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Modernism happened in mathematics, too. Between about 1890 and 1930, the field turned toward dramatic, previously unimaginable abstraction, as radical innovations dispensed with inherited conventions. The recognition of non-Euclidean geometry raised the strange notion that multiple, contradictory realms of mathematics could coexist and remain equally correct, even if some seemed non-representative of the world or nonsensical in human experience. Mathematicians grew frenzied in the effort to resolve newfound paradoxes. The turn of the century saw a series of attempts to rethink math’s most basic foundations, and the 1920s witnessed explosive controversies about the role of language in mathematics. These developments do not merely constitute change in one field; they demonstrate that modernism was a broader cultural phenomenon than literary scholars have realized. This dissertation begins from the shared intellectual history of literary and mathematical modernism: a common attempt to rethink foundational axioms, a common ambivalence toward growing abstraction, and a common interest in and anxiety about form. This convergence, I argue, exposes an interpretive dilemma intrinsic to form in mathematics and writing alike. Modernist shifts in cultural assumptions gave rise to newly abstract forms that revealed form to have been always already abstract—so abstract as to estrange its own cultural function. Formalism, in literature, skates uncomfortably between profundity and superficiality, because attention to literary language in and of itself risks discounting language’s fundamental function: to express, refer, and communicate. But decades before formalism named a school of thought in literary interpretation, formalism—by that name—identified an understanding, and an interpretation, of mathematics. Mathematical formalism has roots reaching back to the 17th century, but after a long history it reached its heyday in the 1920s, when the field’s escalating abstraction had led many to question math’s grounding in the real world, i.e., to doubt its basic capacity for meaning. The influential mathematician David Hilbert argued that such a grounding could be secured via attention to form—in effect, that formalism could rescue mathematical meaning by deferring that meaning, first treating mathematics as though it were only meaningless marks on paper. This dissertation uses Hilbert’s mathematical formalism to explain the apparent contradictions of literary formalism. I demonstrate that writers and mathematicians discovered the same solutions to an urgent common problem, developing technical apparatuses (variables, equivalence relations, axiom systems) through which meaning can develop directly from patterns and rules rather than terms and symbols. Mathematics has a remarkable relationship with reality, because it is both a maddeningly ethereal realm of thought and an undeniably descriptive tool that reliably predicts physical phenomena. I argue that the bond between mathematics and science is not only a metaphor for the relationship between literature and reality, but an imitable model that multiple modernist authors knowingly seized upon and manipulated. It is in this mathematical manner that the most abstract, formal, and seemingly unworldly moments in modernism acquired relevance to history, world, and culture. They did so by describing the world without making reference to the world, expressing the shapes of reality via pattern and form. My first chapter, “Opposition, Polysemy, Pattern: Virginia Woolf’s Mathematical Generality,” begins from the more commonly understood relation between literature and mathematics: one of radical difference. In her early novel Night and Day Woolf resists the rigidity of mathematics, consistently imagining mathematical communication as a negative image of her own art, an inaccessible language that resists translation into English. But this opposition, I argue, also makes visible a likeness between the two fields, glimpsed in the “sacred pages of symbols and figures” that Woolf consistently uses to frame their common signs and signifying processes. The formalism of mathematical writing thus promises to explain many of the escalating formal complexities of Woolf’s later novels. I demonstrate that, even where she makes no direct reference to mathematics, Woolf uses the greater generality of mathematical modes of meaning to reinvent the force of ambiguity in Jacob’s Room and contradiction in To the Lighthouse, before settling into the enormous patterns that establish a formula for “the life of anybody” in The Waves. In mathematics, such generality is most meaningful in conjunction with structured systems—the patterns that make unrestricted meaning possible. I take up the problem of defining and theorizing literary structure in the following two chapters. In “The Metaphor as an Equation: The Formal Abstraction of Ezra Pound’s Precisions,” I trace Pound’s repeated descriptions of image and metaphor as instruments of equation, and I demonstrate that technical mathematical definitions of equation characterize some of Imagism’s starkest innovations. Between 1890 and 1910, mathematical understandings of equation had shifted, reaching beyond number to become newly applicable to shapes, ideas, and words. Pound, I argue, harnessed recent mathematical redefinitions of equality to place a new weight on syntax: in his early poetry, semicolons unseat verbs, and ellipsis makes possible new forms of comparison. Pound’s early metaphors invented a newly symmetric and iterable relationship between tenor and vehicle, one capable of underlying more procreant forms. And form, here, acquires a newly interdisciplinary definition: the patterns and arrangements of relations between things that can exist independent of those things.Pattern has to start from somewhere: a problem that consumed Eliot across his long career. In pure, modern mathematics, absolute knowledge and incontrovertible proof are built from patterns that always stand upon unproven assumptions—axioms. Eliot had examined those axioms in detail. In 1914, as a student at Harvard, Eliot took Bertrand Russell’s graduate course in advanced logic, intensively studying the modernist axiom systems of Gottlob Frege, Alfred North Whitehead, and Russell himself. In “From Axiom to Leap of Faith: T.S. Eliot’s Formal Systems,” I trace Eliot’s ambivalent preoccupation with logical assumption in “Prufrock” and The Waste Land and argue that, in Ash Wednesday and Four Quartets, Eliot ultimately used Russell’s formal systems to reimagine poems as infinitely intricate patterns built from finite starting points: an idea that already existed, in more troubled ways, in his earliest poetry. Eliot conceived of the most complex poetic forms as axiom systems, wherein whole aesthetic universes develop from mere handfuls of beliefs and linguistic links, just as, in mathematical axiom systems, complex meaning and sophisticated beauty develop from strikingly few assumptions, definitions, and rules. Axiom systems refer, intrinsically and everywhere, to themselves, and in this way the patterns loop back upon themselves, at once referring to themselves and anticipating their own analysis. I conclude by turning from poetry and fiction to the literary scholarship and theory engendered by modernism, generalizing the ways in which the philosophy of mathematics reframes the critical stumbling blocks of form and reference. Mathematics was a regular trope, and often a conceptual source, in the first defining works of twentieth-century formalist criticism, evincing a continuing association between the logical analysis of form and the commingling of mathematics and language. The Russian formalists, British practical critics, and American New Critics all saw mathematics as a closed linguistic system, making it uniquely useful as a descriptive analogue for literary form: mathematics offered the prime example of a system that models isolated signifying processes while lying (by general consensus) utterly outside of literature. I argue that the mathematics of form made these literary formalisms a natural and necessary response to literary modernism, determining their near-simultaneous development across different cultural contexts. When modernist form unsettled modernist content, a meta-content emerged: texts by Woolf, Pound, and Eliot all self-consciously refer to the interpretive difficulties they create. During exactly the same era, Hilbert pushed the intuitive sense of mathematics into a separate plane only to find that a new discipline derived from that distinction: metamathematics, which uses mathematics to study mathematics itself. Far from pushing the world out, formalist analysis indirectly drags the world back in, for writers and mathematicians alike. “A Few Sheets of Paper Covered with Arbitrary Symbols”: Formalism, Modernism, Mathematics thus culminates in the argument that the apparent paradoxes of formal literary interpretation—wherein formalist reading sees only the façade yet penetrates to the heart of the thing itself—are not coincidental to how we have practiced literary formalism. In fact, they are logically attendant on any application of formalist analysis, in any field. Formalism depends on the deferral of meaning but then, elaborating its own rules, bypasses that assumption to uncover a meaningful pattern, a meaning in pattern. Modernists from Woolf to Hilbert discovered and elaborated this interpretive loop. Our literary practice of formalism today remains dependent on that logical process of form.

