Georgia Institute of Technology College of Sciences

Gordon and Luecke showed that the knot complements determine the isotopy classes of knots in S^3. In this talk, we will study the topology of various knot complements in S^3: torus knots, cable knots, satellite knots, etc. As an application, we will see some knot invariants using knot complements.

In 1963, Lieb introduced an effective theory to approximate the ground state energy of a system of Bosons interacting with each other via a repulsive pair potential, in the thermodynamic limit. Lieb showed that in one dimension, this effective theory predicts a ground state energy that differs at most by 20% from its exact value, for any density. The main idea is that instead of considering marginals of the square of the wave function, as in Hartree theory, we consider marginals of the wave function itself, which is positive in the ground state. The effective theory Lieb obtained is a non-linear integro-differential equation, whose non-linearity is an auto-convolution. In this talk, I will discuss some recent work about this effective equation. In particular, we proved the existence of a solution. We also proved that the ground state energy obtained from this simplified equation agrees exactly with that of the full N-body system at asymptotically low and at high densities. In fact, preliminary numerical work has shown that, for some potentials, the ground state energy can be computed in this way with an error of at most 5% over the entire range of densities. This is joint work with E. Carlen and E.H. Lieb.

This talk is centered around the symmetry properties of optimizers for the Caffarelli-Kohn-Nirenberg (CKN) inequalities, a two parameter family of inequalities. After a general overview I will explain some of the ideas on how to obtain the optimal symmetry region in the parameter space and will present an application to non-linear functionals of Aharonov-Bohm type, i.e., to problems that include a  magnetic flux concentrated at one point. These functionals are rotationally invariant and, as I will discuss, depending on the magnitude of the flux, the optimizers are radially symmetric or not.

What is the smallest total width of a collection of strips that cover a disk in the plane? How many lines through the origin pairwise separated by the same angle can be placed in 3-dimensional space? What about higher-dimensions?

These extremal problems in Discrete Geometry look deceitfully simple, yet some of them remain unsolved for an extended period or have been partly solved only recently following great efforts. In this talk, I will discuss two longstanding problems: Fejes Tóth’s zone conjecture and a problem on equiangular lines with a fixed angle.

No specific background will be needed to enjoy the talk.