MAGIC Course Proposals for 2007/08

Home , Noncommutative topology, Toda field theory

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MAGIC Course Proposals for 2007/08 A geometric view of classical physics Semester: University: Southampton Contact: Lecturer: Carsten Gundlach Hours: 20 Format: 2 hours a week for 10 weeks Subject classification: Preparation cost: £ Description Differential geometry (5 lectures) â€“ Special relativity (3 lectures) â€“ Thermodynamics (4 lectures) â€“ Fluid mechanics (4 lectures) â€“ Magnetohydrodynamics (4 lectures) Material to be prepared Comments by proposer Based on text by Thorne and Wheeler. It is not meant to be an exhaustive treatment of, e.g., fluid mechanics.

Complex Transforms and the Wiener-Hopf Method Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: David Needham Hours: 20 Format: Subject classification: Preparation cost: £ Description Material to be prepared Comments by proposer

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Dynamical Systems Semester: University: Loughborough Contact: Alexey Bolsinov Hours: 20 Format: Lecturer: Anatoly Neishtadt Subject classification: Preparation cost: £ 1500 Description Linearization of differential equations and maps. Multipliers, Floquet theory, Lyapunov exponents. Normal forms of nonlinear Hamiltonian systems and symplectic maps near fixed point. Bifurcation theory, saddle-node, Poincare-Andronov-Hopf bifurcation, period doubling, Shilnikov bifurcation. Averaging of perturbations, elements of Kolmogorov-Arnold-Moser theory. Splitting of separatrices in perturbed Hamiltonian systems. Smale horseshoe, stochastic layer. Dynamics in slow-fast systems. Adiabatic invariants. Material to be prepared Lecture notes to be prepared Comments by proposer

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Functional Analysis and Spectral Theory Semester: University: Loughborough Contact: Alexey Bolsinov Hours: 10 Format: one semester course Subject classification: Preparation cost: £ 1500 Description Prerequisites: Real analysis, either Fourier analysis and PDE or Linear Differential Equations (preferably both). Suggested Texts: Introductory Functional Analysis with Applications, E. Kreyszig Functional Analysis, W. Rudin Rough Schedule: (chapters in parenthesis are from Kreyszig) Week 1: Motivating examples and metric spaces (chapter 1) Week 2: Banach Spaces (sections 2.1--2.5) Week 3: Operators on Banach Spaces (sections 2.6--2.10) Week 4: Hilbert Spaces (Sections 3.1--3.6) Week 5: Operators on Hilbert spaces (sections 3.7â€”3.10) Week 6: Spectral theory of linear operators in normed spaces (chapter 7) Week 7: Compact linear operators (chapter 8) Week 8: Bounded self-adjoint linear operators on Hilbert spaces, part 1 (sections 9.1--9.6) Week 9: Bounded self-adjoint linear operators on Hilbert spaces, part 2 (sections 9.7--9.11) Week 10: Unbounded self-adjoint linear operators on Hilbert spaces (chapter 10) Material to be prepared Lecture notes to be prepared Comments by proposer

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Lecturer: Dr. E.Hunsicker

Hyperbolic spaces Semester: University: Durham Contact: Michael Farber Hours: 20 Format: 10 weeks, 2 hours per week Subject classification: Preparation cost: £ Description In this course we classify the isometries of the hyperbolic spaces: real, complex and quaternionic hyperbolic space of any dimension and the octonionic hyperbolic plane. The main tool is linear algebra over the real and complex numbers, the quaternions, octonions and Clifford algebras. Material to be prepared Electronic lecture notes will be prepared and made available for the students. Comments by proposer

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Integrable models in Mathematical Physics Semester: University: Loughborough Contact: Alexey Bolsinov Hours: 20 Format: one 2 hour lecture per week for 10 weeks. Lecturer: Sasha Veselov Subject classification: Preparation cost: £ Description Hamiltonian systems, Poisson brackets, Liouville integrability. Kepler system and hydrogen atom. Integrable models of interacting particles on the line: Toda lattice and Calogero-Moser system. Lax pairs and inverse scattering method. Material to be prepared lecture notes to be prepared Comments by proposer

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Lie groups and Lie algebras Semester: University: Loughborough Contact: Alexey Bolsinov Lecturer: Alexey Bolsinov Hours: 20 Format: 2 lectures per week for 10 weeks Subject classification: Preparation cost: £ Description Lie groups, Lie algebras, classical matrix groups GL(n,R), SO(n), SO(p,q), U(n), Lorentz group, Poincare group; exponential map, one-parameter subgroups; actions and basic representation theory, orbits and invariants; Lie-Poisson bracket, dynamical systems with symmetries, applications to Relativity Theory and Hamiltonian Mechanics. Material to be prepared lecture notes to be prepared Comments by proposer

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Mathematical Methods Semester: University: Manchester Contact: Jitesh Gajjar Hours: 20 Format: 2-3 hrs/week min. Subject classification: Preparation cost: £ 5000 The money is needed to write the course and prepare the material by buying out staff time. Description This is a CORE applied module. The aim of the course is to pool together a number of advanced mathematical methods which students doing research (in applied mathematics) should know about. Students will be expected to do extensive reading from selected texts, as well as try out example problems to reinforce the material covered in lectures. A number of topics are suggested below and depending on time available, most will be covered. Course outline: Advanced differential equations (series solution, estimation of behaviour of solutions). Asymptotic methods (WKB, Steepest descent, Matched expansions). Transform methods. (Laplace, Fourier, Mellin, Hankel). Complex analysis, (Contour integrals, conformal mapping techniques, Cauchy-Hilbert integrals, Wiener-Hopf methods). Special functions (hypergeometric). Green’s function methods. Books: Bender & Orsag, ’Advanced mathematical methods for scientists and engineers’ Bleistan & Handlesman, ’Asymptotic expansions of integrals’ Hinch, ’Perturbation methods’ Material to be prepared Printed notes, example sheets and solutions, assessment material. Comments by proposer Prerequisites: Students should have a good knowlege of calculus and some complex variable theory. The lecturer for the course is to be

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announced later.

Nonlinear Waves Semester: University: Loughborough Contact: Alexey Bolsinov Hours: 20 Format: Subject classification: Preparation cost: £ 1500 Description The aim of this module is to introduce students to the major ideas and techniques in the nonlinear wave theory. Material will cover weakly nonlinear waves, Whitham averaging and strongly nonlinear waves, as well as a brief introduction to the questions of existence and stability of nonlinear waves. Weakly nonlinear waves; origin and basic properties of canonical models such as Korteweg-de Vries equation and nonlinear Schrodinger equation; solitary waves; modulational instability of periodic waves; resonant wave interactions: Whitham averaging procedure for nonlinear waves in variable media; existence of small-amplitude solitary waves using dynamical systems concepts; applications to fluids, solids, nonlinear optics and Bose-Einstein condensates. Material to be prepared There is no suitable single textbook for the proposed module. Much of the material is in research papers. We will prepare lecture notes and exercises, which we probably will expand and publish as a textbook later. Comments by proposer

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Lecturers: Prof. R.Grimshaw, Dr. K.Khusnutdinova, Dr. G.El

Operads and topological conformal field theories Semester: University: Leicester Contact: Jeremy Levesley Hours: 10 Format: Subject classification: Preparation cost: £ 2500 New course and costs needed for preparing the e materials. Description Outline: We explain the basics of the Feynman path integral formulation of quantum field theory and how it leads to asymptotic expansions of oscillating integrals and Feynman graphs. We then make a connection to pure mathematics via operads, modular operads and related notions. We will see how operads are applied in various problems in geometry and physics, particularly to the study of moduli spaces of Riemann surfaces. Time permitting, we consider such topics as deformation quantization, the Witten conjecture and/or Chern-Simons theory. Material to be prepared Lecture notes + example sheets + assessment material. Comments by proposer Course proposed by Leicester University (pure) to be given by Prof Andrey Lazarev. Could be either semester.

