Four-Week Courses

Measure Theory
Marty Ross

Period 1 : 16 - 27 Jan 2006

Period 2 : 30 Jan – 10 Feb 2006

• General Measure Theory (Outer measure;     measurable sets; Borel and Radon measures; the Caratheodory criterion for Borel measures)

·      Special Measures on Euclidean Space (Lebesgue measure; Hausdorff measure; the Vitali Covering Theorem; Hausdorff dimension)

·      Integration (Measurable functions; integration and convergence theorems; the Area Formula; iterated integrals and Fubini's Theorem)

·      Functional Analysis (Measures as linear functionals; Lp spaces; the Riesz Representation Theorem)

• Further Topics (Differentiation of measures; the Besicovitch Covering Theorem; the (Generalised) Fundamental Theorem of Calculus; the Co-Area Formula)

 Return to Top

Permutation groups could be argued to be the oldest examples of groups, and their study grew out of the works of nineteenth-century mathematicians such as Camille Jordan and William Burnside. The theory of group actions, or permutation groups, is essentially the study of symmetric objects. Hence the techniques employed in this subdiscipline of group theory, have applications to a diverse range of mathematics.

In this short course, we will concentrate on the basic techniques used in permutation group theory.  This will include the fundamental theory of group actions, transitive groups, primitive groups, and the O'Nan-Scott Theorem.  Other topics include multiple transivity, innately transitive groups, and wreath products.

 Other material that may be helpful:                                             "Finite permutation groups", H. Wielandt. Academic Press, New York - London 1964   

"Permutation Groups", J. D Dixon and B. Mortimer.  Graduate Texts in Mathematics, 163. Springer-Verlag, New York, 1996 

"Permutation Groups", P. J. Cameron.  London Mathematical Society Student Texts, 45.  Cambridge University Press, Cambridge, 1999.

Return to Top

7 lectures per week.

 Return to Top

6 lectures + 1 tutorial each week.

This course will introduce some of the fundamental algorithms which underpin computation with such groups.

Our primary focus will be on effective practical algorithms.

In more detail, we expect to cover some of the following topics:

1. Permutation groups:
* Orbits, stabilisers, backtrack procedures
* Recognition of the alternating and symmetric groups
* Base and strong generating sets
(3 lectures)

2. Finitely-presented groups:
* Coset enumeration
* Presentations for subgroups
* Abelian quotients
(2 lectures)

3. Polycyclic groups:
* Polycyclic presentations and their properties
* Constructing such presentations
(2 lectures)

4. Classification, Automorphisms and Isomorphism:
* Construction and classification of p-groups
* Determining automorphism groups
* Isomorphism testing
(2 lectures)

5. Linear groups:
* Characteristic and minimal polynomials
* Determining the order of an element
* Random-Schreier and other approaches
* Deciding irreducibility of FG-modules
(3 lectures)

There will be two tutorials, each involving practical computations with Magma.

This suggested allocation of break-down in time is a preliminary guide.

Some interest in algorithms is highly desirable, but no practical experience of programming is required.

Any senior undergraduate or introductory graduate algebra book: for example:
Peter Cameron: Permutation Groups
D.L. Johnson: Finitely-presented groups
I.M. Isaacs: Algebra: A graduate course
B. Hartley and T.O. Hawkes: Rings, modules, linear groups

 Return to Top

6 lectures and 1 tutorial per week (total: 28 hours)

Complex algebraic curves and compact Riemann surfaces are pervasive in many branches of mathematics, and they are traditionally approached via either complex analysis or commutative algebra. This course will provide a thorough introduction to the subject giving the flavour of both points of view. We aim to cover the following topics:

Basic definitions and examples:  affine and projective curves, Riemann surfaces, function fields, holomorphic and meromorphic differentials, complex manifolds and algebraic varieties.

Normalisation of plane curves: smooth points, singularities of plane curves, Weierstrab  preparation, resolution of singularities, divisors, intersection numbers and Bezout's theorem,  Riemann--Hurwitz and degree-genus formulas.

Riemann--Roch theorem: Mittag--Leffler problem, finite-dimensionality of H^1(D), Serre duality, Riemann--Roch theorem for curves.

Applications of Riemann--Roch: the canonical map, curves of low genus, Riemann's count.

Abel--Jacobi map: Jacobian of a smooth algebraic curve, Abel's theorem, Jacobi inversion theorem.

A first course on complex analysis (holomorphic functions, Cauchy's theorem, calculus of residues), elementary notions from topology (connectedness, compactness, homotopy of paths) and familiarity with basic algebra (groups, ideals and quotient rings, factorisation in polynomial rings) will be assumed. 

The main references we will follow are:

P. A. Griffiths, Introduction to Algebraic Curves, AMS, 1989

 R. Miranda, Algebraic Curves and Riemann Surfaces, AMS, 1995

Return to Top

5 or 6 lectures per week, and 1 or 2 of tutorial.

When does a problem have an effective algorithmic solution? What does it mean for an algorithm to be effective? How can I prove that my problem is hard?  In this unit we attempt to give rigorous meaning to questions of this type and investigate some possible answers. 

The first half concentrates on concepts related to the million dollar Millennium problem “P=NP?” (see http://www.claymath.org/millennium//).  We give precise definitions of the various classes into which algorithmic problems are categorised, concentrating on the most important from a practical perspective: P, NP, NP-complete, PSPACE.  We look at, and classify some classic examples from graph theory, Boolean algebra, and combinatorics.

