Pseudorandom Vectors Generation Using Elliptic Curves Over Finite Fields

Time:

9:50-11:25am; Sept.5,2022

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.

Discrete Time Simulation of Sequence of Independent Wiener Processes

Time:9:50-11:25am; Sept.6,2022

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.

Discrepancy，Exponential Sums, and Pseudorandomness

Time:9:50-11:25am; Sept.7,2022

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.

Euler, Infinite Series, and Number Theory(Expository talk)

Time:9:50-11:25am; Sept.8,2022

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.

Speaker Biography

莫仲鹏教授，中国香港中文大学数学系学士，美国哈佛大学数学系博士。2007-2010年在美国加州大学柏克莱分校数学系任访问助理教授，2010-2011年在中国香港中文大学数学系任助理教授，2011-2014年在加拿大麦克马斯特大学数学系任助理教授，2014-2016年在中国科学院晨兴数学中心任访问研究员，2016-2018年在美国普渡大学数学系任副教授，2018-2019年在中国台湾省中央研究院数学所任正研究员，2019-至今在苏州大学数学科学学院任特聘教授。他的研究兴趣是：代数数论、自守表示论和朗兰兹纲领，在Mem. Amer. Math. Soc.、Compos. Math.、Comment. Math. Helv.、Math. Res. Lett.等著名数学刊物发表论文10余篇。代表荣誉：加拿大安大略省，Early Researcher Award, 2012；美国李氏基金会，Li Foundation Heritage Prize, 2019；2020年入选江苏特聘教授；2020年入选国家级人才计划青年学者项目。