Math480/540, TOPICS IN MODERN MATH. What are numbers? Part II

1 1 Math480/540, TOPICS IN MODERN MATH. What are numbers?

Sums of squares, sums of cubes, and modern number theory

These are notes which grew out of a talk for general math graduate students with the aim of starting from the questions “Which numbers are sums of two squares?” and “Which numbers are sums of two cubes?” and going on a tour of many central topics in modern number theory. In the notes, I discuss composition laws, class groups, L-functions, modular forms, and elliptic curves, ending with the Birc...

Zermelo and the Early History of Game Theory

In the modern literature on game theory there are several versions of what is known as Zermelo’s theorem. It is shown that most of these modern statements of Zermelo’s theorem bear only a partial relationship to what Zermelo really did. We also give a short survey and discussion of the closely related but almost unknown work by König (1927, Acta Sci. Math. Szeged, 3, 121–130) and Kálmar (1928/2...

Avicenna's Anatomical Legacy as Seen Through the Relevant Topics in Modern Anat-omy

Background: Makhaarej Al-Horouf, the study of speech sounds by Avicenna is a valuable piece of work in the study of speech sounds, which was written about ten centuries ago. It contains six chapters on sound, anatomy of vocal tract, and phonetics. It is amazing to find that Avicenna’s explanations are congruent with the findings of modern scholarship in relevant topics. The study was intended t...

Math 710: Topics in Modern Analysis Ii – L-methods

1. January 3rd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Example 1: The Ohsawa–Takegoshi Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Example 2: The Hörmander(-Skoda) Theorem . . . . . . ...