Introduction to Paradifferential Calculus

Francesco De Anna (Institut de Mathématiques de Bordeaux) – 03/12/15

In this talk I will introduce some main tools and features of the Littlewood-Paley theory and the homogeneous Besov spaces. I will recall some classical lemma, as a Bernstein-type inequality and the Besov-Sobolev relation. Then I will introduce the homogeneous paradifferential calculus, with emphasis on the homogeneous Bony decomposition. The final goal of the talk will be to prove an useful product law between homogeneous Sobolev spaces.

Summability of Multiple Fourier Series, Convex Polytopes and the Ball Multiplier Theorem of Fefferman

George Simpson -20/11/15

This talk is an excursion into multiplier theory and summability methods for multiple Fourier series. By extending the classical Riesz projection and circular Hilbert transform to higher dimensions via the Calderon-Zygmund method of rotations I will present some surprising contrasts between summability via convex polytopes with finite vs infinite faces and end by presenting the celebrated ball multiplier theorem of Charles Fefferman.

Stochastic Mathematical Models of Epidemies (SIS, SIR) and Their Entropy.

Farzad Fatehi – 06/11/15

In this talk at first, I will give you some basic information about random variables and stochastic processes, especially Markov chains. Next, I will introduce the entropy of the different forms of random variables and stochastic processes. After that, I will talk about the applications of stochastic processes in mathematical biology, and why it is better for us to use stochastic models instead of deterministic ones. At the end, I will present some of these stochastic models in biology and their entropy to compare them to see which model is nearer to reality. 

Optimal Constants for Functional Inequalities and the First Eigenvalue of the Laplace-Beltrami Operator 

George Morrison – 23/10/15

In this talk I'll explore the Poincare and logarithmic Sobolev inequalities and look in particular at their optimal constants. In both cases I will show that this constant relates to the first positive eigenvalue of the negative Laplace-Beltrami operator. I will use as evidence Gross' theory of hypercontractivity and explicit examples taken from the sphere. Then, in order to consider a wider class of Riemannian manifold I will introduce the curvature-dimension condition for an operator on a manifold, and use this property to derive the functional inequalities with their optimal constant and as such recover estimates for the first eigenvalue.  

Deriving a Compact Model for the Spread of a Disease through a Random Network

Neil Sherborne – 9/10/15

In this slightly shorter than normal talk I will be discussing the pairwise approximation method for describing the spread of infectious diseases across static networks. In particular I will discuss the so-called multi-stage approach in order to construct a system of ODEs which more realistically describes the distribution which determines the duration of infection. Coupled with this I will illustrate how, with reasonable assumptions, we can improve on previous models to construct a model whose size does not depend on the degree distribution chosen for the network, resulting in a model which can realistically depict the typical behaviour of an SIR epidemic on a social infrastructure which is based on empirical observations. If there is time remaining I will briefly discuss a couple of important characteristics which emerge, and highlight the difficulty that clustering imposes.

The fundamental solution to the heat equation on compact Lie groups.

Charles Morris - 25/09/15

In this talk I am going to discuss how the fundamental solution of the heat equation on a compact Lie group can be expressed as a product over the roots of the Lie group. The terms in this products are fundamental solutions to the heat equation of SU(2), and as such can be identified with classical theta functions. During the process of this talk I will also highlight how certain differential equations on a compact Lie group can be transferred to another equation on the maximal torus where it is often easier to study as this can essentially be thought of as a flat torus in R^n.

Manifolds of non-negative Ricci Curvature and Sobolev Inequalities

Stuart Day – 11/09/15

Let M be a n-dimensional, complete Riemanian manifold with non-negative Ricci-curvature. We prove that if the Sobolev inequality is satisfied on M with the C equal to the optimal constant of this inequality on R^n, then M is isometric to R^n. We also discuss generalisations of this result and further results using the same method of proof.

A Phase Field Approach to a Coupled System

Joe Eyles - 28/08/15

In this talk, I will present a specialised model for tumour growth and a phase field approximation of said model. An explanation of what the phase field is will be given, as well as an example of how to translate a PDE into the phase field paradigm. Some time will be taken to prove a bound on one variable (although bounds for the other variables have been shown). The implementation and problems that arise are discussed, and a possible solution is proposed.