This research paper presents an innovative approach to anomaly detection in maritime data, focusing on the identification of anomalous vessel groups through Automatic Identification System (AIS) data. Traditional AIS anomaly detection systems are challenged by a narrow focus on individual vessel anomalies which leaves out critical collective action information. Our work seeks to overcome this limitation by introducing a graph modeling methodology that emphasizes the collective behavior of vessels, a critical aspect often overlooked in current frameworks. We work towards a system that detects unusual patterns of coordination among groups of vessels, which is particularly relevant for defense contexts where threats such as smuggling, piracy, and adversarial operations are typically carried out by networks of collaborating entities. The research contributes to the field by offering a more comprehensive understanding of maritime traffic through the lens of collective dynamics, enhancing the detection capabilities of AIS monitoring systems. The implications of our methodology are far-reaching, providing a foundational strategy for future research in anomaly detection that could be applied to various domains where group coordination plays a pivotal role in defining anomalous behavior. This paper details our graph-based approach and suggests a trajectory for subsequent investigations into the broader applications of detecting coordinated anomalies.

This paper introduces Prio+, a privacy-preserving system for the collection of aggregate statistics, with the same model and goals in mind as the original and highly influential Prio paper by Henry Corrigan-Gibbs and Dan Boneh (NSDI 2017). As in the original Prio, each client holds a private data value (e.g. number of visits to a particular website) and a small set of servers privately compute statistical functions over the set of client values (e.g. the average number of visits). To achieve security against faulty or malicious clients, unlike Prio, Prio+ clients use Boolean secret-sharing instead of zero-knowledge proofs to convince servers that their data is of the correct form and Prio+ servers execute a share conversion protocol as needed in order to properly compute over client data. This allows us to ensure that clients' data is properly formatted essentially for free, and the work shifts to novel share-conversion protocols between servers, where some care is needed to make it efficient. Our overall approach is simpler than Prio and our Prio+ strategy reduces the client's computational burden by at least two orders of magnitude (or more depending on the statistic) while keeping server costs comparable to Prio. Prio+ permits computation of exactly the same wide range of complex statistics as the original Prio protocol, including high-dimensional linear regression over private values held by clients. We report detailed benchmarks of our Prio+ implementation and compare these to both the original Go implementation of Prio and the Mozilla implementation of Prio. Our Prio+ software is open-source and released with the same license as Prio.

