Mathematical Diversity and Elegance in Proofs: Who will be the next Renaissance man?

It has been claimed by some mathematicians that Bernhard Riemann was the last great ”Renaissance man” of mathematics, in that he was quite knowledgeable of–and, indeed, proved major theorems in–several seemingly disjoint branches of the subject. One could argue for other notable exceptions from the 20th century–perhaps John von Neumann, Bertrand Russell, even Douglas Hofstadter–but the fact remains that it has become genuinely more and more unlikely (converging towards impossible, even) to hold this polymathic position. Consider this hour-long seminar my humble attempt to help us all along the path towards this esteemed and noble title. By the end of the hour, we will have proven some interesting and fundamental results in number theory, graph theory, geometry and combinatorics, using techniques from topology, probability theory, linear algebra, and analysis! Nothing will be particularly difficult or advanced; rather, we seek to celebrate the diversity of mathematics and the beauty and elegance inherent to some theorems and their proofs. “If only I had the theorems! Then I should find the proofs easily enough.” —Bernhard Riemann “I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations.” —David Hilbert