Can one compute a low-dimensional representation of any given data by looking only at its small sample, chosen cleverly on the fly? Motivated by the above question, we consider the problem of low-rank matrix approximation: ...

This thesis gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1) we ...

In machine learning, the choice of space in which to represent our data is of vital importance to their effective and efficient analysis. In this thesis, we develop approaches to address a number of problems in representation ...

A secure multi-party computation (MPC) protocol for computing a function f allows a group of parties to jointly evaluate f over their private inputs, such that a computationally bounded adversary who corrupts a subset of ...

In this thesis I study the Seiberg-Witten equations on the product of a genus g surface [Sigma] and a circle. I exploit S1 invariance to reduce to the vortex equations on [Sigma] and thus completely describe the Seiberg-Witten ...

We study singularity models of mean curvature flow ("MCF") and their generalizations. In the first part, we focus on rigidity and curvature estimates for self-shrinkers. We give a rigidity theorem proving that any self-shrinker ...

In the first part of this thesis, we give a classification of all self-similar solutions to the curve shortening flow in the Euclidean plane R² and discuss basic properties of the curves. The problem of finding the curves ...

This dissertation is concerned with the foundations of a new approach to Floer theory. As opposed to the traditional approach, which can be viewed as a generalization of Morse theory to an infinite dimensional setting, our ...