Reaction-Diffusion Theory Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: David Needham Hours: 40 Format: This is supposed to be a sequence of two courses, 20 lectures each. Subject classification: Preparation cost: £ Description Material to be prepared Comments by proposer

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Solitons in relativistic field theory Semester: University: York Contact: Gustav Delius Lecturer: Ed Corrigan Hours: 10 Format: 1 hour per week Subject classification: Preparation cost: £ 1500 Description Standard solitons in two dimensions (one space, one time)- such as those in the sine-Gordon model - will be reviewed along with a selection of their properties. This leads on to a description of the role of the solitons within quantum field theory, especially their scattering and bound states. There are many questions concerning the generalisation to affine Toda field theory and some of these will also be described. Material to be prepared Comments by proposer

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Viscous Fluid Mechanics Semester: University: Birmingham Contact: Sergey Shpectorov Hours: 30 Format: Subject classification: Preparation cost: £ Description Material to be prepared Comments by proposer

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Computational Complexity Semester: University: Manchester Contact: Nigel Ray Hours: 20 Format: 2 hours of lectures per week, plus supplementary reading from notes provided Subject classification: 03-xx Preparation cost: £ 0 Description An introduction to computational complexity. The aims of the course are: To introduce the main model of computation (Turing Machines) currently being employed in the theory of computation. To introduce the key parameters quantifying computational complexity (deterministic, nondeterministic, random, time, space) and some of the relationships between them. On successful completion of the course unit the student should have realized the following learning outcomes: Be familiar with the way Turing Machines work and their capabilities and limitations, and be able to construct examples to answer simple problems.Be aware of the main parameters quantifying the computational complexity classes and the relationships between them, and be able to classify and compare the hardness of problems in simple cases. Syllabus 1) Turing Machines:Alphabets, languages, Turing Machines, recursive sets and functions, decidability. Multitape and nondeterministic Turing Machines. Universal machines and undecidability. [7 lectures] 2) Complexity Classes: The time and space classification, speed up. Space and time hierarchy theorems. ${\cal P}$ and ${\cal NP}$, ${\cal NP}$-completeness. ${\cal P}={\cal NP}$? and the Traveling Salesman Problem. The theorems of Savitch and Immerman-Szelepscenyi. [10 lectures] 3) Probabilistic and Circuit Complexity. Basic ideas. [3 lectures] Prerequisites: Reasonable background in Pure Mathematics. Some basic acquaintance with Propositional Logic would be useful but not essential.\\ Material to be prepared Materials to be prepared: Complete set of self contained notes + examples sheets + take home tests as downloadable file.\\ Comments by proposer

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Lecturer Prof. J. Paris.

Applications of model theory to algebra and geometry Semester: University: Leeds Contact: William Crawley-Boevey Lecturer: Anand Pillay Hours: 10 Format: One hour lectures using electronically presentation of prepared notes. Subject classification: 03Cxx Preparation cost: £ 2500 Â£2,500 covers the replacement cost of a lecturer. Description The course will discuss and survey some classical and recent applications of model theoretic techniques to various other areas of mathematics. In addition to introducing basic notions of model theory, the course will also introduce in a soft manner notions from algebraic and diophantine geometry as well as valued fields. As such the course is aimed at the general postgraduate audience. But it would also be essential for students aiming to work in model theory and related subjects. The applications will go from elementary things (Ax’s theorem) to more sophisticated ones (definable integration, diophantine geometry over function fields). Lectures 1 to 3: Basics of model theory with examples. (First order structures and theories, compactness, nonstandard models, quantifier elimination.) Lecture 4: Affine algebraic varieties and polynomial maps. Ax’s theorem that an injective morphism from a complex variety to itself is surjective. Lectures 5 and 6: Logic and valued fields. Statement and explanation of the theorem of Ax-Kochen-Ershov on the first order theory of Henselian valued fields. Application to the asymptotic solution of a conjecture of Artin on p-adic solutions of equations. Lectures 7 and 8: Grothendieck groups of first order theories. Algebraically closed valued fields. Formulation and explanation of main theorem (Hrushovski-Kazhdan) of "definable (or motivic) integration" and applications to birational invariants of varieties. Lectures 9 and 10: The model theory of differential fields and the number and structure of points on varieties over function fields. Material to be prepared Lecture notes will be written from scratch in a suitable electronic form. Exercise sheets will also be written. Comments by proposer

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The Algebraic Theory of Quadratic Forms Semester: University: Nottingham Contact: Martin Edjvet Lecturer: Detlev Hoffmann Hours: 10 Format: one lecture/week, 10 weeks (could also be one double lecture/week, 5 weeks) Subject classification: 11E04 11E81 Preparation cost: £ Description Description: - Quadratic forms over general fields and their basic properties: Diagonalization, isometry, isotropy, hyperbolic forms - Witt’s theory: Witt cancellation, Witt decomposition - The Witt ring of a field and their structure for certain fields - Quaternion algebras and their norm forms - The Clifford algebra of a quadratic form - The classical invariants of quadratic forms: dimension, discriminant, Clifford invariant - The fundamental ideal and the filtration of the Witt ring - The Cassels-Pfister theorem - round and multiplicative forms, Pfister forms - The Arason-Pfister Hauptsatz - Merkurjev’s Theorem - A first glimpse of the Milnor conjecture (Voevodsky’s theorem) Material to be prepared Comments by proposer

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Advanced linear algebra Semester: University: Southampton Contact: Christopher Howls Hours: 20 Format: 2 lectures a week for 10 weeks Subject classification: 13C,16D Preparation cost: £ Description Modules over a ring, torsion, free modules, cyclic modules, structure of fintely generated modules over a PID and applications, maybe tensor products. Material to be prepared Comments by proposer

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Compact Riemann Surfaces Semester: University: York Contact: Ian McIntosh Lecturer: Ian McIntosh Hours: 10 Format: 1 lecture per week for 10 weeks Subject classification: 14H, 58 Preparation cost: £ 1500 Course will be prepared from scratch and require at least 2 full working weeks (i.e. at least 70 hours), including typing up and training in video link equipment use. Description Riemann surface as a complex manifold (motivated by multi-valued functions); vector fields and differential forms; basics of integration and singular homology for curves on surfaces; the Abel-Jacobi map and Abel’s theorem; the Riemann-Roch theorem; (maybe get as far as Weierstrass points). Material to be prepared lectures in pdf, exercises. Comments by proposer Pre-requisites: a first course in complex analysis is essential; a first course in differential geometry is highly desirable; a first course in

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topology would be useful.

Matrix Analysis Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: Roy Mathias Hours: 20 Format: One-hour lectures using electronic and white-board presentation of prepared notes and applications. Subject classification: 15Axx Preparation cost: £ 5000 5000 will cover the replacement cost for a lecturer, who will prepare material for the course as specified above. Description This is offered as a core course for Applied. The number of lectures for each section is in []. Matrix theory is an active research filed. It is also an important component in many areas of applied mathematics -- numerical analysis, optimisation, statistics, applied probability, image processing, ... [2] 1. Introduction Matrix products -- Standard product tensor/Kronecker product Schur product Decompostions -- Schur form Real Schur form Jordan form Singular Value decompositions Other preliminaries -- Schur complement -- additive and multiplicative compounds [3] 2. Norms * norms on vector spaces * inequalities relating norms * matrix norms * unitarily invariant norms * numerical radius * perturbation theory for linear systems [4] 3. Gerschgorin’s Thorem, Non-negative matrices and Perron-Frobenius * digaonal dominance and Gerschgorin’s Theorem * spectrum of stochastic and doubly stochastic matrices * Sinkhorn balancing * Perron-Frobenius Theorem * Matrices realted to non-negative matrices -- M-matrix, P-matris, totally positive matrices. [2] 4. Spectral Theory for Hermitian matrices * Orthogonal diagonalisation * Interlacing and Monotonicity of Eigenvalues * Weyl’s and the Lidskii-Weilandt inequalities [2] 5. Singular values and best approximation problems * Connection with Hermitian eigenvalue problem * Lidskii-Weilandt -- additive and multiplicative versions * best rank-k approximations * polar factorisation, closest unitary matrix, closest rectangular matrix with orthonormal columns [3] 6. Positive definite matrices * Characterisations * Schur Product theorem * Determinantal inequalitties * semidefinite completions * The Loewner theory [2] 7. Perturbation Theory for Eigenvalues and Eigenvectors * primarily the non-Hermitian case [2] 8. Functions of matrices * equivalance of definitions of f(A) * approximation of/algorithms for general functions * special methods for particular functions (squareroot, exponential, logarithm, trig. functions) Material to be prepared Hand-written notes need to be typed and hand-drawn diagrams need to be converted to electronic format. Example sheets need to be prepared. Comments by proposer

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Introduction to complex semisimple Lie algebras Semester: University: Southampton Contact: Lecturer: Gerhard Roehrle Hours: 20 Format: 2 lectures a week for 10 weeks Subject classification: 17B (Lie a Preparation cost: £ Description This is an introductory course on the theory of semisimple complex Lie algebras. A good knowledge of linear algebra (including eigenvalues of transformations, bilinear forms, Euclidean spaces, and tensor products of vector spaces) and some abstract algebra (rings and modules) is presupposed. Apart from being important in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining elegant methods and achieving completeness in its basic results. Some of the most active fields in contemporary mathematics are concerned with generalizations of this classical sub ject, such as Kac-Moody Lie Algebras, infinite dimensional Lie Groups, and Quantum groups. The central goal of this course is the classification of simple complex Lie algebras. Along the way we present Weylâ€™s theorem on complete reducibility of finite dimensional representations, as well as the Poincare-Birkhoff-Witt Theorem. Also the basic concepts of root systems, reflection groups, and universal enveloping algebras are introduced. Material to be prepared Comments by proposer