Human Society is complex and involves interaction of many individuals in a society on different levels. Social Science and its fields study how people are organized into groups (Anthropology, Sociology), what are the core motivations of humans (Philosophy), what biological and psychological factors are involved in their behavior (Psychology), how humans compete and cooperate to achieve individual goals and society prosperity (Economics), and how a large society can organize and govern itself (Political Science). Underlying them all are laws of human social world that are the basis of human behavior, organization, interactions and decision making. Just like in Physics these laws can be scientifically studied and expressed using the tools of mathematics.The tools that have been used in the 4 papers included in the dissertation include: Social Network Analysis, Game Theory, Statistics and general tools of mathematical analysis. In the first 3 papers human organization into groups is explored by using Social Network as a base model of society. The first paper studies specific work behavior and organization in software development communities finding complex pattern of organization and providing a formula that can predict success of a project. The 2nd and 3rd papers use abstract mathematical models based on social networks and game theory to study human interactions, cooperation and success. The papers showed how social network structure plays a role in these interactions and helped to understand the very core of human interaction and how it can be modeled with game theory and what are important considerations for parameters used in creating specific game theory and network models.The 4th paper looks at human politics and elections and studies bias that can exist prior to elections with manipulation of electoral district maps to favor specific party, which is known as gerrymandering. Using tools of mathematics - functional analysis and statistics, the paper derives equations and core symmetries of biased and unbiased distribution of voters. These symmetries are analogues to physical symmetries that are the basis of physical laws.

We study the spectral theory of ergodic Schrödinger operators.The focus is on multi-dimensional Schrödinger operators that appear in quantum mechanics to model physical phenomena, e.g., electrons in a lattice or quasicrystal.We present two papers: \emph{On the spectra of separable 2D almost Mathieu operators} and \emph{Spectral estimates of dynamically-defined and amenable operator families}.The second paper is joint work with S.~Beckus.This dissertation explains what was done in those papers and how it closely relates to the research focus---to address specific problems related to the multi-dimensional case.The one-dimensional case is well developed since there are many results and many methods available, e.g., transfer matrices and Schrödinger cocycles.The multi-dimensional case on the other hand is notoriously difficult to study.In part of this dissertation, we consider a separable 2D discrete Schrödinger operator generated by 1D Almost Mathieu Operators.For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval.This complements a result by J.~Bourgain establishing that for fixed couplings the spectrum has gaps for some positive-measure Diophantine frequencies.The result generalizes to separable multi-dimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine.The proof is based on the study of the thickness of the spectrum of the Almost Mathieu Operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.In the other part of this dissertation, we consider a class of multi-dimensional Schrödinger operators.The Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the unimodular second-countable locally-compact Hausdorff group is amenable.Together with S.~Beckus it is proved that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $ \frac{1}{2} $-Hölder continuous.Also, the behavior can be improved resulting in the spectrum map being Lipschitz continuous if the coefficients are instead locally-constant.In 1990, the square root behavior was established by J.~Avron, P.~H.~M.~v.~Mouche, and B.~Simon for the Almost Mathieu Operator or, more generally, the quasiperiodic operators with Lipschitz continuous potentials.Our result extends the square root behavior to a bigger class of operators such as magnetic discrete Schrödinger operators with finite range and with Lipschitz continuous coefficients.

