Hi! Welcome to my site!

I have been teaching mathematics full-time at the Loudoun campus since August 2007, and I am also completing my Ph.D at the University of Virgnia. I have previously held teaching appointments at Virginia Commonwealth University and J. Sargeant Reynolds Community College. I am originally from Egypt, particularly from a region called Shebeen al-Koum (one hour from Cairo), and I speak Arabic and English fluently.

Among my other mathematical interests, I am intrigued by the contribution of Arab/Persian mathematicians, beginning with Muhammad ibn Musa al-Khawarizmi, Abu Kamil, Omar al-Khayyam, and others. The research literature widely suggests that their contribution to mathematics was minimal. We claim that 9th century Arab and Persian mathematicians not only developed a new mathematical discipline (algebra), but also introduced an entirely new way of "thinking" about mathematics. This line of reasoning focused on analytical (i.e. quantitative) methods for exploring mathematics, and thus problems were no longer considered strictly from a geometrical perspective. This development has allowed mathematics to be analyzed from a much more abstract level (i.e. algebra, the calculus, analysis, abstract algebra, topology, etc.) as most problems cannot be represented pictorially.

I really enjoy studying pure mathematics and in particular the areas of abstract algebra, topology, real and complex analysis. My primary focus lies in the area of Galois Theory and group theory.

Galois Theory is a topic studied in abstract algebra, and the fundamental theorem of Galois Theory roughly states that there is a one-to-one correspondence between the intermediate fields of a finite field extension and the corresponding subgroup of automorphisms. Galois Theory can be used to prove that any polynomial with real coefficients of degree 5 or greater is generally unsolvable by radicals. So, Galois Theory gives a nice relationship between groups and fields.

In real analysis, I am interested in the theory behind the construction of the Riemann integral and extending this to the Lebesgue integral. The Riemann integral has many restrictions, in particular functions must be integrated on closed and bounded intervals, limit and integral cannot generally be exchanged, certain discontinuous functions such as the Dirichlet function cannot be integrated by Riemann, and the lack of nice convergence theorems. The Lebesgue integral, however, overcomes most of these restrictions. One first develops the Lebesgue integral by defining a measure (generalizing the notion of length). The most important properties are the convergence theorems: Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem.

Also, I like the notion of the "hierarchy of infinite sets." For example, there are more complex numbers than real numbers, and more real numbers than integers, but there are equally as many integers as positive integers as rational numbers as prime numbers. It was Georg Cantor who introduced the study of cardinal sets and this "hierarchy of infinite sets." By definition, any set that has the same cardinality as the positive integers is infinitely countable. A remarkable connection can be made between cardinal sets and Riemann integrability: If a function has at most a countable number of discontinuities on a closed and bounded interval, then the function is Riemann integrable on the interval. There is a stronger version of this theorem which states that a function is Riemnann integrable on a compact interval if the number of discontinuities on the interval has meaasure zero.

SPRING 2010 COURSES

Math 271 Applied Calculus I
(TF 8:00 - 9:15)   LW 212

Math 241 Statistics I
(TF 9:30 - 10:45)   LR 106

Math 163 Precalculus I
(T 7:00 - 9:40)   RES2 354

Math 151 Math for Liberal Arts I
(M 7 - 9:40)   LW 103

Math 03 Algebra I
(Online Course)

SUMMER 2009 COURSES

Math 174 Calculus with Analytic Geometry II
(TR 7:00 - 10:10)   LR 106

Math 241 Statistics I
(MW 6:30 - 10:10)   RES2 307

Math 285 Linear Algebra
(MW 10:00 - 11:40)   LW 107

Math 299 Introductory Abstract Algebra and Real Analysis
(W 4:00 - 5:10)   LW 107

FALL 2009 COURSES

Math 241 Statistics I
(TF 8:00 - 9:15)   LW 104

Math 151 Math for Liberal Arts I
(TF 9:30 - 10:45)   LR 106

Math 152 Math for Liberal Arts II
(TF 11:00 - 12:15)   LR 269

Math 163 Precalculus I
(M 7:00 - 9:40)   RES2 349

Math 03 Algebra I
(Online Course)