Georgia Institute of Technology College of Sciences

We prove that every rational homology cobordism class in the subgroup generated by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a consequence we show that several natural maps to the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility condition between the determinants of certain 2-bridge knots and other knots in the same concordance class. This is joint work with Daniele Celoria and JungHwan Park.

The KAM (Kolmogorov Arnold and Moser) theory studies
the persistence of quasi-periodic solutions under perturbations.
It started from a basic set of theorems and it has grown
into a systematic theory that settles many questions. 

The basic theorem is rather surprising since it involves delicate
regularity properties of the functions considered, rather
subtle number theoretic properties of the frequency as well
as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of
a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites
(analysis, number theory and geometry). In the second lecture
we will present a complete proof of Moser's twist map theorem
(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.
If there is interest and energy, we will devote a third lecture
to numerical implementations. 

he KAM (Kolmogorov Arnold and Moser) theory studies
the persistence of quasi-periodic solutions under perturbations.
It started from a basic set of theorems and it has grown
into a systematic theory that settles many questions. 

The basic theorem is rather surprising since it involves delicate
regularity properties of the functions considered, rather
subtle number theoretic properties of the frequency as well
as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of
a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites
(analysis, number theory and geometry). In the second lecture
we will present a complete proof of Moser's twist map theorem
(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.
If there is interest and energy, we will devote a third lecture
to numerical implementations.