• Corpus ID: 1382805

New Connections Between Mathematics and Computer Science : Report , Abstracts and Bibliography of a Workshop

@inproceedings{Gunawardena1996NewCB,
  title={New Connections Between Mathematics and Computer Science : Report , Abstracts and Bibliography of a Workshop},
  author={Jeremy Gunawardena},
  year={1996}
}

• J. Gunawardena
• Published 1996
• Mathematics

A workshop on "New Connections between Mathematics and Computer Science" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England from 20-24 November 1995. The workshop was supported by the Engineering and Physical Science Research Council of the United Kingdom, the London Mathematical Society and Hewlett-Packard's Basic Research Institute in the Mathematical Sciences. This document contains a report on the workshop, the abstracts of the talks and the accompanying bibliography. 1 Report on the workshop The interplay between mathematics and computer science has traditionally centered around areas in logic, category theory and discrete mathematics. In recent years new connections between mathematics and computer science have emerged from such unexpected quarters as algebraic topology, differential geometry, dynamical systems and operator algebras. These new developments hold the promise of bringing new insights and powerful mathematical tools to bear on problems in computing. At the same time, such problems have opened new avenues of exploration for the mathematician. To investigate these developments and to introduce them to a wider audience, a workshop on "New Connections between Mathematics and Computer Science" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England from 20-24 November 1995. It was timed to take advantage of parallel programmes in "Semantics of Computation" and "From Finite to Infinite Dimensional Dynamical Systems" at the Issac Newton Institute in the second half of 1995. The workshop was hosted jointly by the two programmes and by Hewlett-Packard's Basic Research Institute in the Mathematical Sciences (BRIMS) in Bristol, England. The workshop was organised by Jeremy Gunawardena of BRIMS, who was also a participant at the "Semantics of Concurrency" programme. Each day of the workshop was devoted to a single theme. The five themes were, in chronological order: • Differential geometry of algorithms, • Developments in complexity theory, • Algebraic topology and distributed computation, • Dynamics of proof and computation, • Geometry of images and computer vision. The choice of these particular themes was influenced in part by the two programmes at the Newton Institute and in part by the prejudices of the organiser. The "Connections between Mathematics and Computer Science" cover a broad spectrum of areas and a number of workshops could have been held under the same title with little overlap among the participants. If there is a coherent thread among the various themes it is that of dynamics and geometry. Four tutorial lectures of one hour each were given in the morning sessions by invited speakers and a small number of half hour contributed talks were given in the late afternoon. The programme was arranged to allow maximum time for discussion among the participants. Particular emphasis was placed on attracting students and, due to generous financial support (see below), it was possible to fund 7 United Kingdom research students to attend the workshop. There were 62 registered participants, excluding some unregistered participants who were already present at the Isaac Newton Institute. By country of affiliation, these broke down as follows: 28 (UK), 10 (USA), 7 (France), 3 (Denmark), 2 (Canada), 2 (Germany), 2 (Holland), 2 (Italy), 1 (Belgium), 1 (Hong Kong), 1 (Japan), 1 (Mexico), 1 (Poland), 1 (Russia). This workshop bought together a very disparate group of people who were not only unfamiliar with each other's work but often operating on the basis of quite different philosophies. In retrospect, and for the benefit of future workshops of a similar nature, it would have been helpful to have devoted some time to basic introductory lectures on dynamical systems and topology (from the mathematical side) and complexity theory and semantics (from the computer science side). Nevertheless, judging from the comments of a number of people, the workshop was stimulating and suggested the emergence of new and fruitful areas of study for both the mathematician and the computer scientist. A independent review of the workshop by John Baez, one of the participants, appears in a weekly world wide web column on "This week's finds in mathematical physics" (accessible at URL:http://math.ucr.edu/home/baez/week70.html) and gives some idea of the scope of material that was presented. Cambridge University Press have expressed interest in publishing a collection of articles based on the workshop. Further information regarding this may be obtained from Jeremy Gunawardena. The next section of this report contains the abstracts of each talk (both invited and contributed) that was presented. The final section contains a bibliography of relevant papers refered to in the abstracts. The workshop was financially supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom, the London Mathematical Society (LMS) and BRIMS. The LMS funding was obtained under the MathFit initiative-Mathematics for Information Technology-for which this was the introductory workshop. MathFit is currently being considered by EPSRC as a managed programme. Further information on the intitiative is available from Professor Ursula Martin, at the Department of Computer Science at the University of St-Andrews, or on the world wide web at URL:http://www.qmw.ac.uk/fVlms/mathfit.html. Further information about BRIMS, including a copy of this report, is available on the world wide web at URL:http://www-uk.hpl.hp.com/brimsj. 