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Envelopes of positive metrics with prescribed singularities
(Université Paul Sabatier, ToulouseAnnales de la Faculté des Sciences de Toulouse, 2016)
We investigate envelopes of positive metrics with a prescribed singularity type. First we generalise work of Berman to this setting, proving C$^{1,1}$ regularity of such envelopes, showing their Monge-Ampère measure is ...
Alpha Invariants and K-Stability for General Polarizations of Fano Varieties
(Oxford University PressINTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2014-09-26)
We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian’s alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system ...
On the rigid cohomology of certain Shimura varieties.
(SpringerResearch in the Mathematical Sciences, 2016)
We construct the compatible system of $\textit{l}$-adic representations associated to a regular algebraic cuspidal automorphic representation of GL$_{n}$ over a CM (or totally real) field and check local-global compatibility ...
On the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ - Dx
(Cambridge University PressMathematical Proceedings of the Cambridge Philosophical Society, 2016-09-09)
We study the variation of the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when ...
The augmented base locus of real divisors over arbitrary fields
(SpringerMathematische Annalen, 2016-07-08)
We show that the augmented base locus coincides with the exceptional locus (i.e., null locus) for any nef $\Bbb R$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness ...
On a Heegaard Floer theory for tangles
(University of CambridgeDPMMS, 2017-03-10)
The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3-sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3-ball. The decategorification ...
The Planar Ising Model and Total Positivity
(SpringerJournal of Statistical Physics, 2017-01-01)
A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let ...
Distinguishing geometries using finite quotients
(Mathematical Sciences PublishersGeometry and Topology, 2017-02-10)
We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, ...
Tautological rings for high-dimensional manifolds
(Cambridge University PressCompositio Mathematica, 2017-04)
We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^*$($M$) of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised ...
An upper bound for the pseudoisotopy stable range
(SpringerMathematische Annalen, 2017-08-01)
We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than (2n - 2). In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, ...