Georgia Institute of Technology College of Sciences

In 1908 Hadamard conjectured that the biharmonic Green function must be positive. Later on, several counterexamples to Hadamard's conjecture have been found and a variety of upper estimates were obtained in sufficiently smooth domains. However, the behavior of the Green function in general domains was not well-understood until recently. In a joint work with V. Maz'ya we derive sharp pointwise estimates for the biharmonic and, more generally, polyharmonic Green function in arbitrary domains. Furthermore, we introduce the higher order capacity and establish an analogue of the Wiener criterion describing the precise correlation between the geometry of the domain and the regularity of the solutions to the polyharmonic equation.

In this talk we will introduce the notion of a cube diagram---a surprisingly subtle, extremely powerful new way to represent a knot or link. One of the motivations for creating cube diagrams was to develop a 3-dimensional "Reidemeister's theorem''. Recall that many knot invariants can be easily be proven by showing that they are invariant under the three Reidemeister moves. On the other hand, simple, easy-to-check 3-dimensional moves (like triangle moves) are generally ineffective for defining and proving knot invariants: such moves are simply too flexible and/or uncontrollable to check whether a quantity is a knot invariant or not. Cube diagrams are our attempt to "split the difference" between the flexibility of ambient isotopy of R^3 and specific, controllable moves in a knot projection. The main goal in defining cube diagrams was to develop a data structure that describes an embedding of a knot in R^3 such that (1) every link is represented by a cube diagram, (2) the data structure is rigid enough to easily define invariants, yet (3) a limited number of 5 moves are all that are necessary to transform one cube diagram of a link into any other cube diagram of the same link. As an example of the usefulness of cube diagrams we present a homology theory constructed from cube diagrams and show that it is equivalent to knot Floer homology, one of the most powerful known knot invariants.

We will develop general theory of quantum invariants based on sl_2 (the simplest Lie algebra): The Jones polynomials, the colored Jones polynomials, quantum sl_2 groups, operator invariants of tangles, and relations with the Alexander polynomial and the A-polynomials. Optional: Finite type invariants and the Kontsevich integral.

Archimedes principle may be used to predict if and how certain solid objects float in a liquid bath. The principle, however, neglects to consider capillary forces which can sometimes play an important role. We describe a recent generalization of the principle and how the standard textbook presentation of Archimedes' work may have played a role in delaying the discovery of such generalizations to this late date.