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Series: Geometry Topology Seminar

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.

Wednesday, May 29, 2019 - 14:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georgia Institute of Technology ,
Organizer: Yian Yao

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.

Thursday, May 30, 2019 - 14:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georigia Inst. of Technology

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.

Series: Geometry Topology Seminar

Roughly, factorization homology pairs an n-category and an n-manifold to produce a vector space. Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (ie, independent of auxiliary choices), continuous (ie, beholds a continuous action of diffeomorphisms), and functorial; the latter is easier to compute.

Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models. For such n-categories, this state sum expression agrees with the observables of the sigma-model — this is a form of Poincare’ duality, which yields some surprising dualities among TQFTs. A host of familiar TQFTs are instances of factorization homology; many others are speculatively so.

The first part of this talk will tour through some essential definitions in what’s described above. The second part of the talk will focus on familiar manifold invariants, such as the Jones polynomial, as instances of factorization homology, highlighting the Poincare’/Koszul duality result. The last part of the talk will speculate on more such instances.