Beyond its normative relevance, the question of how institutions obtain, maintain, utilize, and lose authority is an important historical and conceptual issue. Early American politics in the colonial and revolutionary eras provides fruitful ground for exploring how the authority of institutions affects politics, and how political developments affect institutional authority. Applying game theory to questions of historical political development allows us to clearly identify strategic incentives and generalize from particular cases. In three papers, this dissertation combines historical case studies and formal models to investigate different aspects of legal authority in American political development.In the first paper, I investigate how purely formal constraints on political actors are enforced through coordination among citizens, and conditions under which such coordination fails. Specifically, this paper argues that economic power is a crucial variable for understanding the rule of law. I make this argument in two ways. First, I present a formal model to show how economic inequality undermines the rule of law. In the model, coordination among citizens increases the expense that elites face to implement an unconstitutional policy. But this constraint only binds when elites’ material resources are sufficiently low relative to those of citizens. Second, I relate the model to a historical case. In the early American republic, elites repeatedly expressed concerns about the expense of suppressing armed challenges to the constitutional order, supporting compromise instead. Amid contemporary concerns about the rule of law, this argument indicates the value of renewed attention to economic power.In the second paper, I analyze contrasting political effects of different structures of legal authority. In the process of both European and American state development, legal institutions developed from many separate venues with overlapping jurisdictions to a hierarchical structure with authority concentrated in the lawmaking institutions of the state. How did this change in the structure of legal authority affect political competition among groups in society? This paper presents a formal model to show that concentrating legal authority increases incentives for competing groups to invest in political organization. In turn, this increases political inequality: as all groups invest more effort, groups with a resource advantage are better able to make use of their advantage. The model is consistent with a case comparison of 18th century Pennsylvania and Massachusetts, two British American colonies with contrasting legal structures, patterns of political organization, and legal outcomes. The third paper provides a strategic mechanism by which the boundaries of authority between institutions incrementally change over time. Fundamental political change often occurs gradually, and it is therefore important to understand the strategic underpinnings of gradual institutional change. This paper presents a game in which players are entirely strategic and share proposal power, while showing how one player can gradually accumulate authority over time at the expense of another. The model analyzes a bargaining problem between two actors. Over an infinite number of periods, each actor has a chance each period to propose a bundle of future authority plus present policy, which creates the potential for actors to trade authority for temporary policy concessions. With some degree of noise in the division of authority, a proposal that transfers authority reduces the reservation value of the respondent, creating an incentive for the proposer to trade present policy for future authority. The model shows how ongoing, everyday political interactions gradually rearrange authority relationships among political institutions.

The following outlines my Ph.D thesis on Lagrange Inversion, Rogers-Ramanujan identities, and orthogonal polynomials. Sections $1-3$ serve as a summary of background material, much of which is based on various notes and papers either authored or co-authored by my advisor, Adriano Garsia. The subsequent sections contain related novel research. Specifically, section $4$ outlines the contents of a joint paper \cite{GG20} with Adriano Garsia, where we apply an expanded version of the combinatorial theory of orthogonal polynomials to produce new identities for the Fibonacci Polynomials. Section $5$ then details a purely combinatorial proof of $q-$Lagrange inversion, its application to Rogers-Ramanujan type identities, and more generalized Lagrange inversions involving an additional $t$ statistic.

In this paper we show a version of Johnson’s theorem for $\M(2,\R)$-sequences. In particular, motivated by the work of \cite{zhang2} and \cite{marx}, we show that one can identify the spectrum of Jacobi operator $J_{a,b}:\ell^2(\Z, \R)\rightarrow\ell^2(\Z,\R)$ by those energies $E \in \R$, whose cocycle map $B^{E} \in \M(2,\R)$ does not admit dominated splitting, i.e.$$\sigma(J_{a,b})=\{E\in\R:B^{E}\notin\CD\CS\}.$$ This result extends a well-known Johnson's theorem for Schr\"odinger operators, which identifies the spectrum of Schr\"odinger operator by all energy values $E \in \R$, such that the cocycle map $A^E \in \SL(2,\R)$ is not uniformly hyperbolic. Jacobi operator is a natural extension to Schr\"odinger operator, and dominated splitting generalizes the notion of uniform hyperbolicity.The biggest difference lies in the fact that for Schr\"odinger operator, the cocycle map takes value in $\SL(2,\R)$, whereas for Jacobi operator, the cocycle map takes value in $\M(2,\R)$. In particular, it is allowed to be singular, i.e. have zero determinant. This is one of the main challenges of this paper that we had to overcome. In this paper we also discuss the importance of Jacobi operators, develop definition of dominated splitting for $\M(2,\R)$-sequences, show stability theorem and finish this paper with the discussion of possibility of extending this result to dynamically defined Jacobi operators.

