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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 ...
Efficient advert assignment
(InformsOperations Research, 2016-07-01)
We develop a framework for the analysis of large-scale ad auctions where adverts are assigned over a continuum of search types. For this pay-per-click market, we provide an efficient mechanism that maximizes social welfare. ...
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 ...
RANDOM WALKS ON THE RANDOM GRAPH
(ANNALS OF PROBABILITY, 2018-01)
Homological stability for spaces of embedded surfaces
(Mathematical Sciences PublisherGeometry and Topology, 2017-05-10)
© 2017, Mathematical Sciences Publishers. All rights reserved.We study the space of oriented genus-g subsurfaces of a fixed manifold M and, in particular, its homological properties. We construct a “scanning map” which ...
Maximum likelihood estimation of a multivariate log-concave density
(University of Cambridge, 2010-01-12)
Density estimation is a fundamental statistical problem. Many methods are either
sensitive to model misspecification (parametric models) or difficult to calibrate, especially
for multivariate data (nonparametric smoothing ...
Iwasawa theory for modular forms at supersingular primes
(University of CambridgeDepartment of Pure Mathematics and Mathematical Statistics, 2010-07-06)
Let f=\sum a_nq^n be a normalised eigen-newform of weight k\ge2 and p an odd prime which does not divide the level of f. We study a reformulation of Kato's main conjecture for f over the Zp-cyclotomic extension of Q. In ...
Cliques in graphs
(University of CambridgeDepartment of Pure Mathematics and Mathematical Statistics, 2010-10-12)
The main focus of this thesis is to evaluate $k_r(n,\delta)$, the minimal number of $r$-cliques in graphs with $n$ vertices and minimum degree~$\delta$. A fundamental result in Graph Theory states that a triangle-free graph ...