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Maximal Abelian Sets of Roots
(American Mathematical SocietyMemoirs of the American Mathematical Society, 2017-11)
In this work we let Φ be an irreducible root system, with Coxeter group W. We consider subsets of Φ which are abelian, meaning that no two roots in the set have sum in Φ∪{0}. We classify all maximal abelian sets (i.e., ...
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, ...
On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
(University of CambridgeMathematics, 2017-05-30)
The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, ...
Type theoretic weak factorization systems
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsKing's, 2017-06-01)
This thesis presents a characterization of those categories with weak factorization systems that can interpret the theory of intensional dependent type theory with Σ, Π, and identity types.
We use display map categories ...
Existence of Mori fibre spaces for 3-folds in char p
(ElsevierAdvances in Mathematics, 2017-06-20)
We prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic $p$>5: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal ...
Critical Exponents on Fortuin-Kasteleyn Weighted Planar Maps
(SpringerCOMMUNICATIONS IN MATHEMATICAL PHYSICS, 2017-10-01)
In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain ...