2.Diverse ways of sampling, allocation, models. 3.Random variables, expectation, moments. 4.Important distributions. 5.Limit theorems: law of large numbers, central limit theorem. 6.Random walks and Markov chains. 7.Statistical independence in analysis and number theory.
Page 1 2015-2016 Math 5540H Mathematics 5540H Honors Differential Geometry Spring (even numbered years) 5 credits Catalog Description: Geometry of curves and surfaces in 3-dimensional space, curvature, geodesics, Gauss-Bonnet Theorem, Riemannian metrics. Prerequisite: C or better in 5520H, or in both 2182H and 2568; or credit for 520H, or in both 263.01H and 568; or permission of department. Text: Text vary, for example: Differential Geometry of Curves and Surfaces, DoCarmo Elements of Differential Geometry, R. Milman and G. Rarker Topics List: 1.Geometry of curves; Frenet-Serret equations. 2.Curvature of surfaces, First Fundamental Form, Gauss's Theorema Egregium. 3.Geodesics, exponential map. 4.Isometries, conformal mappings; mapmaking. 5.Gauss-Bonnet Theorem. 6.Riemannian metrics, non-Euclidean geometry.
Page 1 2015-2016 Math 5576H Mathematics 5576H Honors Number Theory Autumn (odd numbered years) 5 credits Catalog Description: Elementary analytic and algebraic number theory, tracing its unifying role in the development of mathematics through history. Prerequisite: C or better in 4182H, or in both 2182H and 3345; or credit for 264H, or for both 263H and 345; or permission of department. Purpose of Course: The intention of this course is to present number theory, the "Queen of Mathematics" through its historical development. Being one of the oldest mathematical disciplines, number theory, in the course of its history, both benefited from and contributed to such major mathematical areas as geometry, algebra and analysis. These courses will be especially beneficial for honor students planning to pursue careers in mathematics, physics, computer science and education, but may be of interest to engineering students as well. Text: Vary, for example: An Introduction to the Theory of Numbers, 6thedition, by Hardy, Wright, Heath & Brown, published by Oxford, ISBN: 9780199219865. An Introduction to the Theory of Numbers, I. Niven, H.S. Zukkerman, H.L. Montgomery Number Theory: An Introduction to Mathematics, Parts A and B, by William A. Coppel, Springer-Velag. Topics List: 1.Review of Egyptian and Mesopotamian Mathematics. Greek tradition. Three classical Greek problems (cube doubling, angle trisection, circle quadrature). 2.Famous irrationalities. 3.Continued fractions and applications thereof (quadratic surds, Pell’s equation, Diophantine approximations, etc.) 4.More on diophantine approximation. Algebraic numbers. Liouville numbers. A glimpse into the Thue-Siegel-Roth Theorem. 5.Uniform distribution modulo one. Weyl criterion. Some important sequences. Pisot-Vijayaraghavan numbers. Formulation and discussion of Margulis’ solution of Oppenheimer’s conjecture.6.Normal numbers. Champernoun’s example. Almost every number is normal. Levy-Khinchine Theorem on normality of continued fractions.