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Schedule - February 12, 2005

List of Faculty Speakers

• Steven J. Miller, Mathematics, Brown University
  From the Manhattan Project to Number Theory: How Nuclear Physics Helps Us Understand Primes (Abstract)

  A fundamental problem in science is to understand spacings between adjacent 'events', where the events range from prime numbers to energy levels of Uranium to eigenvalues of matrices to zeros of 'good' functions. Surprisingly, very diverse systems exhibit similar behavior. In particular, the techniques developed to understand Uranium can be used to study the primes! We'll explore some of these systems, and show the number theory analogues of the nuclear physics tools.

• Richard Stratt, Chemistry, Brown University
  Figuring The Odds On Molecular Motion (Abstract)

  In this talk, we'll explore how probability ideas can be used to understand molecular dynamics in liquids. We'll illustrate these concepts with an example or two chosen from my group's research.

• Jordan Ellenberg, Chemistry, Princeton University
  The Mathematics of SET, or Everything I Know About Combinatorial Geometry I Learned Playing Cards (Abstract)

  The game of Set is a simple but addictive card game played with a special 81-card deck. A standard "folklore question" among players of this game is: what is the largest number of cards that can be on the table which do not allow a legal play? I'll explain how this question, which seems to be about cards, is actually about a certain kind of geometry involving only finitely many points, and I'll talk about what's known about this problem and related ones. If there's time, I'll try to convince you that this problem might have something to do with "algebraic geometry" and if there's even more time, I'll try to convince you that, no, in fact, it really has to do with Fourier analysis.

• Thomas Griffiths, Brain Science, Massachusetts Institute of Technology
  Randomness and Coincidences (Abstract)

  How do people decide that something is random? What makes an event a coincidence? Psychologists, statisticians, and mathematicians often claim that people are bad at reasoning about chance. I will argue that, in fact, careful analysis of our statistical intuitions reveals that they are surprisingly sophisticated. I will discuss how questions about randomness and coincidences can be formulated from the perspective of Bayesian statistics (making connections to contemporary work on these issues in computer science), and describe some simple probabilistic models that can be used to understand human judgments./p>

• Mira Bernstein, Mathematics, Wellesley College
  Games, Hats, and Codes (Abstract)

  Consider two games:

  • (For two players) Start with several piles of pebbles. On her turn, a player removes any number of pebbles from any one pile. Whoever removes the last pebble wins. (This is a very famous game, called Nim.)
  • (For N players) Each player is given a black or white hat, randomly and with equal probability. Each player can see his teammates' hats but not his own. At a signal from the judge, all the players simultaneously guess their own hat colors; they are also allowed to pass and guess nothing. If there is at least one correct guess and no incorrect guesses among the players, the whole team gets a prize. Is there a strategy the players can decide on in advance to maximize their chances of winning? (Try it with N=3.)

  It turns out that the strategies for these two games are closely related to each other, as well as to the theory of error-correcting codes. We will see how the first game can be used to derive an optimal strategy for the second one, and then discuss the ways in which game strategies in general can lead to good codes.

• Mira Bernstein, Brain Science, Massachusetts Institute of Technology
  Probabilistic Models of Human Learning and Perception (Abstract)

  Computation in the brain is all about making smart guesses. In perceiving visual scenes based on retinal images, or learning new concepts from examples, there is never enough information in the inputs to deduce logically the correct structure of the world. Rather, the brain appears to have some expectations about the likely targets of learning or perception -- what kinds of concepts or visual scenes are most likely -- and then to use sophisticated forms of statistical inference in order to make the best possible guesses about what the world is like given the data it observes through the senses. In this talk, I will consider some simple examples of statistical inference in perception and cognition, showing how the basic mathematics of probability theory can be used to explain the remarkable guesses that your brain makes all the time without you having to "think" about it.

• Jimmie Doll, Chemistry, Brown University
  Noise, Random Processes, and Other Orderly Things (Abstract)

  There would appear to be an inherent human fear of disorder. Disorder, by its very nature, tends to be unpredictable and thus a bit dangerous. Perhaps as a consequence of this, science is often thought of (and presented) as the study of order. In the present talk, I would like to explore a number of advances that involve tackling the problem of "noise" in a direct manner. Examples ranging from Brownian motion, minimization theory, Monte Carlo theory, and numerical path integral techniques will be discussed

• Michael Rosen, Mathematics, Brown University
  How Do You Know When A Number Is A Prime? (Abstract)

  Given a large integer N, how can you tell whether N is a primes or not? This is not an easy question to answer. Surprisingly, this is not just an abstract mathematical problem, but one with practical applications. We will review some of the classical approaches to a solution and discuss the beautiful and totally unexpected paper of the Indian mathematician M. Agrawal and his two (undergraduate!) students, N.Kayal and N. Saxena, which gives an algorithm for primality testing which works in polynomial time.