Technion Math Club
All the lectures are in Hebrew unless stated otherwise
- Wednesday, December 31, 2008
Irad Yavneh (Technion)
Title: Multiscale Computation
16:30 // Taub 2 (computer science) - Wednesday, April 22, 2009
Uri Bader (Technion)
Title: The Miracle of the Loaves and Fish
16:30 // Location: Amado 233 - Wednesday, May 20, 2009
Leo Corry Tel Aviv University
Title: Einstein Meets Hilbert on the Way to General Relativity: Who Arrives First?
16:30 // Location: TBA - Wednesday, June 24, 2009
Robert Adler (Technion)
Title: Making Decisions on the Basis of Shape
16:30 // Location: Amado 233
Abstracts
Title: Multiscale Computation
Abstract: What do the following two problems have in common?
(1) Getting a bunch of soldiers to stand in a straight line.
(2) Ranking internet pages by relevance.
A: Both can be solved efficiently by multiscale methods.
Multiscale methods are well-established as an important approach for solving various types of large problems in science and engineering. Based on these two problems – the first borrowed from Alfred M. “Freddy” Bruckstein and the second from Sergey M. Brin and Lawrence E. “Larry” Page – this talk will strive to expose the immense promise and grave challenges associated with multiscale computational techniques.
Title: The Miracle of the Loaves and Fish
Abstract: Recently we came to familiarity with Ponzi schemes – a type of fraud that is doomed to be collapsed eventually because of the finiteness of population/resources/money. Mathematical objects are often infinite, though, and the Ponzi dream might very well come true. In 1924 Banach and Tarski, following Hausdorff, showed that one can decompose a sphere into finitely many pieces and reconstruct from these pieces two copies of the same sphere.
This is now called the Hausdorff-Banach-Tarski paradox, though it is not a paradox – it is a mathematical fact! It shows, among other things, that the collection of subsets of the sphere is too rich to support a reasonable preservation low. This insight is one of the cornerstones of measure theory. Some mathematical objects do admit preservation laws, they are called “amenable”. Others do not. The definition of amenability of groups was given by Von-Neuman, following the HBT paradox. It influenced many branches of mathematics ever since.
Title: Einstein Meets Hilbert on the Way to General Relativity: Who Arrives First?
Abstract: On November 25, 1915, Albert Einstein presented to the Berlin Academy of Sciences the explicit, complete, and correct, generally covariant field equations of gravitation, lying at the heart of his General Theory of Relativity. This was the fourth consecutive week in which he presented, at the weekly meetings of the Academy, what he believed to be the culmination of many years of intense efforts to generalize his principle of relativity so as to cover cases of relatively moving reference systems more general than the inertial ones, and so as to apply to gravitation. In the previous three opportunities, he soon realized that the version he had just presented was still in need of further improvement. After his talk of November 20, Einstein was euphoric about his achievement, which he was ever to consider the most important one of his entire scientific career.
Five days earlier, on November 20, David Hilbert presented in Gottingen his own version of the equations that, in the published version that appeared in print several months later, contained the correct and explicit equations of the theory. According to a view that was commonly accepted for many years, Hilbert had anticipated Einstein in five days incorrectly formulating this important part of the latter’s work. Recent archival research, however, has shown that this was not really the case, and the actual historical situation was much more complex.
Einstein had visited Gottingen in the summer of 1915 and there he presented the current state of his theory, including the problems that were yet to be solved before it could be considered to be complete. Einstein was absolutely delighted by the audience, one of the few in the world that included a considerable number of people who could follow his ideas, ask difficult questions and perhaps even help him overcome some of the difficulties. Above all, the presence of Hilbert in the audience was of great impact for him. A strong personal and scientific friendship developed. Over the following months, and especially over the crucial months of November, they corresponded intensely. This correspondence has been thoroughly analyzed as part of the story of their encounter and their respective talks in November. As this interesting episode involves two of the foremost scientists and one of the most momentous developments of the beginning of the twentieth century, the discovery and publication of the relevant documents aroused much interest and debate within historians of science as well as in broader circles. Still, despite the intrinsic curiosity that it arises, priority disputes (“Who arrived first?”) are not the kind of foremost questions that attract the interest and research efforts of historians of science. This is also the case in the story of Hilbert and Einstein.
The main point of historical interest in the story concerns the first half of the title of this talk: “Einstein meets Hilbert on the Way to General Relativity”. Here we have these two prominent scientists
working out their long-term research programs, each one with his specific aims in mind, working within different disciplines, using different methodologies, and based on different bodies of knowledge as their respective backgrounds. Their scientific and personal encounter raises important and difficult historical questions: What was the essence of their research programs? Where was each of them heading with their work? How did their programs meet and what kind of interaction ensued? How this influenced each of them and their programs? Where did the meeting ultimately lead to?
My talk will address all of these questions and will attempt to provide some of the answers that recent research by several historians has brought to light.