# Alumni Dissertations and Theses

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### The Hilbert Projective Metric, Multi-type Branching Processes and Mathematical Biology: a Model of the Evolution of Resistance

Author:Christopher McCarthyYear of Dissertation:2010Program:MathematicsAdvisor:Yunping JiangAbstract:Bacteria, viruses, or cancer cells, by means of mutation and replication, are sometimes able to escape the selective pressure exerted by treatment. This is called the development, or evolution, of resistance. This dissertation is a study of some of the mathematics underlying a model of resistance put forth by Iwasa, Michor, and Nowak (IMN) "Evolutionary Dynamics of Invasion and Escape" (2003, 2004). In the IMN model the pre-treatment phase is modeled as a determinist dynamical system using Eigen and Schuster's quasispecies theory of evolution. It is assumed that at the start of treatment the system has reached an invariant distribution: the quasispecies equilibrium eigenvector. The equations of the quasispecies theory can be viewed as projections of linear differential equations onto hyperplanes and their asymptotic behavior can be understood via Birkhoff's Projective Contraction Theorem (1957), which is related to the Perron-Frobenius Theorem. An understanding of Birkhoff's contraction theorem requires an understanding of the Hilbert Projective Metric and so we develop an extensive collection of useful related results, some novel, about cones, hyperplanes, and the Hilbert Metric. In the IMN model, the post-treatment phase is modeled as a stochastic multi-type branching process on the various mutant types. The key calculation is the vector of extinction probabilities: the i entry of the vector being the probability that a process, starting with a single mutant of type i, will eventually go extinct (under the selective pressure of treatment). The techniques for calculating these extinction probabilities involve the use of multi-type probability generating functions (PGF's). We prove results about the existence of continuous multi-type PGF's and branching processes. Our proofs involve customizing techniques from the theory of differential equations in complex vector spaces, and then applying results from the theory of several complex variables. We also develop a method to numerically calculate the vector of extinction probabilities. The pre and post-treatment models are fitted together and the probability of a successful treatment is numerically calculated using a combination of standard techniques from numerical analysis together with insights gained from our examination of the mathematical aspects of the model. Our investigation leads to a phenomena somewhat reminiscent of Eigen's error catastrophe theory. Supplementary materials: hyper-linked PDF of dissertation, Matlab m-files.

### Asymptotic Invariants and Flatness of Local Endomorphisms

Author:Nikita MiasnikovYear of Dissertation:2014Program:MathematicsAdvisor:Lucien SzpiroAbstract:For a local endomorphism of a noetherian local ring we introduce 3 asymptotic invariants one of which we call entropy. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.

### Algebraic Models for the Free Loop Space and Differential Forms of a Manifold

Author:Micah MillerYear of Dissertation:2011Program:MathematicsAdvisor:Mahmoud ZeinalianAbstract:Our initial goal is to give a chain level description of the string topology loop product for a large class of spaces. This effort is described in two parts; the first uses Brown's theory of twisting cochains to obtain a model for the free loop space of a manifold and the second constructs a minimal model for the Frobenius algebra of differential forms of a manifold. The first part defines the loop product for closed, oriented manifolds and Poincare Duality spaces. The second part is an attempt to understand the minimal model for the Frobenius algebra of a manifold, with the idea of extending the methods in the first section to define the loop product for open manifolds. Brown's theory of twisting cochains provides a chain model of a principal G-bundle and its associated bundles. The free loop space is obtained by considering the path space fibration, and taking the associated bundle with the based loop space acting on itself by conjugation. Given a twisting cochain, then, we obtain a chain model of LM using Brown's theory. To describe the chain-level loop product in this setting, we need a model for the intersection product in the chains on M. For this, we use the cyclic commutative infinity algebra structure on the homology of M. Such a description would give a chain level description of the string topology loop product for open manifolds. Instead of using the cyclic commutative algebra, we could have used the Frobenius algebra structure. One would expect that the Frobenius infinity algebra can be used to show the necessary relations to define the loop product. Then given the Frobenius infinity algebra on the homology of M for an open manifold, we would have a chain level description of the loop product. The purpose of Section 3 is to gain a better understanding of the Frobenius infinity algebra on the cohomology of M. The Frobenius algebra, induced by the wedge product and Poincare Duality, is well understood; the structure on the level of forms inducing the Frobenius algebra is less well understood. We use the language of operads, dioperads, and properads and Koszul duality to give a definition of Frobenius infinity algebra. We also use descriptions of the transfer of structure using trees and integrating over cells in the moduli space of metrised ribbon graphs. When M is closed and oriented, these tools allow us to build a minimal model for the Frobenius algebra of differential forms on M and to compare it with the cyclic commutative infinity algebra.