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Category Theory Semester: University: Manchester Contact: Nigel Ray Hours: 20 Format: Subject classification: 18-xx Preparation cost: £ 0 Description Syllabus Categories: basic definitions and examples from algebra, logic, set theory, and topology, plus pointed cases,... Functors: many examples in the above contexts. Natural transformations: further examples as above. Adjunctions: theory, plus a detailed discussion of examples such as function set and product in sets, loop and suspension in pointed spaces,... The notes accompanying the course will contain additional examples of a non-superficial kind to illustrate these notions. The lectures will deal with those examples which best suit the interests of the audience. Material to be prepared Detailed printed notes containing many exercises (plus solutions, eventually). Comments by proposer Any comments by proposer: Category theory is used in many parts of pure mathematics and, I believe, is creeping into applied mathematics. To be given by Dr Harold Simmons. It is not an undergraduate subject, but it is something that most postgraduates should have some familiarity with. The course will not teach category theory \lq for its own sake\rq, but as a tool for use in other parts of mathematics.

Sheaf cohomology Semester: University: Southampton Contact: Christopher Howls Lecturer: Bernhard Koeck Hours: 10 Format: 1 or 2 lectures per week for 10 weeks Subject classification: 18F20; 55N Preparation cost: £ Description Elementary sheaf theory; â€“ Cohomology of abelian sheaves; â€“ injective, flabby, acyclic, soft and fine sheaves; â€“ brief survey on sheaf cohomology for topological and differentiable manifolds, for Rie- mann surfaces and algebraic varieties. Material to be prepared Comments by proposer

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An introduction to cyclic homology Semester: University: Southampton Contact: Christopher Howls Lecturer: Jacek Brodzki Hours: 20 Format: 2 lectures a week for 10 weeks Subject classification: 19D Higher algebraic K-theory Preparation cost: £ Description Cyclic type homology theories play a fundamental role in noncommutative geometry by providing an analogue of the de Rham cohomology known from differential geometry. This course will provide an introduction to the formalism of the theory, using the approach of Cuntz and Quillen. Then we will provide a discussion of bivariant theories, including the analytic cyclic homology of Meyer and the local cyclic homology of Puschnigg. The course will end with an outline of the new type of theory: the Hopf-Cyclic cohomology. Material to be prepared Comments by proposer

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Representation Theory Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: Robert Curtis Hours: 20 Format: 20 one-hour lectures Subject classification: 20Cxx Preparation cost: £ Description A review of elementary group theory and linear algebra; an introduction to group representations; FG-modules; reducibility; group algebras; FG-homomorphisms; Maschke’s theorem; Schur’s lemma; irreducible modules and the group algebra; group characters, their inner products and the number of irreducible characters; character tables and the orthogonality relations; lifting characters from a quotient group; tensor products, including the symmetric and antisymmetric parts of $\chi^2$; restriction to a subgroup; induced modules and characters; Frobenius reciprocity; permutation representations and their characters. Material to be prepared Comments by proposer

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Finite Simple Groups and their Geometries Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: Sergey Shpectorov Hours: 20 Format: 20 one-hour lectures Subject classification: 20Dxx Preparation cost: £ Description After a brief refresher on groups of permutations, we will discuss classical groups and related geometries, most prominently, the (projective) special linear group with its action on the projective space and the symplectic group with its action on the symplectic polar space. In the last third of the course we discuss several sporadic simple groups and their geometries, namely, the Mathieu groups and the Witt designs. Time permitting, we will also discuss the Leech lattice and the Conway groups. Material to be prepared Existing notes need to be adapted for mostly electronic presentation Comments by proposer

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Combinatorial Group Theory Semester: University: Nottingham Contact: Martin Edjvet Lecturer: Martin Edjvet Hours: 10 Format: one lecture a week for 10 weeks Subject classification: 20F05 Generators, relations, and presentations Preparation cost: £ Description Free groups; Nielsen-Schreier Theorem; Presentations; Coset enumeration; Presentations of subgroups; Presentations of group extensions;Free products, free products with amalgamation and HNN extensions; Small cancellation theory and van Kampen diagrams. Material to be prepared Comments by proposer

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Finite Reflection Groups Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: Christopher Parker Hours: 20 Format: 20 one-hour lectures Subject classification: 20F55 Reflection and Coxeter groups Preparation cost: £ Description Material to be prepared Comments by proposer

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An introduction to linear algebraic groups Semester: University: York Contact: Steve Donkin Lecturer: Steve Donkin Hours: 10 Format: 1 lecture per week for 10 weeks (by Steve Donkin) Subject classification: 20G15 Linear algebraic groups over arbitrary fields Preparation cost: £ 1500 Description In introduction to algebraic groups, going as far Borel’s Fixed Point Theorem. Affine algebraic varieties, Algebraic groups, Connectedness, Dimension, Varieties in general, Completeness of projective varieties, Borel’s fixed point theorem, Applications: the Lie Kolchin Theorem, conjugacy of Borel subgroups. Material to be prepared lecture notes in suitable form, exercises. Comments by proposer Pre-requisites: a first course in group theory; a course in (commutative) ring theory, some familiarity with point-set topology.

Cohomology of groups Semester: University: Southampton Contact: Christopher Howls Lecturer: Brita Nucinkis Hours: 20 Format: 2 lectures per week for 10 weeks Subject classification: 20Jxx Preparation cost: £ Description Introduction to basic concepts ((co)homology of chain complexes, modules over a ring, exactness, pro jectives and injectives, Ext and Tor..) â€“ Cohomology of groups (pro jective resolutions via topology, Induction, Coinduction, cohomological finiteness conditions, group extensions) â€“ if time permits one or more of the following topics: H2 and group extensions with abelian kernel, cohomological finiteness conditions for soluble groups, virtually torsion free groups, Bredon cohomology... Material to be prepared Comments by proposer

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Hyperbolic manifolds and Kleinian groups Semester: University: Southampton Contact: Lecturer: James Anderson Hours: 20 Format: Subject classification: 30F40 (Kle Preparation cost: £ Description Particularly in light of Perelmanâ€™s proof of Thurstonâ€™s Geometrization Conjecture for 3-manifolds, hyperbolic 3-manifolds are now known to be important to the study of 3-manifolds in general. This course would cover basic ideas about hyperbolic manifolds and Kleinian groups, as well as giving an overview of current state of research in the field. Topics covered would include: Introduction to the basic defintions of Kleinian groups, primarily in dimensions 2 and 3, and hyperbolic manifolds; finiteness theorems for Kleinian groups; deformation theory of Kleinian groups; constructions, including hyperbolic Dehn surgery and Klein-Maskit combination theorems; Kleinian groups in higher dimensions. Material to be prepared Comments by proposer

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Not this October

Erlangen program for geometry and analysis: SL(2,R) case study Semester: University: Leeds Contact: William Crawley-Boevey Lecturer: Vladimir Kisil Hours: 20 Format: Two-hour lectures with 10 minutes break in between using mainly electronic presentation of prepared notes, with some animated demonstrations and free computer algebra system (GiNaC) demo. Subject classification: 30G35, 22E46 Preparation cost: £ 5000 5000 will cover the replacement cost for a lecturer, who will prepare material for the course as specified above. Description The Erlangen program of F.Klein (influenced by S.Lie) defines geometry as a study of invariants under a certain group action. This approach proved to be fruitful much beyond the traditional geometry. For example, special relativity is the study of invariants of Minkowski space-time under the Lorentz group action. Another example is complex analysis as study of objects invariant under the conformal maps. In this course we consider in details SL(2,R) group and corresponding geometrical and analytical invariants with their interrelations. Correspondingly the course has a multi-subject nature touching algebra, geometry and analysis. No special knowledge beyond a standard undergraduate curriculum is required. The considered topics are: * SL(2,R) group and Moebius transformations of the real line. * Complex, dual and double numbers and Clifford algebras with two generators. * Iwasawa decomposition of SL(2,R). * Moebius transformations in the upper half-plane. * Cycles (quadrics) as geometric SL(2,R)-invariants. * Filmore--Springer--Cnops construction and algebraic invariants of cycles. * Linearisation of the Moebius transformations. * Linearised Moebius action in the in the space of function: Hardy and Bergman spaces. * Cauchy integral formula as a wavelet transform. * Cauchy--Riemann and Laplace equation from the invariant vector fields. * Laurent and Taylor expansion over the eigenvectors of rotations. * Functional calculus as an intertwining operator. * Prolongation of representations and functional calculus of non-selfadjoint operators. This course will be a substantial revision of the previous one: http://www.maths.leeds.ac.uk/~kisilv/courses/wavelets.html Material to be prepared The existent material should be converted into a coherent course addressed to PostGrad students. Example sheets and more illustrations need to be prepared. Comments by proposer Previous versions of the course were read at 6th European intensive course on Complex Analysis (Coimbra, Portugal; 2000) and in Leeds (2001). The course is going to be completely revised in the light of new recent results.

