Tue, 05 Mar 2024
11:00
Lecture room 5

Level lines of the massive planar Gaussian free field

Léonie Papon
(University of Durham)
Abstract

The massive planar Gaussian free field (GFF) is a random distribution defined on a subset of the complex plane. As a random distribution, this field a priori does not have well-defined level lines. In this talk, we give a meaning to this concept by constructing a coupling between a massive GFF and a random collection of loops, called massive CLE_4, in which the loops can naturally be interpreted as the level lines of the field. This coupling is constructed by appropriately reweighting the law of the standard GFF-CLE_4 coupling and this construction can be seen as a conditional version of the path-integral formulation of the massive GFF. We then relate massive CLE_4 to a massive version of the Brownian loop soup. This provides a more direct construction of massive CLE_4 and proves a conjecture of Camia.

Mon, 19 Feb 2024
15:30
Lecture room 5

Rough Stochastic Analysis with Jumps

Dr Andy Allan
(University of Durham)
Abstract

Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa Lê, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.

Tue, 16 May 2023

12:00 - 13:30
L3

Abelian Chern-Simons theory on the lattice

Tin Sulejmanpasic
(University of Durham)
Abstract

I will discuss a formulation of an Abelian Chern-Simons theory on the lattice employing the modified Villain formalism. The theory suffers from a well-known problem of having extra zero modes in the Gaussian operator. I will argue that these zero modes are associated with a kind of subsystem symmetry which projects out almost all naive Wilson loops. The operators which survive are framed Wilson loops. These turn out to be topological charges of the associated one-form symmetry, and it has the correct topological spin and correlation functions.

Tue, 08 Nov 2022
16:00
C1

Interacting Systems – where Analysis, PDEs and Probability meet

Amit Einav
(University of Durham)
Abstract

We are surrounded by systems that involve many elements and the interactions between them: the air we breathe, the galaxies we watch, herds of animals roaming the African planes and even us – trying to decide on whom to vote for.

As common as such systems are, their mathematical investigation is far from simple. Motivated by the realisation that in most cases we are not truly interested in the individual behaviour of each and every element of the system but in the average behaviour of the ensemble and its elements, a new approach emerged in the late 1950s - the so-called mean field limits approach. The idea behind this approach is fairly intuitive: most systems we encounter in real life have some underlying pattern – a correlation relation between its elements. Giving a mathematical interpretation to a given phenomenon and its emerging pattern with an appropriate master/Liouville equation, together with such correlation relation, and taking into account the large number of elements in the system results in a limit equation that describes the behaviour of an average limit element of the system. With such equation, one hopes, we could try and understand better the original ensemble.

In our talk we will give the background to the formation of the ideas governing the mean field limit approach and focus on one of the original models that motivated the birth of the field – Kac’s particle system. We intend to introduce Kac’s model and its associated (asymptotic) correlation relation, chaos, and explore attempts to infer information from it to its mean field limit – The Boltzmann-Kac equation.

Thu, 04 Feb 2021
14:00
Virtual

Modeling composite structures with defects

Anne Reinarz
(University of Durham)
Abstract

Composite materials make up over 50% of recent aircraft constructions. They are manufactured from very thin fibrous layers  (~10^-4 m) and even  thinner resin interfaces (~10^-5 m). To achieve the required strength, a particular layup sequence of orientations of the anisotropic fibrous layers is used. During manufacturing, small localised defects in the form of misaligned fibrous layers can occur in composite materials, adding an additional level of complexity. After FE discretisation the model exhibits multiple scales and large spatial variations in model parameters. Thus the resultant linear system of equations can be very ill-conditioned and extremely large. The limitations of commercially available modelling tools for solving these problems has led us to the implementation of a robust and scalable preconditioner called GenEO for parallel Krylov solvers. I will discuss using the GenEO coarse space as an effective multiscale model for the fine-scale displacement and stress fields. For the coarse space construction, GenEO computes generalised eigenvectors of the local stiffness matrices on the overlapping subdomains and builds an approximate coarse space by combining the smallest energy eigenvectors on each subdomain via a partition of unity.

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Mon, 28 Feb 2011

15:45 - 16:45
L3

Stochastic Algebraic Topology

Michael Farber
(University of Durham)
Abstract

Topological spaces and manifolds are commonly used to model configuration
spaces of systems of various nature. However, many practical tasks, such as
those dealing with the modelling, control and design of large systems, lead
to topological problems having mixed topological-probabilistic character
when spaces and manifolds depend on many random parameters.
In my talk I will describe several models of stochastic algebraic topology.
I will also mention some open problems and results known so far.

Thu, 28 Oct 2010

16:00 - 17:00
L3

Distributions of lattices, class numbers and discriminants

Dr M. Belolipetsky
(University of Durham)
Abstract

While studying growth of lattices in semisimple Lie groups we

encounter many interesting number theoretic problems. In some cases we

can show an equivalence between the two classes of problems, while in

the other the true relation between them is unclear. On the talk I

will give a brief overview of the subject and will then try to focus

on some particularly interesting examples.

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