Pointwise Convergence of the Fourier Inversion Formula via the Wave Equation

George Simpson – 14/08/15

A well know problem in harmonic analysis is the convergence, in L^p-norm as well as pointwise a.e., of the partial Fourier sums of L^p functions. Not long ago L. Carleson and immediately afterwards R. Hunt established the pointwise a.e. convergence of Fourier series for L^p functions on the circle, and quite long ago M. Riesz showed the norm converge of Fourier series in L^p for 1<p<\infty. In this talk, I will show how using techniques based on the wave equation and functional calculus, by imposing specific Dini-type conditions, one can obtain pointwise convergence results for the partial sums of multiple Fourier series and integrals of L^p functions. I will present a general condition for pointwise convergence (that involves the spectral measure) and then apply this to the specific cases of the n-dimensional Euclidean space, the unit n-sphere, and the hyperbolic n-space

Heat Kernel Bounds under a Curvature-Dimension Hypothesis

George Morrison – 17/07/15

In this talk I will present a new method for recovering bounds on the heat kernel for an (hypo)elliptic operator L. The result will hold on a general Riemannian manifold and is driven by the geometric properties of curvature and dimension, although the technical details in the proof come mostly for calculus. The conclusion is achieved via the better-known method of proving the ultracontactivity of the heat semigroup associated to our operator L. This is the strongest regularity result possible for the semigroup, and the corresponding kernel bound is easily shown to be equivalent to this. 

On the Action of the Group SL(2, R) on the Hyperbolic Plane

Richard Olu Awonusika - 3/7/15

We discuss the group PSL(2, R) as the group of orientation-preserving isometries of the hyperbolic plane H. Also discussed is the classification of isometries of H, namely hyperbolic, elliptic and parabolic isometries.

Representation Theory on Compact Groups

Charles Morris - 19/6/15

In today's talk I am going to introduce representation theory on compact groups. Once I havecovered the basic tools we move on to discuss results such as Schur's lemma and Schur's orthogonality relations ultimately leading to the Peter-Weyl theorem. Then I will conclude the talk by defining and exploring the Fourier transform on compact groups.  

Error Analysis of a Finite-Element Splitting Method for the Adjusted Cahn-Hilliard Equation for Binary Digital Image Inpainting

Gary Poole - 5/6/15

The adjusted Cahn-Hilliard equation for digital image inpainting is a specific model in the field of PDEs applied to digital image inpainting tasks. The core features of phase-field models (and other curvature models) lend themselves very nicely to these applications. My talk will be a follow-up for the modelling talk on variational approaches I gave at a previous event. In this talk we will aim specifically to provide a numerical framework in the form of a straight-forward time-stepping finite element method for our PDE. We will define the problems we wish to analyse and provide a skeleton outline proof for an error bound in terms of the discretisation parameters of our system.

Homotopy and the Calculus of Variations

Stuart Day - 22/5/15

In this talk I will give a brief introduction to homotopy theory and discuss various methods for enumerating homotopy classes of some function spaces. I will then discuss how homotopy theory can be used to find local minimizers in the Calculus of variations using the Direct method. I’ll give an example that demonstrates how these methods can be used in order to find infinitely many local minimizers. I’ll also discuss the ‘Bubbling off effect’, where the direct method fails to find minimizers in homotopy classes.

p-Norms and Bounds for the Riesz Projection via Plurisubharmonic Functions

George Simpson - 1/5/15

Take the Fourier coefficients of a pth−power integrable complex-valued function on the unit circle T and take points on T raised to an integer k. The Riesz projection is defined as multiplying these two quantities and summing over all positive k (including 0) and cutting away/ignoring all the negative k by mapping the sum onto the Hardy space on T. In this talk, I will cover bounds and norms for the Riesz projection in the paper ”Best Constants for the Riesz Projection” by Brian Hollenbeck and Igor Verbitsky. Specifically, we will cover bounds for weighted L^p spaces, the unit circle and the half-line Fourier multiplier via plurisubharmonic functions and interpolation.

Contractivity Estimates on Markov Semigroups via Logarithmic Sobolev Inequalities

George Morrison - 1/5/15

We can use the tool of logarithmic Sobolev inequalities to gather information about the convergence to equilibrium of a Markov semigroup. We define a Markov semigroup to be the mathematical realisation of some “memoryless” physical phenomenon, the prototypical example being heat diffusion. As time elapses, the entropy (disorder / randomness) of the system decreases to 0, and our logarithmic Sobolev inequalities model relationship between the energy and the entropy of our system. The rate at which this equilibrium is reached depends entirely on the growth of the energy of the system, and we see that sublinear growth recovers ultracontractivity, whereas linear growth only recovers the weaker property of hypercontractivity.