Colloquium 

For the approximate solution of the inverse obstacle scattering problem to reconstruct the boundary of an impenetrable obstacle from the knowledge of the far field pattern for the scattering of time-harmonic acoustic waves, roughly speaking, one can distinguish three groups of methods. Iterative methods interpret the inverse obstacle scattering problem as a nonlinear ill-posed operator equation and apply iterative schemes such as regularized Newton methods or Landweber iterations for its solution. Decomposition methods, in principle, separate the inverse problem into an ill-posed linear problem to reconstruct the scattered wave from its far field pattern and the subsequent determination of the boundary of the scatterer from the boundary condition. Finally, the third group consists of the more recently developed sampling and probe methods. In principle, these methods are based on criteria in terms of an indicator function that decides whether a point lies inside or outside the scatterer. The colloquium will give a survey by describing one or two representatives of each group including a discussion on the various advantages and disadvantages.

Os resultados mais gerais de existência e estabilidade para sistemas de leis de conservação, conhecidos até o momento, foram provados para dados iniciais de variação total pequena. Por isso, diferentemente do que ocorre com a maioria das equações às derivadas parciais, no caso dos sistemas de leis de conservação, com duas ou mais equações, as questões envolvendo soluções espacialmente periódicas tornam-se, em geral, mais difíceis. Faremos uma breve exposição sobre o desenvolvimento da teoria das leis de conservação até o presente, tendo por fio condutor questões envolvendo soluções espacialmente periódicas.

The space of representations of the fundamental group of a surface into a Lie group is a natural object with rich geometry and symmetry. The topology of the surface influences the algebraic structure of the deformation space, with natural families of Hamiltonian flows related to curves on the surfaces. In particular these flows define continuous deformations closely related to the action of the discrete mapping class group of the surface. In the case of the one-holed torus and the four-holed sphere, these deformation spaces are families of affine cubic surfaces with actions of the modular group by polynomial Poisson automorphisms.

This talk shows how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. Discrete calculus is seen to fit into this pattern by reformulating it in terms of commutators. Differential geometry begins here, not with the concept of parallel translation, but with the concept of a physical trajectory and the algebra related to the Jacobi identity that governs that trajectory. We discuss how Poisson brackets give rise to the Jacobi identity, and how the Jacobi identity arises in combinatorial contexts, including graph coloring and knot theory. We give a highly sharpened derivation of results of Tanimura on the consequences of commutators that generalize the Feynman-Dyson derivation of electromagnetism, and a generalization of the original Feynman-Dyson result that makes no assumptions about commutators. The latter result is a consequence of the definitions of the derivations in a particular non-commutative world. Our generalized version of electromagnetism sheds light on the orginal Feynman-Dyson derivation, and has many discrete models. The talk is self-contained and begins with a discussion of how classical discrete calculus embeds in a non-commutative framework such that the Leibniz rule is restored, and the discrete derivatives are represented by commutators. This construction motivates the rest of the talk. See quant-ph/0403012.

We'll survey algebro-combinatorial link homology theories that have the Jones polynomial and other link invariants as their Euler characteristics.

Seventy-five years ago Kurt Gödel overturned the mathematical apple cart entirely deductively; but he held quite different ideas about legitimate forms of mathematical reasoning:

If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.

Kurt Gödel, 1951

This lecture will be a general introduction to Experimental Mathematics, its theory and its “Experimental methodology” that David Bailey and I — among many others — have come to practice over the past two decades. I will focus on the differences between Discovering Truths and Proving Theorems. We shall explore various of the computational tools available for deciding what to believe in mathematics, and — using accessible examples — illustrate the rich experimental tool-box mathematicians now have access to. In an attempt to explain how mathematicians use High Performance Computing (HPC) and what they have to offer other computational scientists, I will touch upon various Computational Mathematics Challenge Problems including $${\int }_0^{\mathrm{\infty }}\cos (2x)\prod _{n=1}^{\mathrm{\infty }}\cos \left(\frac{x}{n}\right)\mathrm{dx}\stackrel{?}{=}\frac{\pi }{8}.$$

This problem set was stimulated by Nick Trefethen”s recent more numerical SIAM 100 Digit, 100 Dollar Challenge (*), which I shall also mention.

Bibliography

• D.H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices Amer. Math. Soc., 52 No. 5 (2005), 502–514. [CoLab Preprint 269].
• Jonathan M. Borwein and David H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A.K. Peters, Natick, MA, 2004.
• Jonathan M. Borwein, David H. Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A.K. Peters, Natick, MA, 2004.

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All resources are available at http://www.experimentalmath.info.

(*) See http://www.cs.dal.ca/~jborwein/digits.pdf.

