# Alumni Dissertations and Theses

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### Dual Graphs and PoincarĂ© Series of Valuations

Author:Charles LiYear of Dissertation:2012Program:MathematicsAdvisor:Hans SchoutensAbstract:Valuations on function fields of dimension two have been studied from the perspectives of dual graphs, generating sequences, Poincare series, and the valuative tree, among others. The goal of this dissertation is to greater unify these various approaches. Spivakovsky's dual graphs are used to calculate the Poincare series of non-divisorial valuations. With Galindo's results in the divisorial case already known, the equivalence of Poincare series with dual graphs is shown. A new elementary constructive proof of minimal generating sequences for non-divisorial valuations is given along the way, using only modest prerequisites from number theory. It is fair to say that the proof of minimal generating sequences is the crux of this dissertation, while the results on Poincare series are all corollaries.

### Problems in additive number theory

Author:Zeljka LjujicYear of Dissertation:2011Program:MathematicsAdvisor:Melvyn NathansonAbstract:In the first chapter we obtain the Biro-type upper bound for the smallest period of B in the case when A is a finite multiset of integers and B is a multiset such that A and B are t-complementing multisets of integers. In the second chapter we answer an inverse problem for lattice points proving that if K is a compact subset of R×R such that K+Z×Z=R×R then the integer points of the difference set of K is not contained on the coordinate axes, Z×{0}U{0}×Z. In the third chapter we show that there exist infinite sets A and M of positive integers whose partition function has weakly superpolynomial but not superpolynomial growth. The last chapter deals with the size of a sum of dilates 2·A+k·A. We prove that if k is a power of an odd prime or product of two primes and A a finite set of integers such that |A|>8k^k, then |2· A+k·A|≥ (k+2)|A|-k^2-k+2.

### Problems in additive number theory

Author:Zeljka LjujicYear of Dissertation:2011Program:MathematicsAdvisor:Melvyn NathansonAbstract:In the first chapter we obtain the Biro-type upper bound for the smallest period of B in the case when A is a finite multiset of integers and B is a multiset such that A and B are t-complementing multisets of integers. In the second chapter we answer an inverse problem for lattice points proving that if K is a compact subset of R×R such that K+Z×Z=R×R then the integer points of the difference set of K is not contained on the coordinate axes, Z×{0}U{0}×Z. In the third chapter we show that there exist infinite sets A and M of positive integers whose partition function has weakly superpolynomial but not superpolynomial growth. The last chapter deals with the size of a sum of dilates 2·A+k·A. We prove that if k is a power of an odd prime or product of two primes and A a finite set of integers such that |A|>8k^k, then |2· A+k·A|≥ (k+2)|A|-k^2-k+2.

### Total Variation of Gaussian Processes and Local Times of Associated Levy Processes

Author:Jonathan LovellYear of Dissertation:2009Program:MathematicsAdvisor:Michael MarcusAbstract:Results of Taylor and Marcus and Rosen on the total variation of Gaussian processes and local times of associated symmetric stable processes are extended to a large class of symmetric Lévy processes. In this extension, the increments variance of the Gaussian process is generalized to a regularly varying function with index 0<α< 2. The result also includes a generalization of the total variation function.

### Total Variation of Gaussian Processes and Local Times of Associated Levy Processes

Author:Jonathan LovellYear of Dissertation:2009Program:MathematicsAdvisor:Michael MarcusAbstract:Results of Taylor and Marcus and Rosen on the total variation of Gaussian processes and local times of associated symmetric stable processes are extended to a large class of symmetric Lévy processes. In this extension, the increments variance of the Gaussian process is generalized to a regularly varying function with index 0<α< 2. The result also includes a generalization of the total variation function.

### 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.

### 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.