This work is divided into two parts which are concerned, respectively, with the combinatorics of the cardinals $\aleph_1$ and $\aleph_2$. The first part of the thesis contains the result due to the author and his advisor, Itay Neeman, that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum; this answers a long-standing open question in forcing. Most of Part 1 appears in our submitted manuscript \cite{GN}. After surveying the relevant background in the first chapter, we proceed in the second chapter to define the notion of a Partition Product. This is a type of iteration built out of smaller ones in specific ways, roughly with memory conditions on the names and with isomorphism and coherence conditions on the various ``memories." We will prove a number of useful facts about partition products in Chapter 2. In Chapter 3, we show how to construct so-called Preassignments of Colors in the context of partition products; this forms the technical heart of Part 1. And finally, in Chapter 4, we show how to construct partition products in $L$; in particular, we construct the partition product which yields the model witnessing our theorem.Part 2 of the thesis addresses questions about a variety of combinatorial principles on $\aleph_2$. In each chapter in Part 2, we will be concerned with showing that some amount of Stationary Reflection holds at $\omega_2$, more specifically showing that various amounts of stationary reflection are compatible with other principles of wide interest. In Chapter 5, we provide an overview of these combinatorial principles and some of their history; we also spend some time collecting standard facts about Mitchell-type forcings which we will use in the subsequent chapters. Chapter 5 concludes with a proof that various Mitchell-type posets and their quotients are proper, a result which we assume is known, but which we have not encountered ourselves elsewhere.Chapters 6 and 7 address questions arising from the recent paper \emph{The Eightfold Way} by Cummings, Friedman, Magidor, Rinot, and Sinapova (see \cite{8fold}). In Chapter 6, we answer an open question asked at the end of that paper by showing that it is consistent, from a Mahlo cardinal, that the Tree Property $(\TP)$ and Approachability $(\AP)$ both fail at $\om_2$, while stationary reflection ($\RP$) holds at $\omega_2$. The authors of \cite{8fold} obtained the consistency of this same configuration from a weakly compact cardinal; our result proves the consistency of this configuration from optimal assumptions. We remark here that we present the original proof discovered by the author of this thesis. Later, the author, working with John Krueger, provided a more streamlined proof of this same result; this proof will appear in the forthcoming paper \cite{GiltonKrueger8}. Chapter 6 also includes an unrelated Easton-style lemma for preserving stationary subsets of countable cofinality; this result is due to the author and Omer Ben-Neria.In Chapter 7, we show that for any Boolean combination, $\Phi$, of $\TP$ and $\AP$, $\Phi$ is consistent with a strong form of simultaneous stationary reflection on $\omega_2$, namely that every stationary $S\seq\om_2\cap\cof(\om)$ reflects almost everywhere. This strengthens some of the results from \cite{8fold}. In Chapter 8, we return to the model from \cite{GiltonKruegerHS}, making good on a promise from the postscript therein. In \cite{GiltonKruegerHS}, the author and John Krueger originally sought to show that stationary reflection on $\om_2$ is consistent with a large continuum, and we built an involved mixed-support iteration to achieve such a model. However, we later learned from I. Neeman that such a model can be constructed by simply adding Cohen reals over the original Harrington-Shelah model (\cite{HS}). In Chapter 8 we will show that after a modification of our original preparatory iteration, we may obtain a model in which $\RP$ and $\AP$ both hold, in which $2^\om>\om_2$, and in which there are neither special Aronszajn trees on $\om_2$ nor weak Kurepa trees on $\omega_1$. This is a configuration which cannot be obtained simply by adding Cohen reals over the original Harrington-Shelah model nor by the methods of disjoint stationary sequences from \cite{GiltonKrueger8}. We hope that this demonstrates the usefulness of such a mixed support iteration.In the final chapter, we provide a list of open questions which we would like to address in future work.

This dissertation is comprised of two papers studying the topology of certain spaces of symplectic embeddings. The first paper shows how given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result which claims that given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, one can prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible.The second paper proves how given two 2n-dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, a certain map from the n-torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map at the level of homology with mod 2 coefficients.

Broadly speaking, this thesis investigates mathematical questions motivated by computer science. The involved topics include communication avoiding algorithms, classical analysis, convex geometry, and low-rank matrix approximation. In total, the thesis consists of four self-contained sections, each adapted from papers the author has been a part of.The first two sections are both motivated by the Brascamp-Lieb inequalities, which are also often referred to as Hölder-Brascamp-Lieb inequalities. These inequalities have featured prominently in recent theoretical computer science work, due to connections to geometric complexity theory, harmonic analysis, communication-avoidance, and many other areas. Moreover, work generalizing the inequalities in various ways, such as to nonlinear versions, has been impactful to the study of differential equations.Section 1 studies the application of Hölder-Brascamp-Lieb (HBL) inequalities to the design of communication optimal algorithms. In particular, it describes optimal tiling (blocking) strategies for nested loops that lack data dependencies and exhibit affine memory access patterns. The problem roughly amounts to maximizing the volume of an object provided some of its linear images have bounded volume. The methods used are algorithmic.Another reason for the interest in these inequalities is because they are an interesting test case for non-convex optimization techniques. The optimal constant for a particular instance of the inequality is given by solving a non-convex optimization problem that is still highly structured. Of particular relevance to this thesis is that it can be formulated as a geodesically-convex problem, considered in the context of the manifold of positive definite matrices of determinant $1$. Even using the methods of Section 1, the procedure is not necessarily polynomial time, and this motivates further study of geodesic convexity.This lead to the work of Section 2, which discusses a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Grunbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least 1/(n+1) of the mass, n being the dimension of the manifold. As an application, the gradient oracle complexity of geodesic convex optimization is polynomial in the parameters defining the problem. In particular it is polynomial in -log(epsilon), where epsilon is the desired error. This is a step toward the open question of whether such an algorithm exists.The remaining two sections of the paper present a different research direction, randomized numerical linear algebra. Numerical linear algebra has long been an important part of scientific computing. Due to the current trend of increasing matrix sizes and growing importance of fast, approximate solutions in industry, randomized methods are quickly increasing in popularity. Sections 3 and 4 in this thesis aim to show that randomized low-rank approximation algorithms satisfy many of the properties of classical rank-revealing factorizations.Section 3 introduces a Generalized Randomized QR-decomposition (RURV) that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theoretical bounds capture the empirical behavior of the factorization.Section 4 concerns a Generalized LU-Factorization (GLU) for low-rank matrix approximation. We relate this to past approaches and extensively analyze its approximation properties. The established deterministic guarantees are combined with sketching ensembles satisfying Johnson-Lindenstrauss properties to present complete bounds. Particularly good performance is shown for the sub-sampled randomized Hadamard transform (SRHT) ensemble. Moreover, the factorization is shown to unify and generalize many past algorithms. It also helps to explain the effect of sketching on the growth factor during Gaussian Elimination.

