This is a new seminar for semester 2, 2016/17. As such it's final form is far from determined but I propose the following as a starting point (from which whoever attends can have input!):

Like PSS or the PG algebra and geometry seminar, the talks will be given by and aimed at postgraduate students. However, in this seminar the speakers to be able to assume some familiarity with common masters level courses with an analytical flavour, for example measure theory, martingale theory, functional analysis or differential equations. This should mean the seminar will be of interest to a cross sections of students from (but not limited to) the Analysis, Probability and Numerical Analysis groups.

I would also like to encourage students to not only talk about their work, but results/inequalties/arguments which are common or helpful in their area or others they are interested in. For example, I may give a talk on the maximal inequality which is a key ingredient to proving ergodic theorems (which is my actual research topic). My hope is that spreading awareness of these ideas could help other attendees in their research. It is also something which does not happen in research group seminars, where many mathematicians will already be familiar with common ideas.

If you think this might be of interest to you then send me an email (it's on my homepage) and I'll add you to the mailing list.

## When, Where and What's On?

In semester 2 of 2016 we will meet every 2 weeks of the semester at 14:15 from Monday 20 Feb until Monday 24 April . This will be in Wolfson lecture theatre , 4W 1.7.

## 2016-17 Semester 2

### Title and Abstract

20 Feb Kieran Jarrett Maximal inequalities and almost everywhere convergence

Results about the almost everywhere convergence of averages play a key role in numerous areas of mathematical analysis. For example the strong law of large numbers and Birkhoff's ergodic theorem, which generalises the strong law, are fundamental results in probability and ergodic theory respectively. In real analysis the Lebesgue differentiation theorem, an extension of the fundamental theorem of calculus, is another result of this type.

It turns out that Birkhoff's theorem and the Lebesgue's differentiation theorem are closely related; both results can be proved using a maximal inequality method. In this talk I will show you what a maximal inequality is, outline how one uses them to prove almost everywhere convergence results and exhibit the approach by proving Birkhoff's ergodic theorem.

6 Mar Pablo Vinuesa Harmonic analysis on compact Lie groups.

In this talk I will give an introduction to harmonic analysis on compact Lie groups. This is a very vast area, in which a lot of work has been done in the past, specially in the '50s. I will discuss the Peter-Weyl theorem and how to construct a basis of $$L^2(G)$$ via this theorem and the representation theory of the group. I will also talk about the Laplace operator in the specific case of $$SU(2)$$, and the heat equation.

20 Mar Owen Pembery Error Analysis for Variational Problems

As we can't solve most PDEs explicitly, we often want to devise numerical algorithms to approximate their solution. Once we've come up with these algorithms, we then want to prove properties about them, for example, do they converge to the true solution of our PDE? One such algorithm is the Finite Element Method, which involves rewriting our PDE as a variational problem, and then looking for an approximate solution to this variational problem. In this talk I'll give an overview of how we move from a PDE to a variational problem and indicate how we perform error analysis. That is, how do we show that our Finite Element solution is the best' solution (for a suitable definition of best')?

3 Apr Xavier Pellet Semi-groups, at the interface between Probability and Analysis

Semi group theory is a huge area for solving PDEs. Furthermore, recently Markov Semi Groups were developed in order to understand some Markov diffusions processes and vice versa.

Keywords : Stochastic differential equation, PDE, Functional analysis, Ergodicity, Numerical analysis and Finance.

Don't panic, my goal for this talk is a real initiation (mini course) to all these tools and to share my passion on this topic.

24 Apr Pite Satitkanitkul Title

Abstract