Theses  Pure Mathematics and Mathematical Statistics
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On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
(20170530)The conjecture of Birch and SwinnertonDyer 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 $\mathbb{C}$, ... 
Exploring Random Geometry with the Gaussian Free Field
(20161001)This thesis studies the geometry of objects from 2dimensional statistical physics in the continuum. Chapter 1 is an introduction to SchrammLoewner evolutions (SLE). SLEs are the canonical family of nonselfintersecting, ... 
On a Heegaard Floer theory for tangles
(20170310)The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3ball. The decategorification ... 
$\textit{K}$Theory of Fermat Curves
(Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 20170110)I investigate the $K_2$ groups of the quotients of Fermat curves given in projective coordinates by the equation $F_n:X^n+Y^n=Z^n$. On any quotient where the number of known elements is equal to the rank predicted by ...