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.
## List of speakers for first semester (2018/19)

### 3rd of December

### 26th of November

### 5th of November

### 22nd of October

### 8th of October

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

### 5th of February

### 19th of February

### 5th of March

### 16th of April

### 23rd of April

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.

Speaker: Ben Robinson.

Speaker: Joel Cawte.

Title: Necessary Conditions for Local Extrema of Constrained problems

Abstract: The method of Lagrange multipliers is a well known tool in finding extrema for constrained problems, both in finite dimensions, and to a lesser extent, infinite dimensions. However, when it comes to distinguishing whether such an extremum is a local minimum/maximum, not much is well known. This talk will (optimistically) go through the finite dimensional case, the generalisation to any Banach Space, and an application to Nonlinear Incompressible Elasticity.

Speaker: Francisco de Melo.

Title: When geometry and analysis meet: Geometric function theory and applications.

Abstract: This talk will be a survey on what I consider to be one of the most beautiful subjects in mathematical analysis, the so-called geometric function theory. Although this is a classical subject, it has recently emerged again as a field of active research, mainly motivated by applications. Among those, is the generalisation of the Christoph-Schwarz Transform to multiply connected domains and the solution of nonlinear free boundary problems in fluid mechanics, the latter being one of the examples to be discussed in the talk. As a pre-requisite, I first undergraduate course on complex analysis is desirable.

Speaker: Joe Harris.

Title: Singularities in Nematic Liquid Crystals.

Abstract: Nematic liquid crystals are classical examples of mesophases intermediate in character between the solid and liquid states of matter. The orientational anisotropy they exhibit makes nematics the working material of choice for a range of optical and industrial applications, notably they form the backbone of the multi-billion dollar liquid crystal display industry. In this talk I will present the work I have done studying radially symmetric solutions in a 2D domain in the Landau-de Gennes Theory and I will present asymptotic approximations of these solutions in the macroscopic and nanoscopic limits.

Speaker: Pablo Vinuesa.

Title: The forbidden and exotic classes.

Abstract: A pseudo-differential operator can be seen as a generalisation of a Fourier multiplier operator. Their corresponding symbols \(a(x,\xi)\) can be classified in symbol classes \(S^m_{\rho,\delta}\), for \( 0\leq\delta\leq\rho\leq 1 \), with condition \( |\partial_x^{\beta} \partial_{\xi}^{\alpha} \,a(x,\xi)|\leq C(1+|\xi|^2)^{\frac{1}{2}(m-\rho|\alpha|+\delta|\beta|)} \). These pseudo-differential operators satisfy nice properties in general; for example, they are bounded in \(L^2\) when \(\rho=\delta\neq 1\). However, there are certain classes for which these "nice" properties do not necessarily hold. For instance, the pseudo-differential operators corresponding to symbols of class \( S^0_{1,1} \), which is known as the "forbidden class", or \( S^m_{1/2\,,\,1/2} \), which is known as the "exotic class". Although thankfully, these classes do offer some other positive properties. In this talk I will discuss these symbol classes and some of their interesting properties.

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.

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.

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.

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.

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