Modulo Errors

Last summer I spent two weeks at the very rewarding Utrecht Summerschool in Mathematics, so I thought I’d spread the word about this year’s course. It’s entitled Topics in Algebra, Analysis and Geometry; analysis is a new inclusion this year (in place of number theory) and will be the main emphasis. Abstracts for the three courses are not yet available, but the titles are QRT and elliptic surfaces, Distributions,and Lie algebras and Integrable Systems.

As last year, the course runs for two weeks in August, with a fairly intensive schedule of lectures and problem classes; when I attended, the students also spent a couple of days preparing a presentation for the final day. The pace is reasonably demanding, and the ideal audience would be students just finishing undergrad and about to enter study for an MSc or PhD (although I went after a year of postgrad study).

There are also social activities organised by both the department and the university – there are fifty courses scheduled across the summer in a wide range of subjects, so you’ll have the opportunity to mix with students from outside of mathematics too. Utrecht itself is a beautiful city – night canoeing through the canals is highly recommended! – and daytrips further afield are also offered.

For further details on the summerschool programme, see here; specifics for the mathematics course are being made available on the department pages. I also took some photos during my stay. Feel free to leave any questions you have in the comments!

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Volterra’s principle resolves a seeming paradox in environmental control- that an attempt to eradicate a pest may increase pest levels, if the intervention also interferes with existing predators. This writeup considers two such examples- the cottony cushion scale insect in the USA, and fishing in the Adriatic Sea – and derives the principle mathematically through consideration of Lotka-Volterra differential equations for predator/prey interaction.

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Attempts to give an overview of how mathematicians deal with, or make use of, a notion of infinity. This is done through describing a series of mathematical ‘playgrounds’- groups, the real numbers, extended/hyper reals, polynomials, limits and projective geometry.

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Lyapunov (Liapunoff) stability is the standard notion of stability and a vital notion in the study of dynamical systems (including applications such as Mathematical Biology). It appears on the syllabus of many courses at Bath. This writeup covers the defintion (with plain-english interpretation), describes the direct method, and proves the Lyapunov stability theorem. Lyapunov functions and asymptotic stability are briefly mentioned.

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Describes the physical interpretation of the brachistochrone, a motivating example of the calculus of variations. Identification of a special case of that method, and its application to the problem.

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Application of the Baire category theorem to something more interesting than functional analysis- cardinality. Demonstrates the uncountability of the reals, and the incompleteness of the rationals.

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Three formulations of the IVT, with proof of equivalence and a non-mathematical explanation of the theorem.

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Techniques for visualising a 4-dimensional spacetime curve: streamlines, particle paths and streaklines. Example calculations for each, and an explanation of what each method demonstrates.

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Description of, and theorems on, conservative functions in vector calculus. Proves the equivalence (in simply connected domains) of being conservative, being irrotational, and having a scalar potential.