Moodle@Warwick

Topology is the study of properties of spaces that are invariant under continuous deformations. An often cited example is that a cup is topologically equivalent to a torus, but not to a sphere. In general, topology is the rigorous development of ideas related to concepts such “nearness”, “neighbourhood”, and “convergence”.

This module covers topological spaces and their properties, homotopy, the fundamental group, Galois correspondence, universal covers, free products, and CV complexes.

The course will follow largely the first chapter of

• Allen Hatcher. Algebraic Topology. Cambridge University Press.

An electronic version of the book is freely available on the author’s web page, and a printed version should be available in the library or the campus bookshop.

The important and pervasive role played by pdes in both pure and applied mathematics is described in MA250 Introduction to Partial Differential Equations. In this module I will introduce methods for solving (or at least establishing the existence of a solution!) various types of pdes. Unlike odes, the domain on which a pde is to be solved plays an important role. In the second year course MA250, most pdes were solved on domains with symmetry (eg round disk or square) by using special methods (like separation of variables) which are not applicable on general domains. You will see in this module the essential role that much of the analysis you have been taught in the first two years plays in the general theory of pdes. You will also see how advanced topics in analysis, such as MA3G7 Functional Analysis I, grew out of an abstract formulation of pdes. Topics in this module include:

• Method of characteristics for first order PDEs.

• Fundamental solution of Laplace equation, Green's function.

• Harmonic functions and their properties, including compactness and regularity.

• Comparison and maximum principles.

• The Gaussian heat kernel, diffusion equations.

• Basics of wave equation (time permitting).

The course will start by introducing the concept of a manifold (without recourse to an embedding into an ambient space). In the words of Hermann Weyl (Space, Time, Matter, paragraph 11):

“The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colours, tones) may be specified by the giving of n quantities, the “co-ordinates,” which are continuous functions within the manifold. This does not mean that the whole manifold with all its elements must be represented in a single and reversible manner by value systems of n co-ordinates (e.g. this is impossible in the case of the sphere, for which n = 2); it signifies only that if P is an arbitrary element of the manifold, then in every case a certain domain surrounding the point P must be representable singly and reversibly by the value system of n co-ordinates.”

Thus the points on the surface of a sphere form a manifold. The possible configurations of a double pendulum (one pendulum hung off the pendulum bob of another) is a manifold that is nothing but the surface of a two-torus: the surface of a donut (a triple pendulum would give a three-torus etc.) The possible positions of a rigid body in three-space form a six-dimensional manifold. Colour qualities form a two-dimensional manifold (cf. Maxwell’s colour triangle).

It becomes clear that manifolds are ubiquitous in mathematics and other sciences: in mechanics they occur as phase-spaces; in relativity as space-time; in economics as indifference surfaces; whenever dynamical processes are studied, they occur as “state-spaces” (in hydrodynamics, population genetics etc.)

Moreover, in the theory of complex functions, the problem of extending one function to its largest domain of definition naturally leads to the idea of a Riemann surface, a special kind of manifold. 

Although it seems so natural from a modern vantage point, it took some time and quite a bit of work (by Gauss, Riemann, Poincare, Weyl, Whitney, …) till mathematicians arrived at the concept of a manifold as we use it today. It is indispensable in most areas of geometry and topology as well as neighbouring fields making use of geometric methods (ordinary and partial differential equations, modular and automorphic forms, Arakelov theory, geometric group theory…)

Some buzz words suggesting topics which we plan to cover include:

-The notion of a manifold (in different setups), examples of constructions of manifolds (submanifolds, quotients, surgery)

-The tangent space, vector fields, flows/1-parameter groups of diffeomorphisms

-Tangent bundle and vector bundles

-Tensor and exterior algebras, differential forms

-Integration on manifolds, Stokes’ theorem

-de Rham cohomology, examples of their computation (spheres, tori, real projective spaces...)

-Degree theory, applications: argument principle, linking numbers, indices of singularities of vector fields

We will also discuss a lot of concrete and interesting examples of manifolds in the lectures and work sheets, such as for example: tori, n-holed tori, spheres, the Moebius strip, the (real and complex) projective plane, higher-dimensional projective spaces, blow-ups, Hopf manifolds… 

The nature of the material makes it inevitable that considerable time must be devoted to establishing the foundations of the theory and defining as well as clarifying key concepts and geometric notions. However, to make the content more vivid and interesting, we will also seek to include some attractive and non-obvious theorems, which at the same time are not too hard to prove and natural applications of the techniques introduced, such as, for instance, Ehresmann's theorem on differentiable fibrations, or that a sphere cannot be diffeomorphic to a product of (positive-dimensional) manifolds. 

This Module is mathematically closely related to, but formally completely independent of MA3D9 Geometry of Curves and Surfaces. 

