Meets Fridays at 2:30pm in CW 130 unless otherwise indicated
Friday, February 3, 2006
Speaker: Sergiy Koshkin
Title: Energy and helicity of knotted vector fields
Abstract: I will introduce a measure of linking between flow lines of a vector field called helicity. Then I will focus on two special types of fields where the meaning of helicity is especially transparent. First is 'fluid knots' that appear in dynamics of ideal fluids and magnetic fields in plasma and constitute the subject of a fast growing field called geometric knot theory. Second is curvature fields of sphere-valued maps that are used to describe interactions of subatomic particles and for which helicity turns into the Hopf invariant. I will also discuss a variational problem of minimizing field energy with fixed helicity and describe the 'knotted solitons' that appear as its solutions. The talk will be accessible to undergraduate students.
Title: Transfinite Induction I
Abstract: The sound of it might send chills down your spine, but transfinite induction is nothing to be afraid of. In fact, it is quite intuitive and easily grasped. I will explain transfinite induction and a little bit about ordinals.
Title: Transfinite Induction II
Abstract: After a brief review, this talk will pick up where the previous one left off. As soon as the necessary concepts are in place, transfinite induction will be used to prove Zorn's lemma from the axiom of choice.
Title: Minimal genus surfaces bounding knots in 4-manifolds
Abstract: Twisting is an operation one can perform on knots or links. This talk will use twisting to obtain bounds on the genus of spanning surfaces to knots. It will also show that the 10_99 knot is not equivalent to a trivial link by a sequence of special local moves.
[Note: This is a joint meeting with the Graduate Student Topology Seminar.]
Title: Ordinal arithmetic
Title: Ordinal arithmetic II
Abstract: This is a continuation of the speaker's previous talk.
Title: A Local Extrapolation Method for 2D Hyperbolic Conservation Laws
Abstract: The main idea of the LEM is to use a combination of the basic method with different step-sizes to eliminate the principle error term of the underlying scheme. It has proven to be a useful method to increase the order of accuracy of the underlying scheme in 1D cases. We will extend LEM for upwind underlying schemes to 2D linear hyperbolic conservation laws.
Title: Cohomology of quantum groups
[Note: This is a joint meeting with the Algebra Seminar.]