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They have a rich and interesting geometry and play a elementary function in a lot of the most important latest developments in quantity theory. A number of subjects from such areas as graph principle, combinatorial algorithms, enumeration, building of combinatorial identities. E. C. Titchmarsh, The principle of the Riemann zeta-function, 2nd ed, (revised by D. R. Heath-Brown) Clarendon Press, Oxford 1986.

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Renormalization has turn into a central philosophy in low-dimensional dynamics. Among its many applications are Kolmogorov-Arnold-Moser theory, interval exchange transformations (where it is recognized as Rauzy-Veech induction), Feigenbaum-type universality, and even fluid dynamics. This course will serve as an introduction to renormalization magic, using the vital thing instance of analytic homeomorphisms of the circle. The prerequisites are a standard course in advanced analysis, and some primary familiarity with the notions of discrete-time dynamics. Contact geometry is a kind of geometry current in all odd dimensions, intently tied to symplectic geometry, and with quite a few relationships to 3-manifold concept, complicated geometry, and mathematical physics.

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