Courses

Four-Week Courses (7 lectures per week, for credit)

Knots and links play a variety of roles in human culture, and, from a mathematical point of view, similarly display surprising and deep interconnections with other branches of mathematics and physics. We will describe some of the combinatorial properties of planar diagrams of knots and links, and ways of determining whether or not two diagrams represent the same link in 3-dimensional space. Many of these techniques are purely algebraic, while others involve notions of 2- and 3-dimensional locally Euclidean spaces (manifolds). In fact, knots and links can be used to represent all possible compact 3- and 4-dimensional manifolds, making contact with research in relativity theory and elementary particle physics. Some of the algebraic structures we will explore have their origins in Lie algebras and representation theory, but familiarity with these notions are not required.

The first part of the course will focus on knots and links as combinatorial objects, and will describe some of the invariants such as the Alexander polynomial, fundamental group, the Seifert form, linking numbers, and the more recent Jones polynomial, Kauffman bracket, Vassiliev invariants and related generalizations.

We then consider the role of knots and links in describing 3-dimensional manifolds, also making (peripheral) contact with with some of the topology and geometry involved in Thurston's Geometrization Conjecture.

If anyone is interested in having Iain `assist with their inquiries': email me at iain at ms.unimelb.edu.au

*C Adams, Why Knot?: An Introduction to the Mathematical Theory of Knots (basic)

*/**C Adams, The Knot Book. (less basic)

**H Burde and H Zieschang, Knots. (classical)

**RH Crowell and RH Fox, Introduction to knot theory. (classical)

K Reidemeister, Knot Theory. (classical)

**Dale Rolfsen, Knots and links. (classical)

**L Kauffman, Knots and physics.

*/**C Livingston, Knot theory.

**/***VO Manturov, Knot Theory.

**/***VV Prasolov and AB Sossinsky, Knots, links, braids and 3-manifolds.

**/***WBR Lickorish, An introduction to knot theory. (appropriate, deeper)

***W Menasco, Handbook of Knot Theory. (deeper)

(20 min Movie) Not knot. Directed by Charlie Gunn and Delle Maxwell. (great fun! - loved by `Grateful Dead' fans)

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The second set of lectures will be devoted to a discussion of qualitative analysis of differential equations. Here the focus will be on problems when not even approximate solutions can be derived: rather we can analyse the equations and thereby deduce how solutions must behave. If time we will discuss equations with particular properties: for example equations with periodic coefficients or problems that can lead onto chaos.

Major topics might include

Asymptotic sequences and series

The WKB method. Transition points.

Singular perturbation problems.

Multiscaling and the method of strained co-ordinates.

Stability and subharmonic resonance.

Phase-planes, indices and the Poincare-Bendixson theorem

Differential equations with periodic coefficients.

Chaotic systems

M. Braun: Differential equations and their applications; an introduction to applied mathematics. Springer (1983)

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J. Neveu, Discrete-Parameter Martingales. New York: Elsevier, 1975.

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E.W. Cheney: Introduction to Approximation Theory, AMS/Chelsey

2) Modern Approximation Theory:

R. DeVore: "Nonlinear approximation", Acta Numerica (1998), pp. 51-150, Cambridge University Press

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In real life one can find explicit solutions of very few PDEs -- and many of these are infinite series whose secrets are complicated to extract. For more than a century the goal is to understand the solutions -- even though there may not be a formula for the solution.

The historic heart of the subject (and of this course) are the three fundamental linear equations: wave equation, heat equation, and Laplace equation along with a few nonlinear equations such as the minimal surface equation and others that arise from problems in the calculus of variations.

We seek insight and understanding rather than complicated formulas.

Strauss, Walter A., "Partial Differential Equations: An Introduction," New York, NY: Wiley, March 3, 1992.

John, Fritz. "Partial Differential Equations", 4th ed., Series: Applied Mathematical Sciences, New York, NY: Springer-Verlag.

Courant, Richard, and Hilbert, David, "Methods of Mathematical Physics," vol II. Wiley-Interscience, New York, 1962.

Evans, L.C., "Partial Differential Equations," American Mathematical Society, Providence, 1998.

Jost, J., "Partial Differential Equations," Series: Graduate Texts in Mathematics, Vol. 214 . 2nd ed., 2007, XIII, 356 p.

Kazdan, Jerry, "Lecture Notes on Applications of Partial Differential Equations to Some Problems in Differential Geometry", available here.

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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, L^p spaces, Riesz Representation Theorems)

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

NOTE: Lecture notes summarising the relevant background on sets and real analysis are available here. Some (but definitely not all) of this material will be reviewed and covered along the way, particularly the material on metric spaces and topological spaces. Before the summer school begins, you should definitely take a good look at the background notes and, if need be, browse through a real analysis text or two.

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One of the main goals of this course is to enhance and enrich the mathematical modeling and problem solving skills of the students. To this end, an important ingredient of the subject is a demonstration that familiar abstract mathematical concepts, constructs and methods can be used effectively in the modelling, analysis and solution of practical decision-making problems.

The major topics covered will include:

Decision-Making Problems

Famous Decision Theoretic Puzzles and Paradoxes

Classical Decision Theory

Mathematical Modeling Paradigms for Decision Analysis

Solution Methodologies

Game Theory

Sequential Decision Processes

Case Studies

French, S., Decision Theory: an introduction to the mathematics of rationality, Ellis Horwood, 1988.

Winston, W.L., Operations Research: Applications and Algorithms, Duxbury, 1994.

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This course is an introduction to Lie groups and Lie algebras, assuming a minimum of prerequisites. To manage with elementary prerequisites we stick to the class of matrix Lie groups (which includes almost all the interesting examples), and mainly to the so-called *classical groups*. The classical groups include the general and special linear groups and groups of "rotations" of linear spaces with real, complex, and quaternion coordinates.

The missing information about a Lie group that its Lie algebra fails to capture is topological. We will explore topological aspects of Lie groups to some extent, depending on the time available.

B. Hall: Lie Groups, Lie Algebras, and Representations (Springer 2003)

A. Baker: Matrix groups (Springer 2003)

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