In flat Minkowski space, the theory of quantization (coherent states) is fundamental for mathematical physics. In the complex wave representation (Fock space), methods from complex analysis such as Toeplitz operators on Hilbert spaces of holomorphic functions play a basic role. In my talk I will concentrate on two important generalizations: First, the incorporation of curved manifolds, such as symmetric Kaehler manifolds and bounded hermitian symmetric domains, in the theory of quantization, leading to the construction of a unified calculus on Bergman spaces generalizing the Toeplitz-Berezin calculus and the Weyl calculus. Second, the more recent ideas concerning super-symmetry, where one considers super-Hilbert spaces of Grassmann analytic functions and still has a deep theory of quantizations invariant under (super) Lie groups.

Physical systems often "relax" to a state of "minimal energy", and finding these minimal energy configurations of the system is a problem in the calculus of variations. Quite often, the solutions, that is, the minimizing configurations, have a special symmetry property that is not an obvious consequence of the energy functional being minimized. How does this symmetry arise? Can we explain it in mathematical and physical terms? A large number of questions in mathematics arise in this way, and it is a very active field of research. In this lecture we shall present several examples, some with recently obtained answers, and others that are still open.

John von Neumann referring to Gödel's incompleteness theorem called it an "achievement ... [that] is singular and monumental ... a landmark which will remain visible far in space and time ..." In spite of this high praise, the incompleteness theorem has had almost no effect on the work of practicing mathematicians. In this talk, I will discuss the relevance of Gödel's work and ideas for the future of mathematics, and will include some recent results of Harvey Friedman on the use of large cardinal axioms in combinatorial mathematics.

Consideram-se as equações de Stokes e de Navier-Stokes nos casos estacionário e de evolução com condições de fronteira não standard. Descrevem-se alguns resultados de existência e de regularidade das soluções para estes problemas.

The lecture is devoted to the boundary value and the initial-boundary value problems for the Navier-Stokes equations, the principal system of hydrodynamics, as well as for some of its generalizations.

It took four years (1985-89) and a lot of design help to make a public exhibition "Mathematics and knots", and this talk will describe and demonstrate some of what we learned in the process, and the extra contacts that resulted. The design team was R. Brown, N.D. Gilbert, T. Porter. Two slogans were used: "advanced mathematics from an elementary viewpoint"; "explaining the methods of mathematics to the general public". In 1996 we got involved in web work, and in 2000 in producing a CDRom, under an EC grant for European Science and Technology Week 2000, coordinated by Professor Mireille Chaleyat-Maurel (Paris), under the title "Raising Public Awareness of Mathematics". The CDrom was launched at Óbidos in November, 2000, (http://www.spm.pt/~spm/Ano2000/lancamentoCDROM.html) with the support of the Sociedade Portuguesa de Matemática.

Curiously, the collaboration with the sculptor John Robinson was a result of a chance encounter through our involvement in Royal Institution Mathematics Masterclasses for 13 year olds, which started in 1985, and have had continued support from Anglesey Aluminium. So the presentation will also explain how the work on popularisation has influenced us. Tim Porter (Bangor) will be involved in the presentation and discussion.

Final questions: Is it a good idea to popularise mathematics to our students? And what would that mean and imply?

We have recently revised our web sites with funding from EPSRC and the continued advice and support of Edition Limitée and John Robinson. The new url is http://www.cpm.informatics.bangor.ac.uk.

The numerical range (also known as the Hausdorff set) of a matrix (or an operator on a Hilbert space) is the set of values of the associated quadratic form on the unit sphere. In this talk, we will discuss various beautiful properties of the numerical range, starting with the classical ones (such as its convexity and the shape in 2-by-2 case) and ending with rather recent: a complete classification in the case of 3-by-3 matrices, curvature of the boundary and its influence on the location of the eigenvalues, flat portions of the boundary (their number and orientation), and an explanation of why some large (in particular, tridiagonal) matrices have elliptical numerical ranges.

In recent years several methods stemming from absolutely pure mathematics find very concrete applications, e.g., Algebraic Topology in Hardware Robotics and Genetics, Elliptic Curves in Coding Theory, Fractal Theory in Geology etc... In this talk a survey of applications of some rather elementary pure mathematics is given. For example, a problem about motions of Robot fingers leads to new quantum groups, or "supershape" defined in terms of Lame ovals becomes an omnipresent notion connecting : Bamboo, architecture, Traffic control, plant morphogenesis, stress flow on a solid body and a proof of the position of the quiescent centre in roots of plants. The exposition is on a general level and almost no knowledge of mathematics is presupposed.

