Lectures on dynamics, fractal geometry, and metric number theory

Dynamics , Geometry and Number Theory

We define the toplogical generating rank dtop(G) of a topological group G as the minimal number of elements of G which generate a dense subgroup of G. I will present joint work with Gena Noskov, partly in progress, about our attempts to compute the topological generating rank for connected Lie groups. The cases that G is semisimple, where dtop(G) is slightly bigger than one, and that G is nilpo...

Lectures on Analytic Number Theory

1. What is Analytic Number Theory? 2 1.1. Generating functions 2 1.2. Operations on series 3 1.3. Some interesting series 5 2. The Zeta Function 6 2.1. Some elementary number theory 6 2.2. The infinitude of primes 7 2.3. Infinite products 7 2.4. The zeta function and Euler product 7 2.5. Infinitude of primes of the form 4k + 1 8 3. Dirichlet characters and L functions 9 3.1. Dirichlet character...

Lectures on Topics in Algebraic Number Theory

2 Preface During December 2000, I gave a course of ten lectures on Algebraic Number Theory at the University of Kiel in Germany. These lectures were aimed at giving a rapid introduction to some basic aspects of Algebraic Number Theory with as few prerequisites as possible. I had also hoped to cover some parts of Algebraic Geometry based on the idea, which goes back to Dedekind, that algebraic n...

Geometry and Number Theory on Clovers

1. INTRODUCTION. In 1826 Abel discovered that the lemniscate, the curve (x 2 + y 2) 2 = x 2 − y 2 pictured in Figure 1, can be divided into n arcs of equal length by straightedge and compass if and only if n is a power of 2 times a product of distinct Fermat primes [1, p. 314]. By an earlier theorem of Gauss, these are exactly the values of n for which a regular n-gon is constructible by straig...

Lectures on Symplectic Geometry