Georgia Institute of Technology College of Sciences

Primary decomposition is a fundamental operation in commutative algebra. Although there are several algorithms to perform it, this remains a very difficult undertaking in general. In cases with additional combinatorial structure, it may be possible to do primary decomposition "by hand". The goal of this talk is to explain in detail one such example. This is joint work with Zekiye Eser; no prerequisites are assumed beyond knowing the definitions of "polynomial ring" and "ideal".

We consider several possibilities on how to select a Filippov sliding vector field on a co-dimension 2 singularity manifold, intersection of two co-dimension 1 manifolds, under the assumption of general attractivity. Of specific interest is the selection of a smoothly varying Filippov sliding vector field. As a result of our analysis and experiments, the best candidates of the many possibilities explored are based on so-called barycentric coordinates: in particular, we choose what we call the moments solution. We then examine the behavior of the moments vector field at the first order exit points, and show that it aligns smoothly with the exit vector field. Numerical experiments illustrate our results and contrast the present method with other choices of Filippov sliding vector field. We further generalize this construction to co-dimension 3 and higher.

The talk involves an explicit formula for the Chern class on K_3(F), F=number field, givenin terms of the cyclic quantum dilogarithm on the Bloch group of F. Such a formula constructsexcplicitly units in number fields, given a complete hyperbolic 3-manifold, and a complex root ofunity, and those units fit in the asymptotic expansion of quantum knot invariants. The existence ofsuch a formula was conjectured 4 years ago by Zagier (and abstractly follows from Voevodsky's work),and the final solution to the problem was given in recent joint work of the speaker with FrankCalegari and Don Zagier. The key ingredient to the concrete formula is a special function, thecyclic quantum dilogarithm, from a physics 1993 paper of Kashaev and others. The connection of thisformula with physics, and with the Quantum Modular Form Conjecture of Zagier continues with jointwork with Tudor Dimofte. But this is the topic of another talk.

The main result to be discussed will be the boundedness from $L^\infty \times L^\infty$ into $BMO$ of bilinear pseudodifferential operators with symbols in a range of bilinear H\"ormander classes of critical order. Such boundedness property is achieved by means of new continuity results for bilinear operators with symbols in certain classes and a new pointwise inequality relating bilinear operators and maximal functions. The role played by these estimates within the general theory will be addressed.