Lecture 1

Title: Pseudorandom Vectors Generation Using Elliptic Curves Over Finite Fields

Abstract: Using the arithmetic of elliptic curves over finite fields, we present an algorithm for the efficient generation of sequence of uniform pseudorandom vectors in high dimension with long period, that simulates sample sequence of a sequence of independent identically distributed random variables, with values in the hypercube $[0,1]^d$ with uniform distribution.

Lecture 2

Title: Discrete Time Simulation of Sequence of Independent Wiener Processes

Abstract: As an application of the pseudorandom vector construction, we describe an algorithm for the construction, of discrete time simulation of uniformly distributed sample path sequence, of a sequence of independent standard Wiener processes (uniform distribution with respect to discrete time simulation of Wiener measure). This has application to the numerical evaluation of Feynman-Kac formulas.

Lecture 3

Title: Discrepancy, Exponential Sums, and Pseudorandomness

Abstract: We justify (at least in a special case) the pseudorandomness of the vectors as described in Lecture 1, by giving explicit discrepancy estimates. The main input is bounds on certain kinds of exponential sums on elliptic curves over finite fields, which is a consequence of the Riemann hypothesis for algebraic curves over finite fields.

Lecture 4 (Expository talk)

Title: Euler, Infinite Series, and Number Theory

Abstract: In this lecture we discuss Euler's work on infinite series, how it shaped the development of modern number theory, particularly concerning zeta functions and modular forms.