This work investigates two regularization techniques designed for identifying critical points of the Yang-Mills energy.In the first half of the dissertation, we define a family of higher order functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for critical dimensions. Consequently, we generalize the results of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Rade and the bubbling criterion in dimension 4 of Struwe in the case where the initial flow data is smooth. This encompasses the contents of the author’s paper solo paper from 2014.In the second half of the dissertation we study an alternate type of regularization. In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over S4, for sufficiently small α values the SO(4)-invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold. This is an overview of the author’s solo paper from 2016.

Given a simplicial complex $\Delta$ and its barycentric subdivision $\sdd$, we explore two homogeneous systems of parameters: one is the elementary symmetric functions inside the $\N$-graded Stanley-Reisner ring of $\Delta,$ $\k[\Delta],$ and the other is written based on a balanced coloring of $\sdd$ which lives inside the $\N^d$-graded Stanley-Reisner ring of $\sdd,$ namely, $\k[\sdd].$ The first is stable under symmetries and the other is stable under colorful automorphisms of $\sdd.$ In this paper, we develop methodology to explore comparing the resolutions of $\k[\Delta]$ and $\k[\sdd]$ over their respective parameter rings. We then prove it in the trivial case, which is simply the coinvariant algebra with alternate grading and one non-trivial case.

Let K be a number field with ring of integers O. Consider the set of n x n alternating matrices with fixed rank, rThe principal ideas behind the proof were first introduced in a paper of Katznelson where the problem of counting matrices is reduced to one of counting lattice points. Because our matrices have entries in O rather than being restricted to the set of rational integers, the lattices of Katznelson are replaced in the present work with O-modules. The generalization from the rational numbers to K renders the standard tools of lattice-theory less directly applicable, and we rely on the Minkowski map and novel arguments to, at various turns, reduce to the lattice case, or abstract results from lattices to O-modules. Ancillary results in our work include a new formula for the discriminant of a torsion-free O-module in terms of its pseudo-basis and a novel structure theorem about the set of alternating matrices whose rows lie in a specified O-module.

This dissertation constructs a Morse-Bott version of embedded contact homology (ECH). The dissertation is comprised of two parts, corresponding to the two papers written by the author as a graduate student at UC Berkeley. The first part explains how to compute ECH in the Morse-Bott setting when certain transversality conditions are met and provided a certain correspondence theorem is true; and gives a large class of examples where the transversality conditions are satisfied. The second part provides the analytic foundations of the first part by giving a proof of the correspondence theorem.

The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew, is a variant of the classical de Rham-Witt complex which provides a conceptual simplification of the classical construction and which is expected to produce better results for non-smooth varieties. In this paper, we introduce a generalization of the saturated de Rham-Witt complex which allows coefficients in a unit-root $F$-crystal. We define our complex by a universal property in a category of so-called de Rham-Witt modules. We prove a number of results about it, including existence, quasicoherence, and comparisons to the de Rham-Witt complex of Bhatt-Lurie-Mathew and (in the smooth case) to crystalline cohomology and the classical de Rham-Witt complex with coefficients.

This paper documents a computer-assisted procedure for rigorously analyzing small hyper-bolic 3-manifolds. Briefly, we will define a compact six-dimensional space P that parame-terizes pairs of elements of Isom(H3 ), and then construct a regular binary space partition(BSP) tree which subdivides P into subregions Pi and whose leaves are — with a few ex-ceptions — labeled with killerwords. These killerwords will encode miniature proofs that Picannot contain any points which correspond to particular choices of pairs of generators ofany torsion-free discrete group of Isom(H3 ). This tree of mini-proofs will then be used to ex-haustively isolate all possible manifolds which can have properties related to the dimensionsof P.This method has been used as a foundation for tackling multiple long-standing problemsat the interface of hyperbolic geometry and low-dimensional topology. In particular: topo-logical rigidity of hyperbolic 3-manifolds; the generalized Smale conjecture for hyperbolic3-manifolds; finding the closed 3-manifold of least volume; and the Gordon conjecture onexceptional Dehn fillings.

