(1) Basic Hypergeometric series and Orthogonal Polynomials

Dr. Zeinab Mansour
Professor of Mathematics - Cairo University

Abstract.

In this mini-course, we begin with an introduction to the q-hypergeometric series, also known as basic hypergeometric series, and derive some elementary summation and transformation results. We then study the q-hypergeometric difference equation, focusing in particular on solutions expressed as power series expansions around 0 and ∞. Next, we introduce the student to the theory of orthogonal polynomials, where we study the existence of orthogonal polynomials via Favard’s Theorem and explore their three-term recurrence relations. We then consider the q-hypergeometric operator in a special case, and demonstrate that there exists a natural Hilbert space—a weighted sequence space—on which this operator is symmetric. The corresponding eigenfunctions turn out to be polynomials, specifically the little q-Jacobi polynomials.

Prerequisites.

Students are expected to be familiar with the following:

1. Sequences and series, power series, and radius of convergence.

2. Linear algebra, particularly:

• Linear independence,

• Orthogonality,

• Inner product spaces.

References.

1. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. (Republished: Dover Publications, Mineola, NY, 2011.)

2. E. Koelink, q-Special Functions, Basic Hypergeometric Series, and Operators, 2018. Available on arXiv: https://arxiv.org/abs/1808.03441

This Mini-Course's Lectures Notes.

(2) Limit Theorems In Probability

Dr. Hany El-Hosseiny
Emeritus Lecturer - Cairo University

Abstract.

Limit Theorems like the central limit theorem or the law of large numbers are basic results in probability. They have a very vast domain of application, and they are also important on the fundamental level as they give an understanding of what is the nature of probability.

In these lectures we give an informative introduction to limit theorems, explaining their nature, their proofs, and some of their applications.

What students should know beforehand

• What is a probability space?

• What is a random variable?

• What do we mean by the law of a random variable?

• Basic examples of probability laws, especially the Gaussian law.

• Expectation, variance, . . . , etc.

The following are some brief references for the main things you should know going into this mini-course.

• Preliminaries from Probability Theory.

• Preliminaries in Metric Spaces and Basic Analysis (this source in particular is most relevant for all our mini-courses, not just this one).

Please go through them, as their content will be assumed during our lectures.

New Lectures Notes:

• The Riemann-Stieltjes Integral.

• Modes of Convergence.

• Examples

• The Characteristic Function

(3) Singularity Formation

Dr. Tarek M. Elgindi (Guest Speaker)
Professor of Mathematics - Duke University

Important Information.

• Dr. Elgindi will be giving two lectures, one on Sunday the 17th and the other on Wednesday the 20th of August. Each lecture will start at 3pm and end at around 4:30pm.

• The lectures will be conducted on Zoom.

• We reserved a room on the AUC Tahrir campus to stream the lectures live. The room is Hill House 406, which is different from the ones we will be having our normal face-to-face lectures in. However, it is not mandatory to stay to attend the lectures in with us on campus.

• Students can look in advance at Sections 1,2, and 3 or the book "Singularities: Formation, Structure, and Propagation" by Eggers and Fontelos. https://doi.org/10.1017/CBO9781316161692.

Please find the link to Dr. Tarek's personal website here.