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Mathematics - University of Connecticut
Heuristics for Prime Statistics AbstractIt's been known since Euclid that there are infinitely many prime
numbers. But more refined questions about primes quickly lead into
unsolved problems.
For instance, how often should we run into primes
of the form n2 + 1? Or how often should we expect to
find twin primes (primes differing by 2, such as 3 and 5 or 101 and 103)?
Questions like these are still open, but probabilistic heuristics
suggest what the answers should be. We will discuss these
heuristics, how they stand up to the numerical evidence, and why
anyone would care about these questions in the first place.
Links: Slides
References:- S. Lang, Math Talks for Undergraduates, Springer-Verlag, New York,
1999; Chapter 1
- P. Ribenboim, The New Book of Prime Number Records, Springer-
Verlag, New York, 1996; Chapter 6
- W. Narkiewicz, The Development of Prime Number Theory: from Euclid
to Hardy and Littlewood,
Springer-Verlag, Berlin, 2000; Section 6.7.
Geophysics - MIT
Measuring Seasonal Changes on Mars Astronomy - Wellesley
Probing the Atmosphere of a Red Supergiant Star with the Hubble Space Telescope AbstractI'll spend about half the talk on the background material of stellar evolution (especially that they lose mass in the red giant phase), and then talk about my research program which uses ultraviolet spectra obtained by Hubble. A hot companion goes into eclipse behind the supergiant, and as it emerges from eclipse, its orbital motion provides a probe of the structure of the supergiant's atmosphere as we observe the effect of the atmosphere on the light passing through it.
Links: Sky and Telescope Magazine, VideoGeology - Brown University
Deep Impact AbstractThe Deep Impact mission collided with a comet on July 4 (2005). This was a unique NASA mission that positioned a probe so that it could collide. The results of this large cratering experiment are providing the first data for interpreting the inside of a medium-age comet. Recently the team reported the first discovery of icy dirt in a few large areas. The presentation will briefly describe some of the first results.
References:- An article by A'Hearn et al concerning the Deep Impact Mission that can be found at either of the following two sources: Science, October 14, 2005, pp. 258-264, and Sky and Telescope (October 2005).
Mathematics - Brown University
Angle defects, total curvature, and the polyhedral Gauss-Bonnet theorem AbstractThe celebrated Gauss-Bonnet theorem relating the curvature of a surface to its topology represents a fundamental tool in differential geometry. In this talk I will discuss a more first-principles "polyhedral" version of this theorem using simple notions from high school geometry, and some basic elements of combinatorial topology (no integration required). I'll then show why most good things in math, as in life, are hyperbolic.
Links: VideoAstronomy - Brown University
Using Gravitational Lensing to Measure Dark Matter and Dark Energy AbstractIn the last decade, we have come to realize that the primary components of the Universe are very different from the matter we are familiar with. The dominant components today are dark matter (an unknown substance that behaves gravitationally like ordinary matter) and dark energy (an even stranger substance with a repulsive gravitational effect), in a roughly 1:3 ratio.
Measuring the distribution in space and the time evolution of the energy densities of dark matter and dark energy will be key to understanding what they are, and one of the most promising ways to address this issue is gravitational lensing.
I'll review the technique of weak gravitational lensing and show how it can be used to measure dark matter and dark energy. I will also present the state of current experiments in this field, and discuss the prospects for advances in the next decade.
Links: VideoMathematics - Brown University
Triangular Billiards Abstract I plan to talk about what happens when you play
billiards on a triangular pool table with a frictionless
infinitesimally small ball. Amazingly, it has been unknown
for 200 years if one can find, for any shaped triangular table,
a way to hit the ball so that it endlessly traces out the
same path. This has seemed to be a completely intractible
problem in dynamics but I will talk about some progress that
I have made on it in the past year using computer-aided
methods. For a lot of my talk I will demonstrate
McBilliards, a videogame-like computer program made by myself and Pat Hooper to investigate this problem.
Links: Video, The McBilliards Program
References:- "Billiards Obtuse and Irrational" by Richard Evan Schwartz
- M. Boshernitzyn, G. Galperin, T. Kruger, S. Troubetzkoy, "Periodic Billiard Trajectories are Dense in Rational Polygons," Trans. A.M.S.
350 (1998) 3523-3535.
- E. Gutkin, "Billiards in Polygons: Survey of Recent Results," J. Stat.
Phys., 83 (1996) 7-26.
- G.A Galperin, A. M. Stepin, Y. B. Vorobets, "Periodic Billiard Trajectories in Polygons," Russian Math Surveys, 47 (1991) pp. 5-80.
- W.P. Hooper, "Periodic Billiard Paths in Right Triangles are Unstable", preprint (2004).
- L.Halbeisen and N. Hungerbuhler, "On Periodic Billiard Trajectories
in Obtuse Triangles," SIAM Review, 42.4 (2000) pp 657-670.
- H. Masur, "Closed Trajectories for Quadratic Differentials with an Application to Billiards," Duke Math J. 53 (1986) 307-314
- H. Masur and S. Tabachnikov, "Rational Billiards and Flat Structures,"
Handbook of Dynamical Systems, 1A (2002) editors: B. Hassleblatt and A.
Katok.
- R. Schwartz, "Obtuse Triangular Billiards I: Near the (2, 3, 6) Triangle," Journal of Experimental Math (2005-6) to appear.
- R. Schwartz, "Obtuse Triangular Billiards II: Near the Degenerate
(2, 2,1) Triangle," preprint, 2005.
- R. Schwartz, "Obtuse Triangular Billiards III: 100 Degrees Worth of
Periodic Trajectories," preprint, 2005
- S. Tabachnikov, "Billiards," SMF Panoramas et Syntheses, 1 (1995)
- S. Troubetzkoy, "Billiards in Right Triangles", preprint 2004.
- W. Veech, "Teichmuller Curves in Moduli Space: Eisenstein Series and an Application to Triangular Billiards", Invent Math, 97 (1992) 341-379
Mathematics - Boston College
The Complex Structure of Elliptic Curves Abstract
Links: VideoMathematics - Harvard University
Hilbert's Third Problem AbstractIn 1900 Hilbert presented a list of twenty-three unsolved problems in a presentation of his personal view of the direction of mathematics. I will discuss the third of these problems: the question of whether given any two polyhedra of equal volume it is possible to dissect one into several polyhedra and rearrange it into the other.
Links: Video |