In this paper we introduce the notion of the osculating space of a minimal surface as a vector bundle formed from the span of the z-derivatives of a conformal parametrization of the surface. We show that a minimal surface being J-holomorphic is equivalent to the dot product being fully degenerate on the osculating space. We then show that any minimal surface lying fully in R^n with osculating space of dimension half the ambient space must either be J-holomorphic or has a complex structure that is not compatible with the Euclidean metric. Considering the dot product as a bilinear form on the osculating space, we prove some strong restrictions on the dimensionality of the null space of this bilinear form. We show that the osculating space behaves nicely with respect to decomposable minimal surfaces and that we can always decompose a minimal surface into a J-holomorphic minimal surface and a minimal surface for which the dot product is non-degenerate on the osculating space. Finally, we prove that complete finite total curvature stable surfaces of genus one in R^5 are holomorphic under the condition that a certain cover of the surface is stable and the normal bundle is not topologically trivial.

This dissertation will show the ways in which generalized geometry elucidates the study of pluriclosed flow. In their 2009 paper, Streets and Tian introduce pluriclosed flow -- a parabolic flow of pluriclosed metrics -- and classify some static solutions. In 2018, Streets expanded this into a geometrization conjecture for compact, complex surfaces. The author is able to use these tools to show an equivalence between pluriclosed flow and a non-linear, coupled Hermitian-Yang-Mills type flow. From there, the author is able to more geometrically prove a result of Streets and Warren -- an Evans-Krylov theorem for pluriclosed flow. The author is also able to use this equivalence to prove long-time existence and convergence of the flow on Bismut-flat manifolds and surfaces of non-negative Kodaira dimension.

Melanoma is an incredibly dangerous form of skin cancer. In 2023 alone, the American Cancer Society estimates that about 7,990 Americans will die from new melanoma cases. Image analysis tools that automate the diagnosis of melanoma will improve dermatologists' diagnostic accuracy and should provide a significantly better detection of melanoma. In this paper, we present an application of modern deep learning architectures using the images of skin lesions in combination with patient-level contextual information to detect the presence of melanoma. More specifically, we used the latest variant of the residual network, the ResNeSt model, to leverage its deep architecture and channel-wise attention mechanisms for melanoma detection. In parallel, we used a multi-layer perceptron model to process patient-level data, and fused its output with the extracted features from the ResNeSt model. To compare and contrast performances, we trained a similar model using a smaller, more generic, convolutional neural network in addition to the larger ResNeSt network. After properly tuning all associated hyperparameters, our primary model obtained a balanced accuracy of 0.853 and a ROC-AUC score of 0.93. The split-attention network was shown to improve the total ROC-AUC by approximately 9%, suggesting that our model was able to successfully leverage the ResNeSt design to extract the most important features and most relevant cross-feature interactions for the detection of melanoma.

In recent years, there has been growing interest in modeling volatility as a stochastic process driven by a non-Markovian process, due to empirical evidence showing that the autocorrelation function of volatility decays as a power function. This paper investigates the use of a time-scaled fractional Ornstein-Uhlenbeck process to model volatility and applies this model to derive an optimal delta hedging strategy. By incorporating non-Markovian processes into our model, we aim to provide insights into the behavior of volatility in financial markets and explore potential benefits for option pricing and risk management strategies.

In this paper, we seek to provide counter examples to two volume comparison lemmas found in \cite{GP2} if we generalize their assumptions to a lower Ricci curvature bound. Second we seek to further understand Riemannian manifolds which contain embedded submanifolds of certain focal radius. First, in papers \cite{GP1}, \cite{GP2} Grove and Petersen discussed the relationship between bounds on sectional curvature, radius, and volume and its effects on the topology of a closed Riemannian manifolds. They prove that for manifolds with a lower sectional curvature bound, an upper radius bound and almost maximal volume, one can give topological equivalence to either $S^n$ or $\mathbb{R}P^n$. It was later proved to be diffeomorphic \cite{PSW1}. Also, Grove and Petersen showed that the limit space of a convergent sequence of manifolds with maximal volume has certain geometric properties. In the proofs of these theorems they use two powerful volume comparison lemmas. We discuss why the methods used in \cite{GP2} cannot be extended in general to manifolds with a lower Ricci curvature bound by looking at some interesting counter examples. Second, in a paper by Guiljaro and Wilhelm \cite{GW1} we see a relationship between closed embedded submanifolds of maximal focal radius and topological type, and seek to understand the necessity of the submanifold being closed by providing counterexamples in which we consider Manifolds with embedded open submanifolds of certain focal radius which do not satisfy the conclusions of the results in \cite{GW1}.

