Primes are not dense in the integers, but they are dense in the almost primes, which obey a good random model. The combination of these three facts gives the desired result: Dense sets of integers contain progressions (by Szemeredi’s theorem). Tao gave a rough sketch of the proof of this theorem, and in it he mentioned that one basically shows first that A can be modeled by some random model, and then that every random model of a dense set must predict the existence of arithmetic progressions. positive upper density) set of integers A contains arbitrarily long arithmetic progressions. The third and final step comes from implementing Szemeredi’s Theorem, which basically says that a “reasonably large” (i.e. by the time one gets to the almost primes - numbers with few prime factors or no small factors) gives one good control and a decent random model. By the time one sieves down to the prime numbers, too many error terms have accumulated. Tao explained the problems with this idea by talking about the simplest sieve, the Sieve of Erathostenes. This is where the idea of sieving comes in: if one could somehow filter out all the numbers that have “exotic properties” then one would get a set of prime numbers and maybe a random model for them. An obvious example of this is that odd numbers are more likely to be primes than even numbers. Now a problem arises from the fact that all numbers are not equally likely to be primes. In fact, using the prime number theorem one can determine the probability that a random number less than N will be prime. First of all, said Tao, it is reasonable to think that prime numbers are distributed somewhat randomly. The proof of the Green-Tao theorem contained three basic ingredients. Examples are the Twin Prime conjecture and Goldbach’s conjecture. Tao explained that this result is particularly interesting because many questions in additive prime number theory – which is concerned with finding additive patterns in the primes – are unsolved and have remained that way for (in some cases) hundreds of years. Their theorem states that the set of all prime numbers contains infinitely many arithmetic progressions of arbitrary length. This work was in fact a contributing factor to Tao receiving the Fields Medal in 2006. Structure and Randomness in Prime Numbers, by Terence Tao (UCLA)Ī crowded room gathered to listen to Terry Tao talk about his recent work with Ben Green on additive prime number theory. "It's like riding a horse without a saddle." Instead, mathematicians are dealing directly with data from science and engineering. "This is the other side, it's not the mathematics of physics," Glimm said. The work described in this special session could be the start of a new approach to research. He discussed new alternative models that are emerging, such as the HOT models (heuristically optimized topologies), which take into account network engineering and architecture. The problem was that those models were created without any attention paid to available data. Willinger said that the initial models of the internet coming out of graph theory, such as the preferential attachment model, do not reflect how the internet really operates. The lecture by Walter Willinger of AT&T Labs-Research focused on modeling of the internet. Smale spoke on a new theory he has been developing to understand how, for example, babies learn to identify visual images and to relate them in the correct way even when they have been exposed to a very few examples. One of the techniques, called "diffusion geometry," works better in many cases than does the traditional statistical tool of principal component analysis. Jones and his Yale colleague Raphael Coifman gave lectures in which they described techniques for creating local coordinate systems that can highlight salient aspects of large data sets. The new applications require quite different approaches and center on managing, organizing, and visualizing large and complex data sets. In the past, applications of mathematics have been dominated by physics problems, with differential equations being among the main tools. The purpose of the session was to focus on the new ways that mathematics is being used in this era of data-rich science and engineering. Jones of Yale University and Steve Smale of the Toyota Institute of Technology at Chicago. Invited Addresses, Sessions, and Other ActivitiesĪMS President James Glimm spearheaded a special session at the Joint Meetings called "The Mathematics of Information and Knowledge." The session was co-organized with Peter W.
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