Moodle@Warwick

Status for Mathematics students: List B for third years. If numbers permit second and fourth years may take this module as an unusual option, but confirmation will only be given at the start of Term 2.

Commitment: 10 two hour and10 one hour seminars (including some assessed problem solving)

Assessment: 10% from weekley seminars, 40% from assignment, 50% two hour exam in June

Prerequisites: None

Introduction
This module gives you the opportunity to engage in mathematical problem solving and to develop problem solving skills through reflecting on a set of heuristics. You will work both individually and in groups on mathematical problems, drawing out the strategies you use and comparing them with other approaches.

General aims
This module will enable you to develop your problem solving skills; use explicit strategies for beginning, working on and reflecting on mathematical problems; draw together mathematical and reasoning techniques to explore open ended problems; use and develop schema of heuristics for problem solving.

This module provides an underpinning for subsequent mathematical modules. It should provide you with the confidence to tackle unfamiliar problems, think through solutions and present rigorous and convincing arguments for your conjectures. While only small amounts of mathematical content will be used in this course which will extend directly into other courses, the skills developed should have wide ranging applicability.

Intended Outcomes

Learning objectives

The intended outcomes are that by the end of the module you should be able to:

• Use an explicit problem solving scheme to control your approach to mathematical problems
• Explain the role played by different phases of problem solving
• Critically evaluate your own problem solving practice

Organisation

The module runs in term 2, weeks 1-10

Thursday 14:00-15:00 OC0.04 (new teaching and learning building)
Friday 15:00-17:00 OC0.04

Most weeks the Thursday slot will be used for the weekly (assessed) problem session, but this will not be the case every week. You are expected to attend all three timetabled hours.

Assessment Details

1. A flat 10% given for ‘serious attempts’ at problems during the course. Each week, you will be assigned a problem for the seminar. At then end of the seminar, you should present a ‘rubric’ of your work on that problem so far. If you submit at least 7 rubrics, deemed to be ‘serious attempts’, you will get 10%.
2. One problem-solving assignment (40%) (deemed to be the equivalent of 2000 words) due by noon on Monday 20th March 2017 by electronic upload (pdf).
3. A 2 hour examination in Summer Term 2017 (50%).

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 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: Dr. David Wood

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 Lectures

Assessment: 100% exam

Prerequisites: MA133 Differential Equations, MA249 Algebra II, MA259 Multivariable Calculus. MA254 Theory of ODEs would be useful, but is not essential.

Leads to:

Content: This module investigates how solutions to systems of ODEs (in particular) change as parameters are smoothly varied resulting in smooth changes to steady states (bifurcations), sudden changes (catastrophes) and how inherent symmetry in the system can also be exploited. The module will be application driven with suitable reference to the historical significance of the material in relation to the Mathematics Institute (chiefly through the work of Christopher Zeeman and later Ian Stewart). It will be most suitable for third year BSc. students with an interest in modelling and applications of mathematics to the real world relying only on core modules from previous years as prerequisites and concentrating more on the application of theories rather than rigorous proof.

Indicative content (precise details and order still being finalised):

1. Typical one-parameter bifurcations: transcritical, saddle-node, pitchfork bifurcations, Bogdanov-Takens, Hopf bifurcations leading to periodic solutions. Structural stability.
2. Motivating examples from catastrophe and equivariant bifurcation theories, for example Zeeman Catastrophe Machine, ship dynamics, deformations of an elastic cube, D_4-invariant functional.
3. Germs, equivalence of germs, unfoldings. The cusp catastrophe, examples including Spruce-Budworm, speciation, stock market, caustics. Thom’s 7 Elementary Catastrophes (largely through exposition rather than proof). Some discussion on the historical controversies.
4. Steady-State Bifurcations in symmetric systems, equivariance, Equivariant Branching Lemma, linear stability and applications including coupled cell networks and speciation.
5. Time Periocicity and Spatio-Temporal Symmetry: Animal gaits, characterization of possible spatio-temporal symmetries, rings of cells, coupled cell networks, H/K Theorem, Equivariant Hopf Theorem.

Further topics from (if time and interest):

Euclidean Equivariant systems (example of liquid crystals), bifurcation from group orbits (Taylor Couette), heteroclinic cycles, symmetric chaos, Reaction-Diffusion equations, networks of cells (groupoid formalism).

