# PG-MA

The PGR Analysis Seminar is a student run seminar series on mathematical analysis. We focus on giving students the opportunity to talk about a topic of interest in mathematical analysis in an informal and friendly atmosphere. It takes place once a fortnight on Mondays at 14:15pm, in the Wolfson Lecture Theatre (4 West, 1.7). The programme for the seminar can be found below.

The analysis group at the University of Bath meets weekly to discuss a wide range of topics. These include operator theory, control theory and PDEs, among many others. More details can be found in this website.

## List of speakers for second semester (2017/18)

### 5th of February

Speaker: Pablo Vinuesa.

Title: Pseudo-differential operators on $$\mathbb{R}^n$$.

Abstract: As a first approach, a pseudo-differential operator can be seen as the generalisation of Fourier multiplier operators. Its "symbol" will depend in the frequency variable and the spatial variable. In contrast a Fourier multiplier only depends on the frequency variable. These symbols can be classified into classes, which are usually called Hormander classes. During this talk I will introduce these operators and discuss the classes of symbols. Some important properties will also be proved.

### 19th of February

Speaker: Owen Pembery.

Title: An Introduction to Bochner Spaces

Abstract: If you've ever considered the solution of a time-dependent PDE, or a stochastic PDE, you've probably been looking at an element of a Bochner space. These are generalisations of $$L^p$$ spaces, where we consider functions that take values in a Banach space. In this talk I'll give an overview of Bochner spaces and explain some of their uses, properties, and idiosyncrasies.

### 5th of March

Speaker: Xavier Pellet.

Title: Studying Chaos with Densities.

Abstract: In this talk I will present some aspect of Chaos by using transformations. More precisely I will introduce two class of operator in order to classify some properties of these transformations: ergodic, mixing and exactness. Baker transformation and Anosov diffeomorphism will support this theory as example. Finally, I will connect these notions to the theory of semi-group.

### 16th of April

Speaker: James Roberts.

Title: An Introduction to Regularity Theory for Energy Minimising Harmonic Maps.

Abstract: Harmonic mappings of Riemannian manifolds are critical points of the Dirichlet energy and their regularity (smoothness) is intimately connected to the topology and geometry of the codomain manifold. For example, if the codomain is non-positively curved then all harmonic maps are smooth in the interior of their domain. In general harmonic maps may have singularities (points of discontinuity) in their domain and it is even possible to construct harmonic maps which are discontinuous everywhere! However, harmonic maps which satisfy additional assumptions such as minimising the energy have a singular set which is small in Hausdorff dimension. In this talk I will introduce the notion of harmonic mappings of Riemannian manifolds and give an overview of the methods required to show regularity.​

### 23rd of April

Speaker: Serena D'Onofrio.

Title: Homogenisation on singular periodic structures.

Abstract: In this talk I will present a method in homogenisation theory for an elliptic operator on periodic sets of lower dimension than the ambient space $$\mathbb{R}^d$$, called singular structures. The particularity of this setting is that each singular object carries a natural measure μ​​ which is supported on the singular structure itself. The goal is to prove that in this case it is possible to obtain an operator-norm convergence estimate for a wide class of periodic measures. The main idea is to use the Floquet transform to pass from the elliptic operator to a family of elliptic operators, in order to obtain a uniform estimate in the space of quasi-periodic functions.

If you have any questions, you can contact me by email: P.Vinuesa@bath.ac.uk