In the second half, we move to the extreme end of difficulty: problems for which we can prove no algorithm can exist.  We find a number of these including examples concerning algebra, matrix multiplication, and tilings of the plane.  Even Sudoku will get a mention! 

The objective is to be able to assess and establish the difficulty of algorithmic problems when they are encountered in other areas of mathematics.

Minimal.  Mathematical maturity is the main requirement, and although some background in algebra and discrete mathematics is desirable; very little is assumed.  This is a mathematics subject, but may be of interest to computer science students with a reasonably strong mathematical background.

Papadimitriou (Computational Complexity, Addison-Wesley, 1994), Garey and Johnson (Computers and Intractability, W. H Freeman and Co., 1979), or Wilf (Algorithms and Complexity, A. K. Peters, 2nd edition, 2002: the 1st edition (1986) is available for free legal download from (http://www.cis.upenn.edu/~wilf/AlgComp3.html).

Return to Top

System Modelling is the process of creating a model which imitates a real world system or even some proposed system. The main focus of this brief course will be on the use of simulation modelling for the modelling and investigation of such systems, although we will also look at some basic analytic queueing models. Both techniques provide the modeller with a rich set of tools, which may be used to model many systems of interest in manufacturing, telecommunications, finance, games, ecosystems, ...

Topics covered will include:

 Introduction to Simulation,  Generation of random variates, Queueing models, Analysis of simulation output.

Familiarity with basic probability theory and Statistics: that is, some background in Sample spaces, events, probability of events  and conditional  probability, independence, random variables, probability mass,  density and distribution functions, expected value and variance, confidence intervals for means and hypothesis testing.

Return to Top

In convex analysis we find a treasure chest of techniques and a mature calculus that facilitates its application.  Many areas of applied mathematics use these techniques and we can only be certain that more applications will arise in the future. 

In the course we will attempt to: 

1. Cover some of the fundamental theory that convex analysis is based on. 

2. Show how these techniques can be applied to a number of real world problems, in particular the formulation and solution of optimization problems. 

Convex analysis comes in two brands, finite dimensional and infinite dimensional, categorized by the dimensionality of the underlying linear space.  We will touch on both in this course but in the main we will treat the case when the underlying space is a Euclidean space inner product space.  We will only touch on an infinite dimensional flavor when the underlying spaces are sequences spaces such as c0 or l¹.  There are some interesting robust control problems that may be framed in these spaces. 

Convex optimization has in recent years matured to a level where it has become a corner stone of optimization theory, a position that was long held only by linear programming (which is just convex optimization over polyhedral functions).  It is hoped that this course will alert more practitioners of applied mathematics to opportunities awaiting those who become proficient with the mathematical language of convex analysis.

Inner product spaces, the Frobenious inner product in the space of real symmetric matrices.  Lagrangian optimality and value functions. Linear programming.  Convex conjugation, Fenchel duality theorem and the minimum norm problem.  Applications to the formulation of optimization problems used to solve real world problems.  Introduction to linear system theory and robust control models.  The Youla parametrization and formulations of the controller design as a convex optimization problems.  Introduction to Lyaponov stability and positive semi‑definite programming.  Other examples of the use of positive semi‑definite programming, if time permits.  Some discussion of solvers under the MatLab environment will be given. 

Undergraduate courses on calculus of functions of several variables (ie partial derivatives, grad and Taylor series), linear algebra (ie matrices, eigenvalues, diagonalization and systems of equations etc)and real analysis (ie limits, sequences and series and uniform convergence).

 Return to Top

Approximately 7 hours per week, including 1 hour per week of tutorials/laboratories as required

Classical cryptography 

Information theoretic security

Computational security

Design and analysis of block ciphers. DES, AES

Algebraic properties of Block Ciphers, Nonlinearity, PN and APN functions over finite fields

Integers, divisibility and number theory

The RSA cryptosystem. Attacks on RSA

The discrete logarithm problem. The El Gamal cryptosystem

Randomized algorithms for discrete logarithms and factoring

Introduction to elliptic curves. The elliptic curve discrete logarithm problem

Signatures, hash functions and authentication

Stream ciphers and correlation attacks

Randomness, testing for randomness, generating randomness.

Nothing beyond undergraduate knowledge in mathematics is expected. More specifically, we assume the following and teach more advanced content as required during the course:  

Basic algebra (groups, rings, fields, etc.)

Basic number theory (integers, divisibility)

Basic statistics and probability theory (expectation, variance, binomial, uniform, gaussian distributions, conditional distributions)

Familiarity with some sort of structured high level programming is helpful but not essential. The mathematical software packages Magma and Maple will be used and introductory tutorials on these packages will be given.

The following books are useful background references and will be available on 2‑hour reserve at the library.

Stinson, Cryptography: Theory and Practice, 2nd Edition, Chapman & Hall/CRC, 2002.

Stinson, Cryptography: Theory and Practice, 1st Edition, CRC Press, 1995

Buchmann,  Introduction to cryptography, Springer‑Verlag, 2001.

Menezes, van Oorschot and Vanstone, Handbook of Applied Cryptography, CRC, 1997, also available freely online at http://www.cacr.math.uwaterloo.ca/hac/

 Return to Top

6 hours of lectures and 1 hour of lab per week

*  FEM for two-point boundary-value problems                              *  General aspects of FEM, mathematical foundations                *  FEM for elliptic problems in 2D                                                     *  Implementation issues                                                                   *  Related topics:  additive Schwarz methods and/or a posteriori error estimates

The main requirement for undertaking the course is a reasonable level of competence in several variable calculus and linear algebra.  A knowledge of Matlab is desirable, but not essential.