Geometric Structures on surfaces and Teichmuller Space Semester: University: Liverpool Contact: Mary Rees Lecturer: Mary Rees Hours: 10 Format: One lecture a week for 10 weeks. Subject classification: 32G15 Moduli of Riemann surfaces, Teichmüller theory Preparation cost: £ 2500 To be determined by the Steering Committee Description Classification of compact Riemann surfaces. Complex and geometric structures on compact surfaces. Teichmuller space. Coordinates on Teichmuller space. Material to be prepared Material on Teichmuller Space. This would be a compilation of an existing course, with some additional material. Teichmuller space has not been covered in the existing course in recent years. Comments by proposer

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Numerical Analysis & Methods Semester: University: Durham Contact: James Blowey Lecturer: James Blowey Hours: 20 Format: One-hour lectures using mainly electronic presentation of prepared notes, with some animated demonstrations. Matlab demonstrations will be used throughout the course. Subject classification: 34A45 Theoretical approximation of solutions Preparation cost: £ Description Elements of approximation theory, orthogonal polynomials, splines, wavelets. Eigenvalue problems (theory and computational aspects). Modern numerical methods for ODE’s (initial value problems, stiff problems). Solution of linear systems (modern iterative methods), computations with sparse matrices. Material to be prepared Much of the material will have to be prepared from scratch. Comments by proposer This syllabus is rather ambitious and as this is a proposed core module a certain amount of liaison will need to take place with other nodes

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who have students who want to take this course.

Linear Differential Operators in Mathematical Physics Semester: University: Liverpool Contact: Alexander Movchan Hours: 10 Format: One lecture a week for 10 weeks Subject classification: 34B Boundary value problems Preparation cost: £ 2500 To be determined by the Steering Committee Description For the existing module see MATH421 SYLLABUS: Generalised derivatives. Definition and simple properties of generalised derivatives. Limits and generalised derivatives. Sobolev spaces. Definition of Sobolev spaces. Imbedding theorems. Equivalent norms. Laplaceâ€™s equation. Laplaceâ€™s equation and harmonic functions. Dirichlet and Neumann boundary value problems. Elements of the potential theory. Generalised solutions of differential equations. Singular solutions of Laplaceâ€™s equation, wave equation and heat conduction equation. Variational method. Weak Solutions. The energy space. Greenâ€™s formula. Weak solutions of the Dirichlet and Neumann boundary value problems. Spectral analysis for the Dirichlet and Neumann problems for finite domains. Heat conduction equation. Maximum principle. Uniqueness theorem. Weak solutions. Wave equation. Weak solutions. Wave propagation and the characteristic cone. Cauchy problems for the wave equation and the heat conduction equation. Material to be prepared This would be a compilation from an existing 36-lecture module Comments by proposer

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Asymptotic methods for PDE Semester: University: Liverpool Contact: Alexander Movchan Hours: 10 Format: 1 lecture a week for 10 weeks Subject classification: 35 Partial differential equations Preparation cost: £ 2500 To be determined at Steering Committee meeting. Description For the existing module see the Liverpool Options Booklet, MATH520 OUTLINE SYLLABUS: â€¢ Asymptotic expansions. Definition and examples. â€¢ Singular and regular perturbations. Definitions and one-dimensional examples of singularly and regularly perturbed boundary value problems. â€¢ Asymptotic behaviour of solutions of boundary value problems in non-smooth domains. Dirichlet and Neumann boundary value problems in domains with conical points on the boundary. Evaluation of coefficients in the asymptotic expansions. â€¢ Asymptotic approximations for solutions of Dirichlet and Neumann problems in domains with small holes. Examples in electro-statics and elasticity (anti-plane shear). â€¢ Asymptotics in thin domains. Dirichlet and mixed boundary value problems for the Laplacian in a thin rectangle. Boundary layer. The limit one-dimensional model. â€¢ Asymptotic analysis of fields in multi-structures. Mixed boundary value problems for the Laplacian in domains with junctions. â€¢ Asymptotic theory of wave propagation. Long-wave approximations. Models of dynamic cracks. Dynamic problems for multi-structures (domains with junctions). â€¢ Asymptotics for heat conduction problems. Thermal crack in a half-plane. Examples. Material to be prepared This would be a compilation from an existing 36 lecture module Comments by proposer

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Ergodic Theory Semester: University: Manchester Contact: Nigel Ray Hours: 10 Format: 10 1hr lectures given by slides prepared in LaTeX, supported by more in-depth printed notes which would also contain the exercises. Subject classification: 37Axx, 37Bxx, 37Cxx, 37Dxx, 28Dxx Preparation cost: £ 0 Description The course will provide an overview of ergodic theory. It will discuss applications to explicit dynamical systems (and consequences of these results to other areas of mathematics such as number theory and hyperbolic geometry) as well as covering abstract theory. Towards the end of the course, techniques that are commonly used in current research will be discussed. Syllabus: [L1] Examples of dynamical systems (maps on a circle, the doubling map, shifts of finite type, toral automorphisms, the geodesic flow) [L2] Uniform distribution, inc. applications to number theory [L3] Invariant measures and measure-preserving transformations. Ergodicity. [L4] Recurrence and ergodic theorems (Poincare recurrence, Kac’s lemma, von Neumann’s ergodic theorem, Birkhoff’s ergodic theorem) [L5] Applications of the ergodic theorem (normality of numbers, the Hopf argument, etc) [L6] Mixing. Spectral properties. [L7] Entropy and the isomorphism problem. [L8] Topological pressure and the variational principle. [L9] Thermodynamic formalism and transfer operators. [L10] Applications of thermodynamic formalism: (i) Bowen’s formula for Hausdorff dimension, (ii) central limit theorems. Material to be prepared Materials to be prepared: Latexed notes, example sheets, material for assessment, computer animations and interactive java applets. Comments by proposer Prerequisites: measure theory, some knowledge of functional analysis. Target audience: the course would be suitable for any beginning graduate student studying pure mathematics or non-linear systems; it may also appeal to beginning graduate students in probability theory. It would be particularly useful for students working in the dynamical systems group at Manchester, and also for students working with the dynamicists at Liverpool and with the number-theorists at York. To be given by Dr Charles Walkden and Prof. Richard Sharp.