In this talk, I will discuss new computational algorithms for infinite-dimensional Lie pseudo-groups. The method is based on the variational bicomplex and a new generalization of the Cartan theory of moving frames. Applications will include practical algorithms for computing complete systems of differential invariants, invariant differential forms and structure equations, classification of syzygies and recurrence relations, solutions to equivalence and symmetry problems for differential equations and variational problems arising in geometry and physics.

In this lecture we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. Our analysis is based on an approach to modular invariants using braided sector induction ( $a$-induction) arising from the treatment of conformal Doplicher-Haag-Roberts framework. Many properties of modular invariants which have so far been noticed empirically and considered mysterious can be rigorously derived in a very general setting in the subfactor context. For example, the connection between modular invariants and graphs (cf. the A-D-E classification for $\mathrm{SU}(2){}_k$) finds a natural explanation and interpretation. We try to give an overview on the current state of affairs concerning the expected equivalence between the classifications of braided subfactors and modular invariant two-dimensional conformal field theories.

This talk is about orthogonal polynomials and their generalizations. The polynomials are obtained by orthogonalization of the sequence of power functions on the unit circle in respect to weighted inner products. The impetus for these results was a theorem of M. G. Krein and its generalization by M. G. Krein and H. Langer. The story started with a theorem of Szego which proved that this type of orthogonal polynomial in the case of a positive weight has all zeros inside the unit circle. M. G. Krein extended this result for the case when the weight is changing signs. M. G. Krein and H. Langer obtained the continuous analog of the latter result. We will present all these theorems together with the one step version. The inverse theorem will also be considered. The next topic will be the matrix orthogonal polynomials and Krein's theorem. Also will be presented other generalizations based on orthogonalization in modules. The theory of Toeplitz and Wiener-Hopf operators plays an important role in these considerations. The talk is based on recent results of A. Ben-Artzi, R. Ellis, I. Gohberg, D. Lay, and L. Lerer, and it is planned for a wide audience.

Concentration phenomena is a common fact in statistical mechanics connected to large deviation theory. It states that some observables do not deviate appreciably from their average in some adequate limit. Recently some new versions of this effect were discovered for independent random variables and more general observables leading to important statistical consequences. Extensions were proved for non independent random variables. We will discuss the result for expanding maps of the interval and present some statistical consequences.

There are many nonlinear initial value problems whose solutions are too complex to be fully resolved numerically; typically, these are problems where one cares only about the behavior of the solutions on one scale, while the dynamics involve many scales. The problem is to find effective equations on the scale of interest and average properly on the other scales.

In optimal prediction methods one assumes that initial data for the degrees of freedom that cannot be resolved are drawn from a known probability distribution; this is a reasonable assumption in many cases. The problem is to make optimal predictions about the behavior of the solutions at later times given partial initial data and partial numerical resolution. An exact equation of motion for the resolved degrees of freedom can be written down, but its solution requires the solution of an auxiliary equation, the orthogonal dynamics equation. This is in general as difficult to solve as the original problem, but in special cases the solution of the orthogonal dynamics equation can be approximated in a straightforward way. The formalism is a generalization of the fluctuation/dissipation theory of irreversible statistical mechanics.

I shall give examples and explain some of the difficulties of these methods.

New computation devices increasingly depend on particular physical properties rather than on logical organization alone as used to be the case in conventional technologies. The laws of physics impose limits on increases in computing power. Two of these limits are interconnect wires in multicomputers and thermodynamic limits to energy dissipation in all computers. Quantum computing is a novel computational paradigm and technology that promises to eliminate problems of latency and wiring associated with parallel computers and the rapidly approaching ultimate limits to computing power imposed by the fundamental thermodynamics. The prospect of quantum computing has created excitement both among researchers and in the popular press by its algorithmic improvements over classical computing: integer factorization in square time (Shor), searching unstructured lists in square root time (Grover), improved communication complexity (Buhrman, Cleve). For some other problems (like binary search) quantum computation gives no super-linear speed-up over classical computing. While fast factoring will break almost all public key cryptosystems in use today, compromizing almost all secure (financial, government, commerce) e-transactions, quantum cryptography (Bennett, Brassard) may possibly come to the rescue. Superiority of quantum computing over classical probabilistic computing are due to the exploitation of interference in parallel quantum superposition and quantum entanglement. The actual realization of quantum computers is a formidable technological and theoretical challenge. Part of this challenge involves quantum information and communication theory (compression, fault-tolerance and error-correcting codes). The great algorithmic challenge is to find more quantum algorithms that improve classical ones (or show that none exist), and to more precisely determine the complexity of quantum computations compared to the classical complexity classes.