James Kilbane

Hello!

Landing page for reading group

This page will form a dumping ground for the notes and pdf's I generate over the course of my PhD in case they are of interest to anyone. My area of research is Banach Space Theory, and hence most of these notes will have that flavour to them. I will also put a copy of my Part III essay for anyone who is interested.

My Part III Essay, on Weyl's theorem, a theorem on compact perturbations and their impact on the spectrum

Lecture notes on function spaces, as lectured by Sophia Demoulini, Easter term 2015.

Notes for incoming Part III's for the functional analysis prerequisites.

Some classical results in Banach Space Theory

Some notes on Grothendiecks inequality a statement about the boundedness of a certain mapping.

A short proof that weakly compact operators can be factorized through a reflexive space.

Some notes on Ultrafilters and Banach Space theory.

Some notes giving a construction of Tsirelsons space and the James space.

A discussion of Kwapiens theorem, a statement relating the type and cotype of a metric space, to whether the space is a Hilbert space.

Some results on metric embeddings

Some notes on Aharoni's Theorem, a statement on embedding metric spaces into the space of sequences that converge to zero

Some more notes on Aharoni's Theorem this time showing the exact constants involved for compact spaces.

An involved look at a result of Kaltons that says any stable metric space can be embedded into a reflexive Banach space.

A combination of results on embeddings into c0, including the best possible constants in Aharoni's theorem and a discussion of some of the non-linear embeddings of c0.

A proof that if we have a Lipschitz mapping from a separable space into a space with the Radon Nikodym property, then the space linearly embeds. Also contains a discussion of the Radon Nikodym Property, and a proof of almost everywhere differentiability of Lipschitz functions in infinite dimensions.

A discussion of the Gorelik Principle, a statement on uniform homeomorphisms between Banach spaces and a proof that l_p and L_p are not uniformly homeomorphic for p > 2. This version has plenty of typos that will be fixed.

A proof of the general bi-Lipschitz embeddability-ey of doubling metrics.

A proof that super reflexive trees are those that do not contain delta trees.

A proof that the an n point subset that isometrically embeds into Lp embeds into lp^m for m = n choose 2. Moreover a proof that in the case 1 < p < 2 that this is (asymptotically) optimal.

A proof that for linfinity that any n point metric spaces embeds into linfinity^{n-c}, for all integers c, whenever n is sufficiently large.

A proof that lp is not a uniform subquotient of lq where the quotient map is Lipschitz for large distances whenever p ≤ q.

A proof that if you have unbounded Szlenk index, or your dual does, then you contain N^h as a tree in a bounded fashion for all h.

A proof of Krivines theorem, a statement that there's always a copy of lpn inside a spreading model if you look far enough.

Schreier set definition

Notes from Gilles Lancien course on Lipschitz Free spaces in summer 2016 at Texas A&M

A proof that c_0 is finitely representable in James space.

A summary of some of the results from Bourgain Milman and Wilson's paper called "On Type of Metric spaces"

Some talks

My talk from the YFAW workshop. This talk was given 3 different times with slightly different material, one at YFAW 2017, one as a CAKE seminar in Cambridge, and once as a talk in Besancon - the notes here are the union of the three talks.