The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

The present thesis focuses on the design and analysis of randomized algorithms for accelerating several linear algebraic tasks. In particular, we develop simple, efficient, randomized algorithms for a plethora of fundamental linear algebraic tasks and we also demonstrate their usefulness and applicability to matrix computations and graph theoretic problems. The thesis can be divided into three parts. The first part concentrates on the development of randomized linear algebraic primitives, the second part demonstrates the application of such primitives to matrix computations, and the last part discusses the application of such primitives to graph problems. First, we present randomized approximation algorithms for the problems of matrix multiplication, orthogonal projection, vector orthonormalization and principal angles computation (a.k.a. canonical correlation analysis). Second, utilizing the tools developed in the first part, we present randomized and provable accurate approximation algorithms for the problems of linear regression and element-wise matrix sparsification. Moreover, we present an efficient deterministic algorithm for selecting a small subset of vectors that are in isotropic position. Finally, we exploit well-known interactions between linear algebra and spectral graph theory to develop and analyze graph algorithms. In particular, we present a near-optimal time deterministic construction of expanding Cayley graphs, an efficient deterministic algorithm for graph sparsification and a randomized distributed Laplacian solver that operates under the gossip model of computation.

The thesis consists of two main results. The first main result is a proof that C^1 rigidity holds for circle maps with a break point for almost all rotation numbers. The second main result is a proof that C^1 robust rigidity holds for circle maps in the fractional linear transformation (FLT) pair family. That is, for this family, C^1 rigidity holds for all irrational rotation numbers. The approach taken here of proving a more general theorem that C^1 rigidity holds for circle maps with a break point satisfying a `derivatives close condition', allows us to obtain both of our main results as corollaries of this more general theorem.

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

The main topic of this thesis is the tropicalizations of Severi varieties, which we call tropical Severi varieties. Severi varieties are classical objects in algebraic geometry. They are parameter spaces of plane nodal curves. On the other hand, tropicalization is an operation defined in tropical geometry, which turns subvarieties of an algebraic torus into certain polyhedral objects in real vector spaces. By studying the tropicalizations, it may be possible to transform algebro-geometric problems into purely combinatorial ones. Thus, it is a natural question, “what are tropical Severi varieties?” In this thesis, we give a partial answer to this question: we obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. Given a regular subdivision of a convex lattice polygon, we construct an explicit parameter space of plane curves. This parameter space is a much simpler object than the corresponding Severi variety and it is closely related to a flat degeneration of the Severi variety, which in turn describes the tropical Severi variety. We present two applications. First, we understand G.Mikhalkin’s correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves. The second application is about Secondary fans. Secondary fans are purely combinatorial objects which parameterize all the regular subdivisions of polygons. We provide a relation between tropical Severi varieties and Secondary fans.

The present thesis focuses on the design and analysis of randomized algorithms for accelerating several linear algebraic tasks. In particular, we develop simple, efficient, randomized algorithms for a plethora of fundamental linear algebraic tasks and we also demonstrate their usefulness and applicability to matrix computations and graph theoretic problems. The thesis can be divided into three parts. The first part concentrates on the development of randomized linear algebraic primitives, the second part demonstrates the application of such primitives to matrix computations, and the last part discusses the application of such primitives to graph problems. First, we present randomized approximation algorithms for the problems of matrix multiplication, orthogonal projection, vector orthonormalization and principal angles computation (a.k.a. canonical correlation analysis). Second, utilizing the tools developed in the first part, we present randomized and provable accurate approximation algorithms for the problems of linear regression and element-wise matrix sparsification. Moreover, we present an efficient deterministic algorithm for selecting a small subset of vectors that are in isotropic position. Finally, we exploit well-known interactions between linear algebra and spectral graph theory to develop and analyze graph algorithms. In particular, we present a near-optimal time deterministic construction of expanding Cayley graphs, an efficient deterministic algorithm for graph sparsification and a randomized distributed Laplacian solver that operates under the gossip model of computation.