, we consider nearly incompressible linear elasticity problems with an uncertain spatially varying Young's modulus. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution.  We introduce a novel three-field mixed variational formulation of the PDE model and and  assess the stability with respect to a weighted norm. The main focus will be  the efficient solution of the associated high-dimensional indefinite linear system of equations. Eigenvalue bounds for the preconditioned system can be  established and shown to be independent of the discretisation parameters and the Poisson ratio.  We also  discuss an associated a posteriori error estimation strategy and assess proxies for the error reduction associated with selected enrichments of the approximation spaces.  We will show by example that these proxies enable the design of efficient  adaptive solution algorithms that terminate when the estimated error falls below a user-prescribed tolerance.

This is joint work with Arbaz Khan and Catherine Powell

Thu, 15 Nov 2018

14:00 - 15:00
Rutherford Appleton Laboratory, nr Didcot

Block Low-Rank Matrices: Main Results and Recent Advances

Mr Théo Mary
(Manchester University)
Abstract

In many applications requiring the solution of a linear system Ax=b, the matrix A has been shown to have a low-rank property: its off-diagonal blocks have low numerical rank, i.e., they can be well approximated by matrices of small rank. Several matrix formats have been proposed to exploit this property depending on how the block partitioning of the matrix is computed.
In this talk, I will discuss the block low-rank (BLR) format, which partitions the matrix with a simple, flat 2D blocking. I will present the main characteristics of BLR matrices, in particular in terms of asymptotic complexity and parallel performance. I will then discuss some recent advances and ongoing research on BLR matrices: their multilevel extension, their use as preconditioners for iterative solvers, the error analysis of their factorization, and finally the use of fast matrix arithmetic to accelerate BLR matrix operations.

Tue, 05 Jun 2018

16:00 - 17:00
L5

Counting rational points and iterated polynomial equations

Harry Schmidt
(Manchester University)
Abstract

In joint work with Gareths Boxall and Jones we prove a poly-logarithmic bound for the number of rational points on the graph of functions on the disc that exhibit a certain decay. I will present an application of this counting theorem to the arithmetic of dynamical systems. It concerns fields generated by the solutions of equations of the form $P^{\circ n}(z) = P^{\circ n}(y)$ for a polynomial $P$ of degree $D \geq 2$ where $y$ is a fixed algebraic number. The general goal is to show that the degree of such fields grows like a power of $D^n$.    

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