Introduction to Linear Analysis Semester: University: Lancaster Contact: Martin Lindsay Hours: 20 Format: 2 hours of lecturing per week for 10 weeks Subject classification: 46 B, C, L; 47 A, L Preparation cost: £ 5000 Description This couse provides an introduction to analysis in infinite dimensions with a minimum of prerequisites. The core of the course concerns operators on a Hilbert space including the continuous functional calculus for bounded selfadjoint operators. There will be an emphasis on positivity and on matrices of operators. The course includes some basic introductory material on Banach spaces and Banach algebras. It also includes some elementary (infinite dimensional) linear algebra that is usually excluded from undergraduate curricula. Prerequisites are: standard undergraduate linear algebra and real and complex analysis, and basic metric space/norm topology. Here is a very brief list of the many further topics that this course looks forward to. Banach space theory and Banach algebras; C*-algebras, von Neumann algebras and operator spaces (which may be viewed respectively as noncommutative topology, noncommutative measure theory and ‘quantised’ functional analysis); Hilbert C*-modules; noncommutative probability (e.g. free probability), the theory of quantum computing, dilation theory; Unbounded Hilbert space operators, one-parameter semigroups and Schrodinger operators. And that is without starting to mention Applied Maths and Statistics applications .... Relevant books G. K. Pederson, Analysis Now (Springer, 1988) [This course may be viewed as a preparation for studying this text (which is already a classic).] Simmonds, Introduction to Topology and Modern Analysis (McGraw-Hill, 1963) [Covers far more than the course, but is still distinguished by its great accessibility.] P.R. Halmos, Hilbert Space Problem Book (Springer, 1982) [Collected and developed by a master expositor.] There are many many other books which cover the core part of this course. Rough Course Contents I Preliminaries (5 lectures) A. Linear algebra, including quotient space and free vector space constructions, diagonalisation of hermitian matrices, algebras, homomorphisms and ideals, group of units and spectrum. B. Banach spaces, including dual spaces, bounded operators, bidual [and weak*-topology], completion and continuous (linear) extension. C. Banach algebras, including Neumann series, continuity of inversion, spectrum, C*-algebra definition. D. Hilbert space geometry, including Bessel’s inequality, dimension, orthogonal complementation, nearest point projection for nonempty closed convex sets. E. Miscellaneous, including Weierstrass Approximation Theorem. II Hilbert space and its operators (9 lectures) A. Sesquilinearity, orthogonal projection; B. Riesz-Frechet, adjoint operators, C*-property; C. Kernel-adjoint-range relation; D. Finite rank operators; E. Operator types: normal, unitary, selfadjoint, isometric, compact, invertible, nonnegative, uniformly positive & partially isometric; F. Fourier transform as unitary operator; G. Hardy space; H. Invertibility criteria; I. Key examples of operators, finding their spectra (shifts and multiplication operators), norm and spectrum for a selfadjoint; J. continuous functional calculus for selfadjoint operators, with key examples: square-root and positive/negative parts. III Further topics (6 lectures) A. Polar decomposition; B. Matrices of operators, positivity in B(h+k), operator space - definition and simple examples; C. Nonnegative definite kernels, Kolmogorov decomposition; D. Tensor products; E. Hilbert-Schmidt operators; F. Topologies on spaces of operators (WOT, SOT, uw); G. Compact and trace class operators, duality; H. Double Commutant Theorem; I. Dilation and von Neumann’s inequality; J. Two projections in general position. Coverage of topics beyond A-D in III will depend on availability of time. Material to be prepared Lecture notes, exercises

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Comments by proposer

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Lecturers: N. Laustsen, J.M. Lindsay

Introduction to Operator Algebras Semester: University: Southampton Contact: Lecturer: Jacek Brodzki Hours: 20 Format: 2 lectures per week for 10 weeks Subject classification: 46L, in pa Preparation cost: £ Description The theory of Operator Algebras provides tools that are of importance in analysis, geometric group theory, noncommutative geometry, mathematical physics. This course will start with the properties of operators on Hilbert spaces to motivate the need for a general framework which would capture abstractly their most useful properties. We shall then discuss the basics of C*-algebras, including spectra, Gelfand-Naimark theorem, functional and holomorphic calculus. From this we shall provide an introduction to noncommutative topology, introducing important topological invariants of C*-algebras (K-theory and K-homology). A similar introduction to von Neumann algebras will be given stressing their relation with measure theory. We shall provide examples which are important in the current research. Material to be prepared Comments by proposer

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Symplectic geometry Semester: University: Durham Contact: Wilhelm Klingenberg Hours: 20 Format: Two hours per week for 10 weeks Subject classification: 53D Symplectic geometry, contact geometry Preparation cost: £ Description Symplectic Linear algebra, Normal forms for symplectic forms: Darboux Theorem, Lagrangian fibrations, Symplectic group actions and moment maps, Hamiltonian actions Material to be prepared Comments by proposer

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Manifolds and homology Semester: University: Sheffield Contact: Neil Strickland Lecturer: Neil Strickland Hours: 20 Format: One hour per week for ten weeks. Subject classification: 55N Homology and cohomology theories Preparation cost: £ 1000 A nice round number, placed here just to check that the form works. I will do something more careful later. Description The course will cover the cohomology of topological spaces, with a heavy emphasis on interesting examples, most of which are manifolds. Material to be prepared We will adapt the notes that are already available at http://www.shef.ac.uk/courses/manifolds. We will fill in some gaps, write problem sheets and solutions, and convert some of the notes to lecture slides. Comments by proposer

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This course is the best thing since sliced bread.

Differential topology and Morse theory Semester: University: Durham Contact: Michael Farber Hours: 20 Format: 2 hours per week for 10 weeks Subject classification: 57Rxx Preparation cost: £ 5 The cost will mainly cover substitution teaching for the time spent on preparation of the lectures and the lecture notes and actual lecturing. Description The course will describe basic material about smooth manifolds (vector fields, flows, tangent bundle, foliations etc), introduction to Morse theory, various applications. Material to be prepared Electronic lecture notes will be prepared and made available for the students. Comments by proposer This course will be of interest for students working in topology, geometry, mathematical physics and some other fields. The course may be

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redelivered in the future.

Curves and Singularities Semester: University: Liverpool Contact: Mary Rees Lecturer: Peter Giblin Hours: 10 Format: 1 hour per week for 10 weeks Subject classification: 58K Theory of singularities and catastrophe theory Preparation cost: £ 2500 To be determined by Steering Committee. Description For the existing course, see MATH443 OUTLINE SYLLABUS: Curves, and functions on them. Classification of functions of 1 real variable up to R-equivalence. Regular values of smooth maps, manifolds. Applications. Envelopes of curves and surfaces. Unfoldings of functions of 1 variable. Criteria for versal unfolding. Material to be prepared This would be an adaptation of an existing 36 lecture course. Introductory material would be used. Comments by proposer

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Discrete Optimization in Operational Research Semester: University: Manchester Contact: Jitesh Gajjar Hours: 20 Format: Ten 2 hour lectures, in either the Easter break or June. Subject classification: 62-xx Preparation cost: £ 5000 Material to be prepared from scratch. Costs to buy out replacement for lecturer’s preparation time. Description This course will present an overview of optimization models and algorithms at the core of a broad range of operational research techniques via a problem-oriented set of ten 2 hour lectures. Each lecture session will include illustrations of the theory through worked examples. The course will emphasize the importance of networks and graphs in formalizing combinatorial optimization problems and will highlight the use of relaxation techniques and tolerances in developing algorithms for NP-hard optimization problems. Text Network Flows: Theory, Algorithms, and Applications by Ravindra K. Ahuja,Thomas L. Magnanti,James B. Orlin. Prentice-Hall, 1993. Lectures 1. Introduction. Modelling and problem formulation. 2. Review of linear programming. Duality theory. 3. Integer and mixed integer programming. Cutting planes. 4. Convex optimization. Basic results. Kuhn-Tucker theory. 5. Dynamic programming. Applications e.g. to scheduling. 6. Optimization on networks and graphs. Location problems. 7. Relaxation. Branch and bound. 8. Combinatorial optimization. Travelling salesman and vehicle routing problems. 9. Tolerances. Research issues. 10. Algorithm development. Material to be prepared Lecture notes + slides + example computer problems. Comments by proposer

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To be given by Dr Tso.

Multivariate Analysis Semester: University: Manchester Contact: Jitesh Gajjar Hours: 20 Format: Ten 2 hour lectures during the Easter break. Subject classification: 62-xx Preparation cost: £ 5000 Material to be prepared from scratch. Costs to buy out replacement for lecturer’s preparation time. Description This course aims to present the core mathematical foundations of multivariate statististical techniques in a problem-oriented set of ten 2 hour lectures. Each lecture session includes an illustration of the theory and techniques through worked examples. The course will highlight reduced-rank regression as an important unifying model and will emphasize the key role of matrix approximation results such as the Eckart-Young theorem in justifying optimality properties of the various techniques. Text W.J. Krzanowski. Principles of Multivariate Analysis. Oxford. Lectures 1. Introduction. Multivariate probability density functions, moments and characteristic function. 2. Data matrix. DeÂ…nition and basic properties. Sample moments. Maha- lanobis distance. 3. Review of matrix results. Eigenanalysis and spectral decomposition . SVD and Eckart-Young Theorem. 4. Principal components analysis. Optimality properties. The biplot. 5. Other dimensionality reduction techniques. e.g. Canonical correlations analysis, factor analysis. 6. Reduced-rank multivariate regression. Least squares analysis. 7. Multivariate normal (MVN) distribution.and Wishart distribution. 8. HotellingÂ’s T2 distribution. Tests of hypothesis and conÂ…dence intervals based on MVN distribution. 9. Discriminant analysis and clustering. Dendrograms. Procrustes rotation. 10. Distance and similarity matrices. Seriation and multidimensional scaling. Material to be prepared Lecture notes + slides + numerical examples. Comments by proposer

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To be given by Dr Tso.

