Now showing items 1-10 of 13
Torelli Theorems and Isogeny Theory for Irreducible Symplectic Varieties in Positive Characteristics
The first part treats the a class of higher dimensional analogues of K3 surfaces, called K3^n-type varieties, in positive characteristics. The notion of K3^n-type varieties is well understood in complex hyperkähler geometry. ...
Complex rank 3 vector bundles on complex projective 5-space
This work concerns two aspects of the study of complex rank $3$ topological vector bundles on complex projective five-space. The first aim is to classify such bundles: to give complete, computable algebraic invariants. ...
Correlated Random Matrices
In this thesis, we investigate the appearance of random matrix statistics for correlated random matrices. The Wigner-Dyson-Mehta Universality Conjecture regarding the statistics on eigenvalue differences is one of the ...
Perverse mod p Sheaves on Affine Flag Varieties
We prove a mod p analogue of the geometric Satake equivalence and we give a geometric construction of central elements in affine mod p Hecke algebras. To accomplish these goals we apply techniques from F-singularities to ...
Some Analytical Results on the Seiberg-Witten Equations and the Bogomolny Equations
This thesis contains some analytical results on the Bogomolny equations and the Seiberg-Witten equations. Chapter one studies the Bogomolny equations. In this chapter, first I briefly introduce Taubes' analytical approach ...
Nearby Cycles and Dualities in Geometric Langlands Program
In this thesis, we study nearby cycles on certain Vinberg-style degenerations in the geometric Langlands program. We relate them to various exotic dualities in this field, such as the (local and global) geometric second ...
Geodesic planes in hyperbolic 3-manifolds
Let P be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold M. Assume that P^*=M^*\cap P is nonempty, where M^* is the interior of the convex core of M. Does this condition imply that P is either ...
Clustering via Deep Dictionary Learning
Clustering, the process of discovering hidden groups in data, is a fundamentally important problem in statistical data analysis with applications ranging from image segmentation to sequence analysis to anomaly detection. ...
A Lighting-Invariant Approach to Local Shape from Shading
Shape from shading is a classical problem in computer vision, in which the depth field of an object or a scene is reconstructed from a pattern of intensities in an image. This can be thought of in some sense as the inverse ...
Absolute Hodge Cycles in Prismatic Cohomology
In this thesis we construct the notion of absolute Hodge cycles in prismatic cohomology for abelian schemes. We show that they are compatible with their de Rham and p-adic components under comparison isomorphisms.