Chaos is a real-world phenomenon that arises in many different contexts, making it difficult to tell exactly what chaos is. Yorke will give examples of the aspects of chaos.
James A. Yorke earned his bachelor's degree from Columbia University in 1963. He came to the University of Maryland for graduate studies, in part because of interdisciplinary opportunities offered by the faculty of the Institute for Physical Sciences and Technology (IPST). After receiving his doctoral degree in 1966 in Mathematics, Yorke stayed at the University as a member of IPST. Today he holds the title of Distinguished University Professor and also is a member of the Mathematics and Physics Departments.
Professor Yorke's current research projects range from chaos theory and weather prediction and genome research to the population dynamics of the HIV/AIDS epidemic. He is perhaps best known to the general public for coining the mathematical term "chaos" with T.Y. Li in a 1975 paper entitled "Period Three Implies Chaos," published in the American Mathematical Monthly. "Chaos" is a mathematical concept in nonlinear dynamics for systems that vary according to precise deterministic laws but appear to behave in a random fashion.
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In the world of discrete mathematics, we encounter a bewildering variety of topics with no apparent connection between them. There are block designs in combinatorics, finite projective planes in geometry, round-robin tournaments and map colorings in graph theory, (0, 1)- matrices in linear algebra, quadratic residues in number theory, error-correcting codes on the internet, and the torus at the doughnut shop.
In a paper by de Smit and Hendrik Lenstra (Notices of the AMS, April 2003), it is shown that well known mathematical results about elliptic curves imply that what Escher was trying to achieve in this work has a unique mathematical solution. This discovery opened up the way to filling the void in the print. With help from artists and computer scientists, a completion of the picture was constructed at the Universiteit Leiden. The white hole turns out to contain the entire image on a smaller scale, which in the Dutch language is known as the Droste effect, after the Dutch chocolate maker Droste. In the talk, the mathematics behind Escher's print and the process of filling the hole was explained and visualized with computer animations.
When you send your credit card number over the Internet, cryptography helps to ensure that no one can steal the number in transit. Julius Caesar and Mary Queen of Scots used cryptography to send secret messages, in the latter case with ill-fated results. More recently, cryptography is used in electronic voting, and it is also used to "sign" documents electronically. While cryptography has been used for thousands of years, public-key cryptography dates only from the 1970's. Some recent exciting breakthroughs in public-key cryptography include elliptic curve cryptography, pairing-based cryptography, and identity-based cryptography, all of which are based on the number theory of elliptic curves. This talk will give an elementary introduction to cryptography, including elliptic curve and pairing-based cryptography.
The notion of `holonomy' in mechanical systems has been around for over one hundred years and gives insight into daily operations as mundane as steering and parallel parking and in understanding the behavior of balls (or more general objects) rolling on a surface with friction. A sample question is this: What is the best way to roll a ball over a flat surface, without twisting or slipping, so that it arrives at at given point with a given orientation?