 Autumn 2015

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 LittlewoodPaley theory and the homogeneous Besov spaces. I will recall some classical lemma, as a Bernsteintype inequality and the BesovSobolev 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 CalderonZygmund 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 LaplaceBeltrami 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 LaplaceBeltrami 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 curvaturedimension 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 socalled multistage 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 nonnegative Ricci Curvature and Sobolev Inequalities
Stuart Day – 11/09/15
Let M be a ndimensional, complete Riemanian manifold with nonnegative Riccicurvature. 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.
 Summer 2015

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^pnorm 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 Dinitype 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 ndimensional Euclidean space, the unit nsphere, and the hyperbolic nspace
Heat Kernel Bounds under a CurvatureDimension 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 betterknown 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 orientationpreserving 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 PeterWeyl theorem. Then I will conclude the talk by defining and exploring the Fourier transform on compact groups.
Error Analysis of a FiniteElement Splitting Method for the Adjusted CahnHilliard Equation for Binary Digital Image Inpainting
Gary Poole  5/6/15
The adjusted CahnHilliard equation for digital image inpainting is a specific model in the field of PDEs applied to digital image inpainting tasks. The core features of phasefield models (and other curvature models) lend themselves very nicely to these applications. My talk will be a followup 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 straightforward timestepping 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.
pNorms and Bounds for the Riesz Projection via Plurisubharmonic Functions
George Simpson  1/5/15
Take the Fourier coefficients of a pth−power integrable complexvalued 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 halfline 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.