### An unstable variant of the Morava Change of Rings theorem for K(n) theory

Author:Dustin MulcaheyYear of Dissertation:2011Program:MathematicsAdvisor:Robert ThompsonAbstract:We formulate a very general criteria for a base change comonads for Ext computations. We then use this criteria to prove a generalized version of the Morava change of rings theorem from stable homotopy theory.

### Conformally Natural Extensions of Continuous Circle Maps

Author:Oleg MuzicianYear of Dissertation:2012Program:MathematicsAdvisor:Jun HuAbstract:Conformally natural and continuous extensions were originally introduced by Douady and Earle for circle homeomorphisms, and later by Abikoff, Earle and Mitra for continuous degree ±1 monotone circle maps. The first main result of this thesis shows that conformally natural and continuous extensions exist for all continuous circle maps. The second main result shows that if

f is a continuous circle map and is M-quasisymmetric on some arc on the unit circle S1 , then such an extension off is locally K-quasiconformal on a neighborhood of the arc in the open unit disk D, where the neighborhood and K depend only on M.### Problems in Additive Number Theory

Author:Brooke OroszYear of Dissertation:2009Program:MathematicsAdvisor:Melvyn NathansonAbstract:The first chapter deals with the following problem: Let f (n) be a growth function, and A be a sequence with f (n) < a

_{n}Uf (n), U constant. Under what conditions is it possible to construct another sequence with b_{k}asymptotically equal to Bf (k), which has A as a subsequence? The next two chapters deal with the possible sizes of generalized sum sets on finite sets of integers. The final chapter discusses counting relatively prime subsets of the natural numbers.### Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal

Author:Norman PerlmutterYear of Dissertation:2013Program:MathematicsAdvisor:Joel HamkinsAbstract:This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other. The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. An inverse limit exists if and only if a natural source exists. If the inverse limit exists, then it is given by either the entire thread class or by a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, it is consistent that there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved by forcing in both directions under fairly general assumptions but not in all cases. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter. The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding j: V --> M such that M is closed under sequences of length equal to the clearance of the embedding. This clearance is defined as the supremum, over all functions f from κ to κ, of j(f)(κ). Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to a Woodin-for-supercompactness cardinal. There are no excessively hypercompact cardinals.

### String Topology & Compactified Moduli Spaces

Author:Katherine PoirierYear of Dissertation:2010Program:MathematicsAdvisor:Dennis SullivanAbstract:The motivation behind this work is to solve the master equation dX = X*X in a chain complex which is a direct sum of homomorphism complexes of tensor powers of a chain complex P, where P computes H(LM,M), the S^1-equivariant homology of the free loop space LM of a manifold M, relative to constant loops. Here, we solve a modification of this equation: dX = X*X + A and suggest an avenue for modifying the solution of the second equation to obtain a solution of the master equation. The solution of the second equation is constructed by building a pseudomanifold of string diagrams which has prescribed boundary. The string topology construction describes the action of cellular chains of the pseudomanifold on P. Further, the pseudomanifold is homeomorphic to a compactification of the moduli space of Riemann surfaces. A second smaller compactification is defined over which string topology operations conjecturally extend.