Intodusction to Numerical Analysis Semester: University: Leicester Contact: Jeremy Levesley Hours: 10 Format: Lectures which are of a motivatory type to explore the issues, not provide details. Exercises and matlab computer practicals will enage students with the details. Subject classification: 65-01 Instructional exposition (textbooks, tutorial papers, etc.) Preparation cost: £ 2000 This is the replacement cost for approximately 20 undergraduate lectures, which includes the cost of lectures plus the preparation of good computer practicals. Description Lecture 1: Introduction to numerical analysis: scope, relevance and philosophy, divided differences as approximation of derivatives. Lecture 2: Quadrature and extrapolation. Cubature. Lecture 3: Numerical Solution of ODEs (one-step methods, multistep methods, stability, consistency, Dalquist’s theorem, convergence). Discussion on popular methods in practice. Lecture 4: Numerical solution of systems of ODEs. Stiff ODEs. Applications to molecular dynamics (?). Lecture 5: Introduction to Numerical Solution of PDEs. Finite difference methods (FDM) for elliptic problems. Finite difference methods for parabolic problems. Von Neumann analysis/bounds on the Courant number. Discrete maximum principles. Lecture 6: Finite diffrence methods for hyperbolic PDEs. CFL condition. Brief introduction to Finite Volume Methods for PDEs. Numerical Fluxes for non-linear Conservation Laws. Lecture 7: Finite Element Methods (FEM) for PDEs. Basic principles. Discussion on FEM vs FDM. The issues of complex geometries and of stability. Lecture 8: Introduction ot Numerical Linear Algebra. Motivation. Direct Methods. Iterative Methods. CG and GMRES. Brief introduction to Preconditioning. Lecture 9: Introduction to Numerical Methods for Stochastic Differential Equations (SDEs). Monte carlo method. Lecture 10: Multiresolution and Wavelets. Application to image compression. Material to be prepared Lecture notes and computer practicals. Comments by proposer

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Object-oriented scientific programming in C++ Semester: University: Manchester Contact: Jitesh Gajjar Hours: 10 Format: The course will have three main parts: 1) Self-study course (with support via message board and a small number of access grid sessions) of procedural C++ programming during semester one. 2) One-week residential intensive course on object-oriented programming before the start of semester two. 3) Guided reading on an object-oriented implementation of the finite element method with weekly access-grid-based mini-lectures and discussions during semester two. Subject classification: 65K04, 65N04, 65N04 Preparation cost: £ 2500 Description Overview: ------The aim of the course is to introduce students to the fundamental concepts of object-oriented programming and their practical application in large-scale scientific computing. The course will consist of three main parts. During the first semester students will participate in an introductory self-study course on procedural programming in C++. The course materials will be provided online and support will be provided via a message board and two or three interactive question and answer sessions on the access grid. The second part of the course --- an intensive, one-week, residential course in Manchester --- will introduce the key concepts of object-orientation, motivated by suitable examples from applied mathematics/scientific computing. In the final part of the course, students will study the finite-element-based discretisation of linear and nonlinear PDEs and how the methods may be implemented within an object-oriented framework. Students will be introduced to oomph-lib, the object-oriented multi-physics finite-element library, available as open-source software at http://www.oomph-lib.org. During this part of the course students will study selected tutorials from oomph-lib’s extensive online documentation and work on the associated exercises. The weekly access grid session will then be used to provide a brief review of the material, followed by an interactive discussion of the topics and/or any problems the students encountered in the exercises. While students will be encouraged to download and install oomph-lib on their own computers, they will retain Manchester computer accountsthroughout the course to facilitate help with compilation/installation problems. Part 1: Self-study course: Procedural programming in C++ ------Introduction to programming. The basic work cycle. - A "Hello world" program. Syntax and scope: Braces, semicolons and comments. Standard headers for input/output. cout and cin. - C/C++ data types.Operators. Mathematical functions. - Elementary control structures: for and while loops, if and case statements. - Functions. Pass by value and by (const) reference. - C-style arrays: allocation/deallocation with new/delete and the concept of pointers. - Input/output streams for files. - Structures as "collections of related data". - How to debug code.Part 2: One week residential intensive course in Manchester. ------Object-oriented programming in C++ ------About 2-3 hours of lectures per day, interlaced with intensive computing practicals, the latter organised as pair-programming exercises. Need a linux cluster with blackboard/whiteboard, OHP and data projector. - Syllabus: Day 1: Quick introduction to the departmental linux system. Template (meta-)programming and the STL. Fundamental STL containers: vectors, maps, lists, sets, complex etc. Standard algorithms, sorting and searching, relative costs for the different containers. Day 2: Namespaces and C++ classes to encapsulate data. Simple examples of classes. STL containers of classes. Motivation from large projects and linear algebra objects. A couple of matrix classes and conversion routines. Day 3: Inheritance, run-time polymorphism and virtual functions. Use the example of matrices in different storage formats. Generic algorithms: direct and iterative solvers using only interfaces defined in a base class. Navigating class hierarchies, static and dynamic casting. Day 4: Big projects, code organisation into *.h and *.cc files. The use of libraries and Makefiles. How templates really work and template header files. Make a library of the matrix objects developed on days 2 and 3. Add an additional storage format in a driver code and link against the library. Day 5: Generalisation to templated classes and classes containing other objects. (Matrices as containers templated by the type of the entries). Introduction to oomph-lib and its documentation and linking against the library. Explore the use of oomph-lib by using the library’s own matrix classes and solvers. Overview of the final part of the course and the arrangements for access grid sessions and message board. Part 3: The finite element method: an object-oriented approach ------Week 1: Discussion of the overall course structure. Introduction to scientific computing. Real world problem --> mathematical model --> discretisation (if necessary) --> solution = determination of a number of discrete "unknowns" from a large system of linear or nonlinear algebraic equations. Conceptual overall framework: Newton’s method. Week 2: Introduction to the discretisation of elliptic ODEs/PDEs. The weak form; Galerkin’s method. Show that it leads to a system of (non)linear algebraic equations that can be solved by Newton’s method. Integration by parts. Problems: How to construct basis/test functions that vanish on Dirichlet boundaries with complicated shapes; Jacobian matrix tends to be dense. Weeks 3&4: Finite elements to overcome the problems outlined above: Sparsity and geometric flexibility are built-in. Global finite element basis functions. Breaking things up into an element-by-element procedure: Distinguishing between nodes and equation numbers. Element-by-element Week 5: Obvious extension to higher dimensions and object orientation: The Node, Element, Mesh and Problem objects and their (natural!) role in the overall procedure. Week 6: Implementation in oomph-lib. Discuss/demonstrate solution of 1D/2D/3D Poisson problems with existing oomph-lib objects. Week 7: More detailed discussion of the overall data structure. The Not-So-Quick Guide. Week 8: Spatial adaptivity (demonstration and brief overview of ideas) Week 9: Time-stepping (BDF schemes): Theory and implementation in oomph-lib, demonstrated for unsteady heat equation Week 10: Vector valued problems (Navier-Stokes). Week 11: Overview of other capabilities. How to write new elements. Material to be prepared - Material for self-study course in semester one. - Setup of message board. - Material for intensive course. - Minor tinkering with oomph-lib’s existing online tutorials. Comments by proposer The one-week intensive course in the week before semester two will be the most important element of the course. Students must get "hands-on" experience of object-oriented programming and an intensive residential course is the best (the only!) way to achieve this. However, to fully benefit from the intensive course, students must already be familiar with the procedural aspects of C++. The self-study course offered during semester one aims to give students the opportunity to acquire such knowledge. For those who are already familiar with other procedural programming languages (FORTRAN or C, say) the self-study course will simply act as a conversion course. For students who are completely new to programming, the self-study course provides an introduction to classical procedural programming. A sufficient number of exercises will be provided to allow the students to (self-)assess whether they are likely to benefit from the second and third parts of

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the course. To be given by Dr A Hazel and Prof. M. Heil.

