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The Calderón problem for connections
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsTrinity College, 20171003)This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the DirichlettoNeumann (DN) ...

Symmetry in monotone Lagrangian Floer theory
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsTrinity College, 20171001)In this thesis we study the selfFloer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ ...

Fano Varieties in Mori Fibre Spaces
(International Mathematics Research Notices, 20160101)

Bounding cohomology for low rank algebraic groups
(University of CambridgeDPMMSHomerton, 20170801)Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in ...

Infinitesimal Models of Algebraic Theories
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsHomerton College, 20171125)Smooth manifolds have been always understood intuitively as spaces that are infinitesimally linear at each point, and thus infinitesimally affine when forgetting about the base point. The aim of this thesis is to develop ...


Type theoretic weak factorization systems
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsKing's, 20170601)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 ...

RANDOM WALKS ON THE RANDOM GRAPH
(ANNALS OF PROBABILITY, 201801)


On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
(University of CambridgeMathematics, 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
(University of CambridgeDPMMSPeterhouse, 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
(University of CambridgeDPMMS, 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 CambridgeUniversity of CambridgeChurchill College, 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 ...

Computations in monotone Floer theory
(Department of Pure Mathematics and Mathematical Statistics, University of CambridgeUniversity of CambridgeDepartment of Pure Mathematics and Mathematical Statistics, 20160628)Floer theory is a rich collection of tools for studying symplectic manifolds and their Lagrangian submanifolds with the help of holomorphic curves. Its origins lie in estimating the numbers of equilibria in Hamiltonian ...

Spectral methods and computational tradeoffs in highdimensional statistical inference
(Department of Pure Mathematics and Mathematical Statistics, University of CambridgeUniversity of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsFaculty of MathematicsSt John's College, 20161004)Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we ...

Categories of spaces built from local models
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsTrinity Hall, 20160628)Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most wellknown ...


Semicontinuity of stability for sheaves and variation of Gieseker moduli spaces
(De GruyterJournal für die reine und angewandte Mathematik, 2016)We investigate a semicontinuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion ...


On Short Time Existence of Lagrangian Mean Curvature Flow
(SpringerMathematische Annalen, 2016)We consider a short time existence problem motivated by a conjecture of Joyce in [8]. Specifically we prove that given any compact Lagrangian L ⊂ C^n with a finite number of singularities, each asymptotic to a pair of ...