SGS XX: Sunday, April 29, 2012
The Georga Institute of Technology

Abstract: Branched covers are a useful method of constructing and understanding 3-manifolds. But how can they be used to construct contact manifolds? What happens to contact structures under branched covering maps? We will discuss recent progress in the construction of contact structures via branched covers, emphasizing the search for universal transversal knots. Recall that a topological knot is called universal if all 3-manifolds can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transversal knot in the standard contact structure on S3 from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.

Abstract: What does it mean to take a random closed polygon in space? What is the expected geometry of a random n-gon? This talk describes an attractive and natural measure on the space of length-2 closed n-gons which comes from a map originally constructed by Knutson and Haussmann from the manifold V_2(C^n)/SU(2) to the moduli space of length-2 closed n-gons in space up to the action of the Euclidean group E(3). This map pushes forward the standard Riemannian metric on the Stiefel manifold V_2(C^n) to a natural probability measure on polygon space. With respect to this probability measure, we can make a number of interesting explicit calculations for random polygons. For instance, the expected value of the radius of gyration of a random closed n-gon of length 2 in 3-space is 1/2n. This talk covers joint work with Tetsuo Deguchi (Ochanomizu University) and Clay Shonkwiler (UGA).

Abstract: (Joint with Pengfei Guan) In this talk, we will present a new type of mean curvature flow. For any closed star-shaped smooth hypersurface, this flow exists for all time t>0 and exponentially converges to a round sphere. Moreover, we will show that all the quermassintegrals evolve monotonically along this flow. Consequently, we prove a class of isoperimetric type of inequalities including the classical isoperimetric inequality on star-shaped domains. We will also present a fully non-linear analogue of this flow. More specifically, we study a fully non-linear parabolic equation of a function on the standard sphere and discuss its long-time existence and exponential convergence. As applications, we recover the well-known Alexandrov-Fenchel inequalities on bounded convex domains in Euclidean space.

Abstract: Capillary surfaces are interesting geometric objects which turn out to be important in microgravity environments (e.g. in space) and in tiny devices (e.g. electronic, "lab on a chip"). This talk will focus on the mathematical theory of capillary surfaces in vertical cylinders and sketch the 2010 proof of the 40 year old "Concus-Finn conjecture." Related open questions will be mentioned.

Abstract: It's now been 101 years since Otto Toeplitz asked the simple question: Does every closed curve in the plane with no self-intersections contain four points which form the vertices of an inscribed square? This talk will be an introduction for a general audience to the various mathematical ideas which prove and generalize parts of the square peg theorem, complete with some good pictures and animations. The key idea for undergraduates is that the simple act of solving a system of two equations in two unknowns (something that everyone who took high-school math has probably done) is actually the jumping off point for a wonderful journey into modern mathematics. Click here for the poster for this event.