Numerical Methods for Differential Equations Semester: University: Birmingham Contact: Sergey Shpectorov Lecturer: Daniel Loghin Hours: 20 Format: One-hour lectures using mainly electronic presentation of prepared notes, with some animated demonstrations. Matlab demonstrations will be used throughout the course. Subject classification: 65Lxx, 65Mxx, 65Nxx Preparation cost: £ 1500 The amount will cover expenses incurred by (i) preparation (in electronic form) of problem and solutions sheets adapted to the scope stated below (ii) the development of a small matlab numerical toolbox which is to be used in the computer sessions included in the syllabus. It is expected that one or two graduate students will be involved in developing and adapting existing material which can be found on the current course webpage http://web.mat.bham.ac.uk/loghin/teaching/msm4a10a/ Description Part I: Numerical Solution of Ordinary Differential Equations. 1. Revision of standard results -- Initial value problems: existence and uniqueness -- First-order ODE. Higher-order ODE. First order systems -- Stability -- Examples: Hamiltonian mechanics 2. One-step methods for initial value-problems: -- Euler’s method and variants -- Error analysis of explicit and implicit one-step methods -- Runge-Kutta methods: explicit and implicit methods. -- Stability Part II: Numerical Solution of Partial Differential Equations 3. Elements of function spaces -- Lp-spaces -- Weak derivatives, Sobolev spaces -- Applications: weak solutions of PDE 4. Variational formulation of PDE -- Positive-definite problems. Lax-Milgram Lemma. -- Symmetric problems. Galerkin orthogonality. Optimality results. -- Examples: elliptic problems of second and fourth order. -- Indefinite problems. Mixed variational problems; the Babuska theory. Saddle-point problems; the Brezzi theory. -- Examples: Stokes equations, Oseen problem, Darcy equations. 5. Finite element spaces -- Shape functions. Piecewise Lagrange polynomials. Piecewise Hermite polynomials. -- Numerical integration. -- Piecewise polynomial approximation. Interpolation error. 6. Error analysis of finite element methods. -- A priori error analysis. Optimal estimates in the energy norm. L2-estimates. -- A posteriori error analysis. The duality approach. Adaptive algorithms. 7. Time-dependent problems -- Finite element forumlation of parabolic problems -- A priori error analysis Material to be prepared Existing (electronic) lecture notes need to be adapted, together with the example/problem sheets. Same for the Matlab demonstrations. Comments by proposer This course has run for the first time in 2006-07 as a fourth year option on the MSci course. It is expected to become a core course as part of

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a joint MSc programme with the Department of Economics.

Integrable systems Semester: University: Manchester Contact: Jitesh Gajjar Hours: 10 Format: One hour per week. Subject classification: 70H06, 35Q53, 17B65, 17B80 Preparation cost: £ Description Description: 1) Finite dimensional Hamiltonian systems: Poisson brackets, canonical transformations. Notion of integrability. Noether Theorem. Action angle variables for the pendulum. Arnol’d Liouville theorem. Example: solution of the Euler--Poinsot rigid body by elliptic integrals. 2) Hamiltonian systems on coadjoint orbits: Lie algebras. Kostant--Kiriillov Poisson brackets. Lax pairs. Hamiltonian structure of Lax equations. Example: Integrability of the Manakov system on ${so}(n)$. 3) Infinite dimensional Integrable systems: Pseudo--differential operators. Lax pairs for KdV. Material to be prepared 2 or 3 example sheets with solutions. Exam will be seminar style, i.e. each student will have a choice of a research paper on a subject related to the exam topic. They’ll have to pepare a seminar of 30 minutes on the selected article. Comments by proposer Prerequisites: analytical mechanics (hamiltonians and lagrangians) and Lie algebras (adjoint and coadjoint action). Target audience: students aiming to take a PhD in Integrable Systems. Course to be given by DR Marta Mazzocco

Continuum Mechanics Semester: University: Keele Contact: Yibin Fu Hours: 20 Format: One-hour lectures using mainly electronic presentation of prepared notes Subject classification: 74-xx, 76-xx Preparation cost: £ 5000 The 5000 pounds will cover the replacement cost for a lecturer, who will prepare material from scratch. Description This is offered as a core course for Applied. The objective is to derive in a rational way the governing equations for both solids and fluids and to solve a few illustrative problems. It is intended that, by the end of the course, students will have the knowledge necessary for the in-depth study of various phenomena in linear elasticity, nonlinear elasticity, rheology, and fluid mechanics. Syllabus Vector and tensor theory: Vector and tensor algebra, tensor product, eigenvalues and eigenvectors, symmetric, skew-symmetric and orthogonal tensors, polar decompositions, integral theorems. Kinematics: The notion of a continuum, configurations and motions, referential and spatial descriptions, deformation and velocity gradients, stretch and rotation, stretching and spin, circulation and vorticity. Balance laws, field equations and jump conditions: Mass, momentum, force and torque, theory of stress, equations of motion, energy, jump conditions. Constitutive equations: Basic constitutive statement, examples of constitutive equations, observer transformations, reduced constitutive equations, material symmetry, internal constraints, incompressible Newtonian viscous fluids, isotropic elastic materials, viscoelastic materials, rheological models such as Reiner-Rivlin fluid and Bingham fluid. Advanced formulations: Elementary continuum thermodynamics, variational formulations, conjugate measures of stress and strain, Hamiltonian formulations. Illustrative problems. Linear Elasticity: a cylinder deformed by its own weight, surface wave propagation; Nonlinear Elasticity: inflation of spherical and cylindrical shells Fluid Mechanics: a selection of steady flows bounded by plane boundaries. Reference books: P. Chadwick, Continuum Mechanics, Dover (1999). S.C. Hunter, Mechanics of Continuous Media, Ellis Horwood (1976). I-S, Liu, Continuum Mechanics, Springer (2002). C. Truesdell, The Elements of Continuum Mechanics, Springer (reprinted 1985). P.G. Drazin and N. Riley, The Navier-Stokes equations: a classification of flows and exact solutions, Cambridge University Press (2006). Material to be prepared Lecture notes + example sheets + solutions. Comments by proposer

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Offering from Keele University. Lecturer Prof. Y. Fu.

Waves and Mathematical Modelling Semester: University: Liverpool Contact: Alexander Movchan Hours: 10 Format: 1 lecture a week for 10 weeks Subject classification: 74J Waves Preparation cost: £ 2500 To be determined by the Steering Committee Description For the original modules, see MATH427 OUTLINE SYLLABUS: Hyperbolic PDEs. Definitions. Characteristics. Formulation of problems. Dâ€™Alembertâ€™s formula. Outline of inviscid fluid dynamics. Linear theory. Plane waves. Reflection and transmission at a plane interface. Spherical waves. Dipole fields. Scattering by a solid sphere. Vibration of an infinite volutme. Poissonâ€™s formula. Introduction to non-linear theory. Systems of quasi-linear first-order partial differential equations. Characteristics. Riemann invariants. Model examples. Conservation laws, weak solutions and shocks. Definitions and model examples. Water wave theory. Governing equations. Linearised model. Dispersive waves. Model examples. Non- linear equations, solitary waves. Material to be prepared This would be a compilation from an existing 36 lecture module Comments by proposer

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Hydrodynamic Stability Theory Semester: University: Leeds Contact: Alastair Rucklidge Hours: 20 Format: One-hour lectures using mainly electronic presentation of prepared notes, with some animated demonstrations. Subject classification: 76Exx Preparation cost: £ 5000 5000 will cover the replacement cost for a lecturer, who will prepare material for the course as specified above. Description This is offered as a core course for Applied. The number of lectures for each section is in []. [2] 1. Introduction Derivation of the Navier-Stokes equations Boundary conditions Non-dimensionalisation Additional forces and equations: Coriolis force, buoyancy Boussinesq approximation [2] 2. Basics of stability theory Swift-Hohenberg equation as a model Linear stability. Dispersion relation. Marginal stability curve. Weakly nonlinear theory. Normal form for pitchfork bifurcation Global stability [4] 3. Rayleigh-Benard convection Basic state. Linear theory. Normal modes. Marginal stability curve. Weakly nonlinear theory. Modified perturbation theory. Global stability for two-dimensional solutions Truncation: the Lorenz equations [2] 4. Double-diffusive convection Rotating convection, plane layer and spherical geometry Linear theory: real and complex eigenvalues, Takens-Bogdanov point. Taylor-Proudman theorem. Thermosolutal convection. Linear theory: real and complex eigenvalues. Salt fingers. [6] 5. Instabilities of parallel flows Instabilities of invicid shear flows. Linear theory. Squire’s theorem. Rayleigh’s equation. Plane Couette flow. Rayleigh’s inflexion point criterion. Howard’s semi-circle theorem. Examples: Kelvin-Helmholtz, bounded shear layer. Role of stratification. Role of viscosity, global stability. Shear flow instabilities of viscous fluids. Orr-Sommerfeld equation. Examples: plane Couette flow, plane Poiseuille flow, pipe flow, Taylor-Couette flow. Problems with normal mode analysis. Pseudo-spectrum and non-normality. Absolute and convective instabilities. Finite domain effects. [3] 6. Introduction to pattern formation Stripes, squares and hexagons. Weakly nonlinear theory. Mode interactions. Oscillatory patterns: standing and travelling waves. Long-wave instabilities of patterns: Eckhaus, Benjamin-Feir, etc. [1] 7. Introduction to the transition to turbulence Supercritical vs subcritical bifurcations. Fully developed turbulence. Turbulent cascade. Energy spectrum. Isotropic turbulence. Mean plus fluctuations, Reynolds stress. Closures: eddy viscosity, Subgrid-scale modelling, similarity models Material to be prepared Hand-written notes need to be typed and hand-drawn diagrams need to be converted to electronic format. Animated demonstrations need to be prepared. Example sheets need to be prepared. Comments by proposer We developed this course as new PG course in 2006-07, with the intention of offering it as a core MAGIC course in 07-08. Either semester is OK. The content is mainly in place (apart from animated demonstrations and example sheets), but it is all in hand-written form at the moment.

