Summer 2025
Math Workshops and Classes
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The American University in Cairo | Department of Mathematics and Actuarial Science
Algebraic geometry is a branch of mathematics that studies algebraic varieties. A typical question would be to classify all curves in the plane which are the solution sets of equations like P(x,y)=0 where P(x,y) is a polynomial in two variables, up to a certain notion of equivalence.
Being a very old subject, many tools have been developed over the years in order to answer some of those questions (and to introduce new ones). Sheaves and sheaf Cohomology have been brought into algebraic geometry by Cartan and Serre, who created a new language that is extremely useful in the study of algebraic geometry. We shall follow a hybrid approach, focusing on the geometry of complex projective varieties.
As of today, it is clear that algebraic geometry is a beautiful and a useful subject, that has applications in many areas of science such as Mathematical Physics and Computer Science. The goal of this workshop would be to survey some of the methods of the algebraic geometry.
In one of the analysis mini-courses, 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.
In the other mini-course, we discuss limit theorems in probability. 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.
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