P. Vishe
DescriptionLast few years have seen resolution of some of old and famous problems in number theory. The Oppenheim conjecture, formulated by Oppenheim in 1929 deals with representaions of numbers by real quadratic forms in several variables. More precisely, the question posed by Oppenheim was following:Let Q(x_1,...,x_n) be a quadratic polynomial in n variables and with real coefficients. Then given that this form is ''nice enough'', given any \epsilon>0, can you find integers x_1,...,x_n such that |Q(x_1,...,x_n)|<\epsilon? Namely, it asks whether 0 is approximated by values of an indefinate quadratic form evaluated at integer points? This is clearly a number theoretic problem and initial progress was made by the number theorists. However, the reality turned out to be quite different. This conjecture was famously resolved by G. Margulis in late 90's using dynamics on the space of lattices and the action of unipotent flows. This won him the Fields medal in 2000. This exciting project aims to study Marguils' proof of the Oppenheim conjecture. We will start by studying dynamics on hyperbolic spaces (Upper half plane (see picture above), SL(2,R), spaces of lattices in R^n), followed by learning about horocycle and geodesic flows on these spaces, the famous Ratner's theorem for unipotent flows leading upto the proof of Oppenheim conjecture. This process beautifully combines elements from both Algebra and Number theory as well as real as well as complex analysis. An affinity for Analysis and Number theory is a must. Prerequisites: Knowledge of Algebra 2 is necessary.
Knowledge of Analysis III/IV is recommended. Familiarity
of Geometry III/IV may be useful as well but not
essential. Textbooks: Y. Lima, Lectures on Ratner's Theory,
http://www.mat.ufc.br/~yuri/root_ratner.pdf M. Bachir
Bekka and Matthias Mayer, Ergodic theory and topological
dynamics of group actions on
homogeneous spaces, Vol. 269, Cambridge
University Press,2000 A. Katok,
B. Hasselblatt, Introduction to the Modern Theory
of Dynamical Systems,Cambridge University
Press, 1995.
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email: Pankaj Vishe