Advanced Topics in Electromagnetism Semester: University: Southampton Contact: Lecturer: Giampaolo D’Alessandro Hours: 10 Format: 1 lecture per week for 10 weeks Subject classification: 78-xx, 35Q Preparation cost: £ Description There are many problems in electromagnetism that are of fundamental importance for society in general (e.g. lasers, precision measurements, optical communications and processing) and science in particular (quantum optics, light-matter interactions, light mediated assembly of complex materials). Many of these require refined mathematical modelling and the application of advanced analytical and numerical tools. This course aims to cover some aspects of advanced electro-magnetic theory with emphasis both on the mathematical techniques needed to deal with them and the physical context where they arise. The course comprises two separate moduli that may vary from year to year. Each modulus will be covered in five lectures; the students will be required to do extensive reading before and after each lecture. â€“ Scattering solutions of the wave equation â€“ Optical fibres â€“ Paraxial optics â€“ Polarisation and anisotropic media â€“ Semiclassical ray optics â€“ Covariant electromagnetism â€“ Multiple scattering of light by particles â€“ Optics of liquid crystals â€“ Light propagation in non-standard media Material to be prepared Comments by proposer If desired, this could easily be extended to a 20 lecture course. Prerequisite: The students will be assumed to be familiar with Maxwellâ€™s equations, scalar and vector potentials, the wave solution of Maxwellâ€™s equations, Maxwellâ€™s equations in linear, isotropic media,

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dipole and multipole potentials.

Entanglement criteria Semester: University: York Contact: Gustav Delius Lecturer: Stefan Weigert Hours: 10 Format: 1 lecture per week for 10 weeks Subject classification: 81P68 Quantum computation and quantum cryptography Preparation cost: £ 1500 Course will be prepared from scratch and require at least 2 full working weeks (i.e. at least 70 hours). Description Entanglement plays an important role as a resource in the rapidly developing field of quantum information. After a brief introduction to the uses of entangled states, this course intends to survey various tools to detect the presence (or not) of entanglement, for pure and mixed states of mostly bipartite systems. The topics covered will include Bell inequalities and Schmidt numbers, entanglement witnesses, the PPT criterion, realignment, and various other criteria detecting entangled states. Material to be prepared summary of lectures, exercises Comments by proposer Pre-requisites: good working knowledge of linear algebra including tensor products; good working knowledge of quantum mechanics

Introduction to quantum integrable systems Semester: University: York Contact: Gustav Delius Hours: 10 Format: 1 lecture per week for 10 weeks. Lecturer: Evgeny Sklyanin. Subject classification: 81R12 Relations with integrable systems Preparation cost: £ 1500 Course will be prepared from scratch and require at least 2 full working weeks (i.e. at least 70 hours), including typing up and training in video link equipment use. Description One-dimensional nonrelativistic particles with point (delta-function) interaction. Conserved quantities, integrability. Bethe wave function, Bethe equations. Repulsive case: Young functional, enumeration of solutions of Bethe equations. Thermodynamic limit. Ground state of the Bose gas, excitations: particles and holes. 6-vertex model and its equivalence to XXZ spin chain. Transfer matrix. Commutativity of transfer matrices, R-matrix. Yang-Baxter equation. Algebraic Bethe Ansatz. Thermodynamic limit. Ground state, excitations. (In reality the course may be reduced to a smaller number of topics.) Material to be prepared lectures in pdf Comments by proposer

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Pre-requisites: basics of quantum mechanics, linear algebra.

Quantum Field Theory in Curved Spacetime Semester: University: York Contact: Gustav Delius Hours: 10 Format: 1 lecture per week for 10 weeks. Lecturer: Dr Bernard Kay Subject classification: 81T20 Quantum field theory on curved space backgrounds Preparation cost: £ 1500 Course preparation will require at least 2 full working weeks (i.e. at least 70 hours), including training in video link equipment use. Description Quantum Field Theory in Curved Spacetime is a theoretical arena in which certain aspects of the interaction of quantum matter fields with gravity are studied in the presence of certain simplifying approximations, which allow one to avoid many of the complications and difficulties of full quantum gravity. It is the appropriate theoretical arena for understanding particle-creation effects in the early universe as well as the evaporation of black holes (Hawking effect). Yet the understanding of the basics requires only a relatively straightforward extension of the theory of free quantum fields. The course will aim to give a pedagogical exposition of what that extension consists of and to help students to approach the classic papers -- including the paper by Hawking on the Hawking effect. The course should also be of interest to students who wish to widen their general understanding of the mathematical and conceptual structure of (flat-space-time) quantum field theory and it should also help to bridge the gap between introductory courses on quantum (field) theory and the (still active) mathematically oriented research literature on quantum field theory in curved spacetime. Material to be prepared I shall make sure students get a copy of the material presented in the lectures in some suitable form, but the details will depend on the technology which is to be used to deliver the lectures, which is still to be decided. There will be some suitable exercises. Comments by proposer Pre-requisites: A first course on General Relativity at MMath level and also good prior knowledge of Quantum Mechanics. Prior acquaintance with the geometry of black holes (at least the Kruskal extension of the Schwarzschild metric) would be desirable. Alternatively, students would have to study this topic for themselves and/or one of the lectures could be devoted to it. An acquaintance with the elements of Quantum Field Theory (what is relevant is mainly only the theory of free fields) would be desirable but not absolutely essential. On the pure-mathematical side, there are many areas of differential geometry and also of distribution theory and classical linear partial differential equations and of functional analysis (especially operators on Hilbert spaces) and operator algebras, knowledge of which would be helpful. But the course will only assume the typical, and partly informal, acquaintance with these subjects that a student might be expected to have acquired during their

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basic studies of general relativity and quantum (field) theory.

Topics in Relativistic Astrophysics Semester: University: Southampton Contact: Lecturer: Ian Jones Hours: 10 Format: 1 lecture per week for 10 weeks Subject classification: 83-xx, 83C Preparation cost: £ Description Linearised theory (2 lectures): Linearized Einstein equations â€“ Detection of GWs (1 lecture): Transverse traceless gauge; GW Interferometers â€“ Generation of GWs (2 lectures): The quadrupole formalism; Example: binary systems â€“ Relativistic fluids (2 lectures): Formulation; Newtonian limit â€“ Kerr black holes (3 lectures): Basic properties; Orbits of particles; Ergoregion/Penrose process Material to be prepared Comments by proposer Follow on course for students who have already studied some course in general relativity.

String Theory Semester: University: Liverpool Contact: Mary Rees Hours: 20 Format: 2 lectures a week for 10 weeks Subject classification: 83E30 String and superstring theories Preparation cost: £ 2500 To be determined by the Steering Committee. Description Outline syllabus Review of relativistic particles and harmonic oscillators. Bosonic strings: action principle, equations of motion and their solution. Quantization of bosonic strings, determination of particle spectrum, photons and gravitons. Dimensional reduction, non-abelian gauge symmetry, T-duality and D-branes. Outlook: Superstrings Material to be prepared This would be a reorganisation of an existing course which has previously been taught in 12 90-minute sessions Comments by proposer

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The lecturer would be Dr. Thomas Mohaupt

Markov Decision Processes with applications to Reliability Semester: University: Liverpool Contact: Mary Rees Hours: 20 Format: 2 lectures a week for 10 weeks Subject classification: 90C40 Markov and semi-Markov decision processes Preparation cost: £ 2500 To be determined by Steering Committee Description For the existing course see MATH461 OUTLINE SYLLABUS: â€¢ Introduction: Revision of Probability. â€¢ Reliability: Functional relationships between the Distribution function, Density function, Hazards function and Reliability function. Mean time to failure. System reliability. â€¢ Simulation: Random numbers, pseudo-random numbers generation. Random numbers from specific probability distributions. â€¢ MDP: Finite and infinite horizon, dynamic programming approach. Application to Reliability, Queues, Quality control, Inventory etc. Material to be prepared This would be a compilation from an existing course which is offered in a 36 lecture format. Comments by proposer

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The lecturer would be Dr. Alexei Piunovsky

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