The first year

Research Study Groups

A key part of the training will be an innovative feature aimed at preparing students for research collaborations and teamwork with skills that cannot be taught in a traditional classroom environment. Modern interdisciplinary research requires the participants to be able to understand and interpret work of a mathematical and statistical nature that is expressed in an unfamiliar specialist language, to glean from that work the important mathematical and statistical problems and directions of research, to formulate tractable problems and see how a research team with complementary skills can come together to solve the problem, and to be able to communicate sophisticated mathematical concepts and approaches to collaborators without advanced mathematical training. None of these skills and abilities can be taught in a lecture. We therefore propose a novel component that will lead students through interdisciplinary research experiences and focus their attention on the skills they need to develop.

Year 1 Course Descriptions

Students will take a selection of courses chosen from the list below, as well as graduate level courses offered by the Departments of Mathematics and Statistics. The choice of optional courses will be tailored to the needs of the students. (A), (C), (P) and (S) denote the different MASDOC themes.

• Analysis (A) Hilbert and Banach space methods. L_p spaces, Sobolev spaces, Hölder spaces. Orthonormal bases and frames. Fourier analysis and wavelets. Applications to linear PDEs.

• Nonlinear PDEs (A) The study of nonlinear PDEs requires a variety of techniques including semigroups, variational analysis and viscosity solutions. The students will be familiarised with the most important nonlinear phenomena and the ideas needed for their mathematical analysis.

• An Invitation to Graduate Probability (P) Probabilistic conditioning and independence, stochastic models in space and time, probabilistic limits and approximations, simulation and probability. Surveys the core elements of modern probability theory at a level that is both suitable to provide generalists with a good overview of what can be done, and also an attractive taster for the module “Probability: Theory and Examples”.

• Probability: Theory and Examples (P) This is an assignment-driven module in advanced probability. The intention is to train aspiring probabilists to a high level of mathematical fitness, self-reliance, and collaboration using a module based primarily around compulsory assessments, ensuring an absolutely firm grounding in the foundations of probability.

• Scientific Computing (C) This course is aimed at developing practical skills in scientific computation. Topics covered include: High performance scientific computing including parallel computing, both OpenMP and MPI, and programme development; Computational linear algebra and optimization for both dense and sparse problems. The module will be delivered both through lectures and directed computer lab work. Students will complete group projects, presented to the whole class addressing issues of design, performance and validation.

• Numerical Analysis and PDEs (C) Variational formulation of boundary value problems and the Galerkin method. Finite element spaces; Approximation Theory; Galerkin orthogonality and Aubin-Nitsche; Error Analysis; Nonlinear PDEs; Surface Finite Elements; Finite Differences; Stability, Consistency and Convergence.

• Statistical Methodology and Computation (S) Introduces students to the language and principles of likelihood and non-likelihood based statistical methodology, Classical and Bayesian inference paradigms, and important Computational Statistics methods including MCMC, EM algorithm, Monte Carlo methods for Maximum Likelihood and related problems.

• Statistical Inference for Stochastic Models (S) This course is based on a series of mini-workshops led by a statistics research area specialist, in which students will present material from a list of topics. Each theme will cover motivation (data type, scientific question(s)), models and their properties, inference methodology, statistical issues and some pointers to research questions.

Years 2, 3 and 4

On satisfactory progression from the first year, PhD supervisors and a project are agreed for each student. Each student has a first and a second supervisor. Elements of the training programme to aid the completion of a thesis and the development of research skills are:

• Annual MASDOC conference
• Weekly seminars by leading researchers
• International engagement through graduate schools and short research visits to international partners
• MASDOC Forum: engagement with emerging "Hot Topics"
• Further training through a tailored choice of graduate courses, TCC and APTS courses
• Journal club: a weekly event for all PhD students in which one student presents an account of a significant paper in the literature
• Informal presentations of research to peers
• Mentoring of MASDOC first year MSc students
• All MASDOC PhD students will be enrolled on the Warwick Interdisciplinary Science Transferable Skills Certificate