Aims: Understand how steady states can be dramatically affected by smoothly changing one or more parameters, how these ideas can be applied to real world applications and appreciate this work in the historical context of the department.

Objectives:

Books:

There is no one text book for this module, but the following may be useful references:

• Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Guckenheimer/Holmes 1983
• Catastrophe Theory and its Applications, Poston and Stewart, 1978
• The Symmetry Perspective, Golubitsky and Stewart, 2002
• Singularities and Groups in Bifurcation Theory Vol 2, Golubitsky/Stewart/Schaeffer 1988
• Pattern Formation, an introduction to methods, Hoyle 2006.

This module investigates how solutions to systems of ODEs (in particular) change as parameters are smoothly varied resulting in smooth changes to steady states (bifurcations), sudden changes (catastrophes) and how inherent symmetry in the system can also be exploited. The module will be application driven with suitable reference to the historical significance of the material in relation to the Mathematics Institute (chiefly through the work of Christopher Zeeman and later Ian Stewart). It will be most suitable for third year BSc. students with an interest in modelling and applications of mathematics to the real world relying only on core modules from previous years as prerequisites and concentrating more on the application of theories rather than rigorous proof.

Indicative content (precise details and order still being finalised):

1. Typical one-parameter bifurcations: transcritical, saddle-node, pitchfork bifurcations, Bogdanov-Takens, Hopf bifurcations leading to periodic solutions. Structural stability.
2. Motivating examples from catastrophe and equivariant bifurcation theories, for example Zeeman Catastrophe Machine, ship dynamics, deformations of an elastic cube, $D_4$-invariant functional.
3. Germs, equivalence of germs, unfoldings. The cusp catastrophe, examples including Spruce-Budworm, speciation, stock market, caustics. Thom’s 7 Elementary Catastrophes (largely through exposition rather than proof). Some discussion on the historical controversies.
4. Steady-State Bifurcations in symmetric systems, equivariance, Equivariant Branching Lemma, linear stability and applications including coupled cell networks and speciation.
5. Time Periocicity and Spatio-Temporal Symmetry: Animal gaits, characterization of possible spatio-temporal symmetries, rings of cells, coupled cell networks, H/K Theorem, Equivariant Hopf Theorem.

Further topics from (if time and interest):

Euclidean Equivariant systems (example of liquid crystals), bifurcation from group orbits (Taylor Couette), heteroclinic cycles, symmetric chaos, Reaction-Diffusion equations, networks of cells (groupoid formalism).

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

Examinable topics from the module;

The first part concerning basics of Ising model in the extent covered in lectures

is close to Chapters 3.1-3.7 of the book Statistical Mechanics of Lattice systems 

by S. Friedli and Y. Velenik

Including, in particular, following Statements with proofs:

Existence of the free energy (preassure) in the thermodynamic limit and its properties

Fekete’s  Lemma

Correlation inequalities: GKS I, II, FKG

Existence of magnetizatin in the thermodynamic limit and its properties

Peierls argument

Analyticity of the free energy in d=1

Kramers-Wanier duality

Convergence of the abstract cluster expanion and its applications

High and Low temperature expansion

In the second part we followed chosen sections from the PCMI lecturea

by H. Duminil-Copin (major parts of Section 1 and 2)

In particular,

Definitions of order parameters and transition/critical temperatures.

Overview of proven and conjectured statement for Ising, Potts, and O models.

Coupling between random cluster (RC) a Potts models (Edwards-Sokal measure)

Equivalence of RC and Potts model order parameters.

The percolation transitiuon for random cluster models.

Sufficient condition for monotonicity for percolation models

Propositions for RC mofrls:

comparison between boundary conditions,

monotonicity,

FKG inequality

Existence of a sharp phase transitions for RC and Potts models

Burton-Kean theorem

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).

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).

2018 Cohort MD153 (18/19)

The GMC requires Medical Schools to allocate approximately 10% of the curriculum to the SSC, with the remainder being allocated to the core. In contrast to the prescriptive, competency-based focus of Blocks within the core curriculum, SSCs provide opportunities to acquire intellectual and attitudinal attributes in preparation for professional life and life-long learning.

2017 Cohort

The SSC 2 Block will give you the opportunity to undertake a project (research, audit or service evaluation) and attend a series of lectures and seminars supporting the development of your enquiry and research skills. 

MD360 (16/17) OLD

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