I hope to give most of the lectures live in MS.01 acc. to the stated timetable. If I can, I will use Lecture capture and post some short video clips for those of you working from home. I set up this moodle page as storage for these posts.
Topological Data Analysis (TDA) is an approach to data analysis based on techniques from algebraic topology. Topology is the study of properties of sets that are invariant under continuous deformations; it is concerned with concepts such as ``nearness'', ``neighbourhood'', and ``convergence''. Nowadays, topological ideas are an indispensable part of many fields of mathematics, ranging from number theory to partial differential equations. Algebraic topology, in particular, aims to understand topological properties of spaces through algebraic invariants. The premise of topological data analysis is that data there is an underlying topological structure to data. Familiar examples include clustering, where the aim is to subdivide data into different clusters, or ``connected components'', and connectivity in networks. In this module we introduce persistent homology, a powerful method for studying the topology of data. We discuss the theoretical foundations, as well as computational and algorithmic aspects and various applications. While the course is mainly theoretical in nature, you are encouraged to experiment using a range of available software and applications. The lecture material will be available as video recording and slides, and exercises will be published semi-regularly.
Intended Learning Outcomes
Upon completion of this module you should be able to:
- understand how topological information can be extracted from discrete data;
- use persistent homology to compute persistence diagrams and barcodes;
- explain the different parts of the persistent homology pipeline and the computational challenges involved;
- evaluate the stability and robustness of persistent homology computations;
- summarize different approaches to the topology of data and discuss applications
Literature
- Steve Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS 2015
- Herbert Edelsbrunner and John Harer. Computational Topology, An Introduction. AMS 2010
- Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod & Heather A Harrington. A roadmap for the computation of persistent homology. 2017
More specialised sources and papers will be made available in time.
The purpose of this module is to provide rigorous training in probability theory for students who plan to specialise in this area or expect probability to feature as an essential tool in their subsequent research. It will also be accessible to students who never got into probability theory beyond core-module level taught in the first year and who are eager to get acquainted with basic probability theory, in particular, the aim is to appeal to but not limited to students working in analysis, dynamical systems, combinatorics & discrete mathematics, and statistical mechanics. To include these two different groups of students and to accommodate their needs and various background the module will cover in the first two weeks a steep learning curve into basic probability theory (see part I below). Secondly, the written assessment, 50 % essay with 16 pages, can be chosen either from a list of basic probability theory (standard textbooks in probability and graduate lecture notes on probability theory) or from a list of high-level hot research topics including original research papers and reviews and lecture notes (see below). List of possible essay topics see attached pdf - file.
Overview
There is much active mathematical research into aeroacoustics (the study of sound in aircraft engines). This field is closely followed, and often contributed to (sometimes helpfully) by engineers in both academia and industry (e.g. Airbus, Boeing, NASA, etc). The aim of this course is to give an overview of the mathematical techniques needed to understand the current research problems, and read current papers in the area. This could lead on to several possible PhD projects, including in asymptotics, numerical analysis, and stability theory.
Aims
The application of wave theory to problems involving the generation, propagation and scattering of acoustic and other waves is of considerable relevance in many practical situations. These include, for example, underwater sound propagation, aircraft noise, remote sensing, the effect of noise in built-up areas, and a variety of medical diagnostic applications. This course would aim to provide the basic theory of wave generation, propagation and scattering, and an overview of the mathematical methods and approximations used to tackle these problems, with emphasis on applications to aeroacoustics. The ultimate aim is for students to understand the underlying mathematical tools of acoustics sufficiently to read current research publications on acoustics, and to be able to apply these techniques to current research questions within mathematics, engineering and industry.
Learning Outcomes
- Reproduce standard models and arguments for sound generation and propagation.
- Apply mathematical techniques to model sound generation and propagation in simple systems.
- Understand and apply Wiener-Hopf factorisation in the scalar case.
Approximate Syllabus
- Some general acoustic theory.
- Sound generation by turbulence and moving bodies (including the Lighthill and Ffowcs Williams Hawkings acoustic analogies).
- Scattering (including the scalar Wiener-Hopf technique applied to the Sommerfeld problem of scattering by a sharp edge)
- Long-distance sound propagation including nonlinear and viscous effects.
- Wave-guides.
- High frequencies and Ray Tracing.
Reading List
- D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, et al, "Modern Methods in Analyticial Acoustics", Springer 1992.
- M. Howe, "Acoustics & Aerodynamic Sound", Cambridge 2015 (available online through Warwick Library).
- S.W. Rienstra & A. Hirschberg, "An Introduction to Acoustics", (available online).