Books:

Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer (1983) (esp. Chapters 1, 2, 4)

M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces, Springer (1988) (esp. Chapters 2-6)

Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, Publish or Perish, Inc. (2005) (esp. Chapters 1-8)

John Lee, Introduction to smooth manifolds, Springer (2012)

Loring W. Tu, An Introduction to Manifolds, Springer (2011)

Lecturer: Christian Böhning

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 lectures plus assignments

Assessment: Assignments (15%), 3 hour written exam (85%).

Prerequisites: 
A background in algebra (especially MA249 Algebra II) is essential. The module develops more specialised material in commutative algebra and in geometry from first principles, but MA3G6 Commutative Algebra will be useful. More than technical prerequisites, the main requirement is the sophistication to work simultaneously with ideas from several areas of mathematics, and to think algebraically and geometrically. Some familiarity with projective geometry (e.g. from MA243 Geometry) is helpful, though not essential.

Leads To: 
A first module in algebraic geometry is a basic requirement for study in geometry, number theory or many branches of algebra or mathematical physics at the MSc or PhD level. Many MA469 projects are on offer involving ideas from algebraic geometry.

Content: 
Algebraic geometry studies solution sets of polynomial equations by geometric methods. This type of equations is ubiquitous in mathematics and much more versatile and flexible than one might as first expect (for example, every compact smooth manifold is diffeomorphic to the zero set of a certain number of real polynomials in R^N). On the other hand, polynomials show remarkable rigidity properties in other situations and can be defined over any ring, and this leads to important arithmetic ramifications of algebraic geometry.

Methodically, two contrasting cross-fertilizing aspects have pervaded the subject: one providing formidable abstract machinery and striving for maximum generality, the other experimental and computational, focusing on illuminating examples and forming the concrete geometric backbone of the first aspect, often uncovering fascinating phenomena overlooked from the bird's eye view of the abstract approach.

In the lectures, we will introduce the category of (quasi-projective) varieties, morphisms and rational maps between them, and then proceed to a study of some of the most basic geometric attributes of varieties: dimension, tangent spaces, regular and singular points, degree. Moreover, we will present many concrete examples, e.g., rational normal curves, Grassmannians, flag and Schubert varieties, surfaces in projective three-space and their lines, Veronese and Segre varieties etc. 

Books:

• Atiyah M.& Macdonald I. G., Introduction to commutative algebra, Addison-Wesley, Reading MA (1969)
• Harris, J., Algebraic Geometry, A First Course, Graduate Texts in Mathematics 133, Springer-Verlag (1992)
• Mumford, D., Algebraic Geometry I: Complex Projective Varieties, Classics in Mathematics, reprint of the 1st ed. (1976); Springer-Verlag (1995)
• Reid, M., Undergraduate Algebraic Geometry, London Math. Soc. Student Texts 12, Cambridge University Press (2010)
• Shafarevich, I.R., Basic Algebraic Geometry 1, second edition, Springer-Verlag (1994)
• Zariski, O. & Samuel, P., Commutative algebra, Vol. II, Van Nos- trand, New York (1960)

https://www.mathematical-modelling.science/index.php/lectures/warwick-2020-2021

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).

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

1. Steve Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS 2015
2. Herbert Edelsbrunner and John Harer. Computational Topology, An Introduction. AMS 2010
3. 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.

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).

MB ChB Phase 3 spans year 3 AND 4.

2017 Cohort are supported through year 4 via the MD30X-19/20 module Moodle space.

The two laboratory mini-projects conclude your MSc year and prepare you for your PhD projects. MD979 is the first one. Due to the highly inter-disciplinary nature of the IBR DTP, you have a great variety of choice for these projects. You will have an 11-week period for each of the two mini-project modules (MD979 and the subsequent MD980). Normally there would be one experimental biology project and one either theoretical biology project (e.g., bioinformatics, computational biology) or experimental project in chemistry, physics or engineering. The supervisor pool is accordingly expanded to include colleagues across the departments of the Faculty of Science in addition to WMS.

The projects will either be developed by the you together with an academic from the supervisor pool, or initially by the supervisors alone, who will submit projects directly to the mini-project call for presentation to the student cohort at a mini-project “fair”. You should be aware that all projects will first be vetted in the IBR DTP management team before inclusion in the course.

This short course is designed to develop the skills required to manage patients with a range of chronic diseases such as diabetes, dementia, musculoskeletal conditions, depression and mental illness, and end of life care within primary care.

The course will introduce you to the knowledge and skills that are needed for the delivery and organisation of high quality care, responsive to the physical, emotional, psychological and social needs of people with chronic conditions within the multidisciplinary setting.

The course will provide a sound foundation and understanding of economic concepts and their relevance to decisions around the allocation of healthcare resources, allowing you to critically appraise health economics studies and work effectively with health economists in your team.  It will provide you with a toolkit of methods useful for research and healthcare management.