2 Abstracts of talks The combinatorics of abstract data types Mike Atkinson, Department of Computer Science, University of St-Andrews [e-mail] An abstract data type may be regarded as a machine whose instruction set is the set of allowed operations which characterise the data type. The theory of machines has traditionally been concerned with the relationship between input sequences to the machine and output sequences. We shall discuss the combinatorial aspects of this relationship in the case of various data types regarded as machines. For references, see [2, 3, 58J. n-Categories in logic, topology and physics John Baez, Department of Mathematics, University of California at Riverside [e-mail] Loosely speaking, an 'n-category' is a structure with objects, morphisms between objects, morphisms between morphisms or '2-morphisms', and so on up to n-morphisms. While only well-understood for low n at present, n-categories appear to provide a framework for understanding 'processes between processes', thus making precise certain analogies between logic, topology, and physics. In categorical logic, the process of 'weakening' consists of replacing equational laws by specified natural isomorphisms, which then must satisfy certain equations of their own to be manipulated with some of the same facility as the original equations. (For example, the product of real numbers is strictly associative, while the tensor product of vector spaces is only associative up to natural isomorphism.) To iterate this process we need some notion of n-category. On the other hand, homotopy theory may be regarded as the study of the n-category associated to a space whose objects are points of the space, whose morphisms are paths, whose 2-morphisms are paths between paths, and so on. The n-categorical coherence laws arise naturally from the topology in this context. Similarly, in topological quantum field theory, n-dimensional cobordisms may be represented as n-morphisms, and important equations such as the Yang-Baxter and Zamolodchikov equations can also be seen as n-categorical coherence laws. Here we sketch some of the main ideas involved, illustrating them with many examples. For references, see [5, 6J. Complexity of trajectories in rectangular billiards Yuliy Baryshnikov, School of Mathematics, University of Hull [e-mail] Take a cube as a billiard domain and a generic trajectory in it. Apparently, nothing more simple can be imagined in billiards but even here some complexities hide. Write down the number i each time the trajectory hits a wall parellel to i-th coordinate hyperplane and count the number of different words of length n in the resulting infinite word. This number as function of n is called the complexity of the trajectory. A classical result by Morse and Hedlund, [55J, is that the complexity in dimension 2 does not depend on trajectory (and equals to n + 1). This was generalized recently, [1], to three dimensions. I prove the independence in all dimensions and find the function, [7J. The infinite words arise actually in great many areas like computer sciences (digitalized straight lines, finite languages), game theory and quasicrystalls. The proof hints at some possible underlying structures like combinatorial theory of infinite-dimensional polytopes or multidimensional generalisations of chain fractions. Discrete computations and Hamiltonian and gradient flows Anthony M. Bloch, Department of Mathematics, University of Michigan [e-mail] In this talk I will discuss some of the connections between the theory of gradient flows and integrable Hamiltonian systems and various discrete computations. In particular I will discuss how problems of least squares fitting, the sorting of numbers and linear programming can be couched as discrete versions of smooth Hamiltonian and/or gradient flows. I will explain a natural dualism that arises between gradient flows and integrable or solvable systems and the importance of the geometry of the moment map. The latter gives rise to a convex polytope which plays a crucial role in the computations and especially in understanding linear programming by interior point methods. I will discuss some of the theory of integrable systems with particular reference to the Toda lattice system which plays a key role in this work. I will also discuss some extensions of these ideas to a