### ON POLYNOMIAL ROOTS APPROXIMATION VIA DOMINANT EIGENSPACES AND ISOLATION OF REAL ROOTS

Author:OMAR RETAMOSO URBANOYear of Dissertation:2015Program:MathematicsAdvisor:VICTOR PANAbstract:Finding the roots of a given polynomial is a very old and noble problem in mathematics and computational mathematics. For about 4,000 years, various approaches had been proposed to solve this problem (see cite{FC99}). In 1824, Niels Abel showed that there existed polynomials of degree five, whose roots could not be expressed using radicals and arithmetic operations through their coefficients. Here is an example of such polynomials:$$x^5-4x-2.$$ Thus we must resort to iterative methods to approximate the roots of a polynomial given with its coefficients. There are many algorithms that approximate the roots of a polynomial(see cite{B40}, cite{B68}, cite{MN93}, cite{MN97}, cite{MN99}, cite{MN02}, cite{MN07}). As important examples we cite Quadtree (Weyl's) Construction and Newton's Iteration (see cite{P00a}). Some of the algorithms have as their goal to output a single root, for example, the absolutely largest root. Some other algorithms aim to output a subset of all the roots of the given polynomial, for example, all the roots within a fixed region on the complex plane. In many applications (e.g., algebraic geometric optimization), only the real roots are of interest, and they can be much less numerous than all the roots of the polynomial (see cite{MP13}). Nevertheless, the best numerical subroutines, such as MPSolve 2.0 cite{BF00}, Eigensolve cite{F02}, and MPsolve 3.0 cite{BR14}, approximate all real roots about as fast and as slow as all complex roots. The purpose of this thesis is to find real roots of a given polynomial effectively and quickly, this is accomplished by separating real roots from the other roots of the given polynomial and by finding roots which are clustered and absolutely dominant. We use matrix functions throughout this thesis to achieve this goal. One of the approaches is to approximate the roots of a polynomial $p(x)$ by approximating the eigenvalues of its associated companion matrix $C_{p}$. This takes advantage of using the well-known numerical matrix methods for the eigenvalues. This dissertation is organized as follows. Chapter 1 is devoted to brief history and modern applications of Polynomial root-finding, definitions, preliminary results, basic theorems, and randomized matrix computations. In Chapter 2, we present our Basic Algorithms and combine them with repeated squaring to approximate the absolutely largest roots as well as the roots closest to a selected complex point. We recall the matrix sign function and apply it to eigen-solving. We cover its computation and adjust it to real eigen-solving. In Chapter 3, we present a "matrix free" algorithm to isolate and approximate real roots of a given polynomial. We use a Cayley map followed by Dandelin's (Lobachevsky's, Gr"{a}ffe's) iteration. This is in part based on the fact that we have at hand good and efficient algorithms to approximate roots of a polynomial having only real roots (for instance the modified Laguerre's algorithm of [DJLZ97]). The idea is to extract (approximately) from the image of the given polynomial (via compositions of rational functions) a factor whose roots are all real, which can be solved using modified Laguerre's algorithm, so we can output good approximations of the real roots of the given polynomial. In Chapter 4, we present an algorithm based on a matrix version of the Cayley map used in Chapter 3. As our input, we consider the companion matrix of a given polynomial. The Cayley map and selected rational functions are treated as matrix functions. Via composition of matrix functions we generate and approximate the eigenspace associated with the real eigenvalues of the companion matrix, and then we readily approximate the real eigenvalues of the companion matrix of the given polynomial. To simplify the algorithm and to avoid numerical issues appearing in computation of the high powers of matrices, we use factorization of $P^{k}-P^{-k}$ as the product $prod_{i=0}^{k-1}(P-omega_k^iP^{-1})$ where $omega_k=exp(2pisqrt {-1}/k)$ is a primitive $k$th root of unity.

### Geometrical aspects of linear differential equations over compact Riemann surfaces with reductive differential Galois group

Author:Camilo Sanabria MalagonYear of Dissertation:2010Program:MathematicsAdvisor:Richard ChurchillAbstract:Suppose L(y) = 0 is a linear differential equation with reductive Galois group over the function field of a compact Riemann surface. We prove that any solution to the equation can be written as a product of a solution to a first order equation and a solution to the pullback of an equation of a special form (a "standard equation"). We classify standard equations using ruled surfaces. We relate the symmetries of L(y) = 0 to the outer-automorphisms of the differential Galois group.