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Analysis of Some PDEs over Manifolds
(2013-02-14)
In this dissertation I discuss and investigate the analytic aspect of several elliptic and parabolic partial differential equations arising from Rimannian and complex geometry, including the generalized Ricci flow, Gaussian ...
D-Modules on Spaces of Rational Maps and on Other Generic Data
(2012-12-13)
Fix an algebraic curve X. We study the problem of parametrizing geometric data over X, which is only generically defined. E.g., parametrizing generically defined maps from X to a fixed target scheme Y. There are three methods ...
Mapping Class Groups, Homology and Finite Covers of Surfaces
(2012-12-20)
Let S be an orientable surface of genus g with n punctures, such that \(\chi(S) = 2 − 2g − n < 0\). Let \(\psi \epsilon Mod(S)\) denote an element in its mapping class group. In this thesis, we study the action of \(\psi\) ...
Alternate Compactifications of Hurwitz Spaces
(2012-12-19)
We construct several modular compactifications of the Hurwitz space \(H^d_{g/h}\) of genus g curves expressed as d-sheeted, simply branched covers of genus h curves. They are obtained by allowing the branch points of the ...
Mathematical Models of Cancer
(2013-02-22)
Major efforts to sequence cancer genomes are now occurring throughout the world. Though the emerging data from these studies are illuminating, their reconciliation with epidemiologic and clinical observations poses a major ...
Towards an Instanton Floer Homology for Tangles
(2012-08-10)
In this thesis we investigate the problem of defining an extension of sutured instanton Floer homology to give an instanton invariant for a tangle. We do this in three separate steps. First, we investigate the representation ...
The Arithmetic of Simple Singularities
(2012-08-10)
We investigate some arithmetic orbit problems in representations of linear algebraic groups arising from Vinberg theory. We aim to give a description of the orbits in these representations using methods with an emphasis ...
Restrictions of Steiner Bundles and Divisors on the Hilbert Scheme of Points in the Plane
(2012-09-18)
The Hilbert scheme of \(n\) points in the projective plane parameterizes degree \(n\) zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert ...
Symplectic Rational Blow-Up and Embeddings of Rational Homology Balls
(2012-09-18)
We define the symplectic rational blow-up operation, for a family of rational homology balls \(B_n\), which appeared in Fintushel and Stern's rational blow-down construction. We do this by exhibiting a symplectic structure ...