I will introduce contact structures, describe the basic dichotomy of tight versus overtwisted contact structures and give some classification results. If time permits, I will outline some deeper connections between the contact geometry and the topology of the underlying manifolds.
I will introduce contact structures, describe the basic dichotomy of tight versus overtwisted contact structures and give some classification results. If time permits, I will outline some deeper connections between the contact geometry and the topology of the underlying manifolds.
The factorial function is ubiquitous in mathematics. So why restrict it to just the integers? I will describe Manjul Bhargava's generalization of the factorial function. We will then see some examples and applications (number theoretic and otherwise) which will provide evidence that this is the "correct" generalization.
I will present E. Artin's classical answer to Bhargava's 'generalization' of the factorial function. It will also be a kind of teaching experiment for me, be warned. At the end, because I think I may finish early, perhaps we can discuss possibilities for extending some of the theorems in Artin or Borges' "The Library of Babel." And I should of course mention: pizza.
I will begin with a quick survey of some major results in the ergodic theory of (real, semisimple connected) Lie group actions. I will then talk about Lie groups over local fields. A natural way to study group actions in this setting is to consider flows on \Gamma/X where X is the associated Bruhat-tits building, and \Gamma is a lattice. I will try to explain the geodesic and horocycle flow in this setting. As usual, everything will be defined and this assumes no pre-requisites.
Sphere packing problems have been considered for centuries, yet many have resisted solution until the advent of computers. Why are solutions so difficult to obtain? That is but one of many questions that will be avoided during this talk. Instead, I will briefly describe a few solved and unsolved problems involving the kissing number and density of regular and irregular sphere packings in various dimensions, finite and infinite. Familiarity with Euclidean space, unit spheres, and pizza will be assumed. (Familiarity with sausages will not.)
P.S. It seems that we are not approaching a solution to the graduate student packing problem: what is the maximum number of graduate students that can be packed into a graduate student seminar?
Is it possible to tile a square with smaller squares, all of different sizes? The corresponding problem for cubes is impossible. This seemingly simple question in elementary geometry generated a lot of interest earlier this century. It was known that a 33 x 32 rectangle could be tiled, as well as a 177 x 176 rectangle. I'll show how you can construct such tilings using electical networks, and settle the question for squares.
The talk should be accessible to high school students, but will hopefully be interesting to grads.
Small quantum cohomology is a deformation of ordinary cohomology and has a lot of interesting and important applications in mathematics and physics. I will talk about the basics of the general theory and step by step computations of small quantum cohomology of complex projective space. As time permits I also compute the equivariant quantum cohomology of the same space.
A little knowledge of algebraic geometry will be enough for this talk.
The theory of Fewnomials gives an effective generalization of real algebraic geometry. I will outline results from real algebraic geometry, define fewnomials, and then state the fewnomial results before stating some open problems. Come with guesses for just what fewnomials are! Or just come to hear about Prince Khovanskii, laugh at my bad drawings, and eat pizza.
The standard way that people shuffle a deck of playing cards is what Bayer and Diaconis call a 2-riffle shuffle, or a dovetail shuffle. Question: how many times must one shuffle a deck of 52 cards so that the deck is 'random'? According to their work, the answer is about 8.55, but the more famous answer is that seven shuffles is usually 'good enough'. I will survey their results and describe the connection between shuffling and descent algebras.
Every graduate student of Mathematics is invited (encouraged!) to give a talk. You most certainly do NOT have to present original work. The talks should be accessible to any bright graduate student, at any level of their studies. You can present a topic that most people may not have heard about, or things people may have heard about but have no idea what it is.
Come to hear new topics, learn new methods, build camaraderie with your fellow graduate students... and most importantly - eat some free pizza!
Page created: Sep. 7th, 2003
Last updated: Sep. 6th, 2004
Maintained for the 2004-2005 academic year by Becci Torrey (rtorrey@brandeis.edu)