Lundberg, Erik

Due to the phenomenon of gravitational lensing, light from distant sources may appear as several images. The “image counting problem from gravitational lensing" refers to the question of how many images might occur, given a particular distribution of lensing masses. A common model treats the lensing masses as a finite collection of points situated in a finite collection of planes. The position of the apparent images correspond to the critical points of a real-valued function and also as solutions to a system of complex rational equations. Herein, we give upper bounds for the number of images in a point mass multiplane ensemble with an arbitrary number of masses in an arbitrary number of planes. We give lower bounds on the number of solutions in a closely related problem concerning gravitational equilibria. We use persistence homology to investigate two different stochastic ensembles. Finally we produce a multiplane ensemble, related to the maximal one plane ensemble, that produces a large number of images.

An acyclic orientation of a graph is an assignment of a direction to each edge in a way that does not form any directed cycles. Acyclic orientations of a complete bipartite graph are in bijection with a class of matrices called lonesum matrices, which can be uniquely reconstructed from their row and column sums. We utilize this connection and other properties of lonesum matrices to determine an analytic form of the generating function for the length of the longest path in an acyclic orientation on a complete bipartite graph, and then study the distribution of the length of the longest path when the acyclic orientation is random. We use methods of analytic combinatorics, including analytic combinatorics in several variables (ACSV), to determine asymptotics for lonesum matrices and other related classes.

This thesis is composed of three main parts. Each chapter is concerned with
characterizing some properties of a random ensemble or stochastic process. The
properties of interest and the methods for investigating them di er between chapters.
We begin by establishing some asymptotic results regarding zeros of random
harmonic mappings, a topic of much interest to mathematicians and astrophysicists
alike. We introduce a new model of harmonic polynomials based on the so-called
"Weyl ensemble" of random analytic polynomials. Building on the work of Li and
Wei [28] we obtain precise asymptotics for the average number of zeros of this model.
The primary tools used in this section are the famous Kac-Rice formula as well as
classical methods in the asymptotic analysis of integrals such as the Laplace method.
Continuing, we characterize several topological properties of this model of
harmonic polynomials. In chapter 3 we obtain experimental results concerning the
number of connected components of the orientation-reversing region as well as the geometry
of the distribution of zeros. The tools used in this section are primarily Monte
Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology
software known as Perseus. While the results in this chapter are empirical rather
than formal proofs, they lead to several enticing conjectures and open problems.
Finally, in chapter 4 we address an industry problem in applied mathematics
and machine learning. The analysis in this chapter implements similar techniques to
those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or
Control Area Network) is a network for allowing micro-controllers inside of vehicles
to communicate with each other. We propose and demonstrate the e ectiveness of an
algorithm for detecting malicious tra c using an approach that discovers and exploits
the natural geometry of the CAN surface and its relationship to random walk Markov
chains.

The study of random polynomials and in particular the number and behavior
of zeros of random polynomials have been well studied, where the rst signi cant
progress was made by Kac, nding an integral formula for the expected number of
zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations
of the formula to complex polynomials and random elds show an interesting
dependency of the number and distribution of zeros on the particular method of randomization.
Three prevalent models of signi cant study are the Kostlan model, the
Weyl model, and the naive model in which the coe cients of the polynomial are
standard Gaussian random variables.
A harmonic polynomial is a complex function of the form h(z) = p(z) + q(z)
where p and q are complex analytic polynomials. Li and Wei adapted the Kac integral
formula for the expected number of zeros to study random harmonic polynomials and
take particular interest in their interpretation of the Kostlan model. In this thesis we
nd asymptotic results for the number of zeros of random harmonic polynomials under
both the Weyl model and the naive model as the degree of the harmonic polynomial
increases. We compare the ndings to the Kostlan model as well as to the analytic analogs of each model.
We end by establishing results which lead to open questions and conjectures
about random harmonic polynomials. We ask and partially answer the question,
\When does the number and behavior of the zeros of a random harmonic polynomial
asymptotically emulate the same model of random complex analytic polynomial as
the degree increases?" We also inspect the variance of the number of zeros of random
harmonic polynomials, motivating the work by the question of whether the distribution
of the number of zeros concentrates near its as the degree of the harmonic
polynomial increases.