In 1980, Murray Marshall proved that the category of Quaternionic Structures is naturally equivalent to the category of abstract Witt rings. This paper develops a combinatorial theoryfor finite Quaternionic Structures in the case where $1=-1$,by demonstrating an equivalence between finite quaternionic structures and Steiner Triple Systems (STSs) with suitable blockcolorings. Associated to these STSs are Block Intersection Graphs (BIGs)with induced vertex colorings. This equivalence allows for a classification of BIGs corresponding to the basic indecomposable Witt rings via their associated quaternionic structures. Further, this paper classifies the BIGs associatedto the Witt rings of so-called elementary type, by providingnecessary and sufficient conditions for a BIG associated toa product or group extension.

Inspired by the paper of Klagsbrun, Mazur and Rubin [5], this thesis investigates the disparity of 2-Selmer ranks of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. In the first part, we calculate the density of quadratic twists of E with even 2-Selmer ranks under two different counting methods. First we count twists by elements inside a large convex body of the Euclidean space that contains the integer lattice of K. The second counting method is counting quadratic twists E^L by the norms of the finite part of conductors of quadratic extensions L/K. Under both counting methods we give an explicit formula for the densities, which are finite products of local factors. In the second part of the paper we give a method that uses Tate’s algorithm to calculate the size of the cokernel of the local norm maps of E at places over 2, assuming that E has good reduction. With this method we can extend Kramer’s early work on the cokernel of the local norm maps, and compute the local factors mentioned above in some additional cases.

In 1934, H. Whitney presented a series of papers which discussed how to determine whether a function or a jet of order m is the restriction of a Cm-function on Rn. In the first paper of the series, Whitney's Extension Theorem was proved. In the latter, Whitney answered special cases of the following questionQuestion(Whitney's Extension Theorem, WEPn,m) Let f be a continuous function from a closed subset E of Rn to R.How can we determine whether f is the restriction of a Cm-function on Rn?In this dissertation, we work in o-minimal expansions of real closed ordered fields. Definable versions of Whitney's Extension Theorem and Whitney's Extension Problems will be discussed in this context. Definable set-valued maps are also studied; a definable version of Michael's Selection Theorem will be proved and used, in combination with a definable version of Whitney's Extension Theorem, to give a positive answer to a definable version of WEPn,1.In addition to the above problems, we also discuss smoothing problems. This is inspired by a series of papers by A. Fischer.In this series, a construction of a definable Cm-approximation of a definable locally Lipschitz function is provided. Here, we also work in an o-minimal expansion of a real closed field and relax the condition further to just continuous.

We examine the Carlson-Simpson Lemma (VW(k,l)), which is the combinatorial core of the Dual Ramsey Theorem, from the perspective of Reverse Mathematics. Our results include the following:Working in the system BSigma_2^0, we carry out the construction of a failure of the ordered version of the Carlson-Simpson Lemma OVW(k,l), which was introduced in a paper by Miller and Solomon. This observation implies that we can construct such a recursive counterexample in the model of SRT_2^2 that was discussed in a recent paper by Chong, Slaman and Yang. It follows that SRT_2^2 does not prove OVW(k,l) over RCA_0.We also show that the strength of the principle VW(k,l) is independent of the number of colors l being used.By proving that VW(k,l) is not conservative over RCA_0 for arithmetical sentences, we conclude that VW(k,l) is not provable from any theory that is conservative over RCA_0 for arithmetical sentences.

This dissertation consists of three papers I have written or helped write during my time at UC Berkeley. The three papers all fall under the common theme of exploring connections between free probability and planar algebras.In Chapter 2, an amalgamated free product algebra, $\cN(\Gamma)$, is constructed for any connected, weighted, loopless graph $\Gamma$, and its isomorphism class can be nicely read off based on the weighting data on the graph. This construction was heavily influenced by an algebra appearing in in work of Guionnet, Jones and Shlyakhtenko, and it is used, along with some standard embedding arguments, to show that the factors of Guionnet, Jones, and Shlyakhtenko appearing in are isomorphic to $L(F_{\infty})$ when the planar algebra is infinite depth.Chapter 3 is the paper ``Rigid $C^{*}-$tensor categories of bimodules over interpolated free group factors" which was co-authored with Arnaud Brothier and David Penneys. In this paper, we establish an unshaded planar algebra structure (with multiple colors of strings) which can be used to model a countably generated rigid $C^{*}-$tensor category, $C$. We use this to construct a category of bifinite bimodules over a $II_{1}$ factor $M_{0}$, and we show that this category is equivalent to $C$. Finally, we use the work in Chapter \ref{chap:graph} to show that the factor $M_{0}$ is isomorphic to $L(F_{\infty})$.Chapter 4 is a note regarding multishaded planar algebras. The problem studied consists of placing the ``Fuss-Catalan" potential on a specific kind of subfactor planar algebra $Q$, which is the standard invariant for and inclusion $N \subset M$ with an intermediate subfactor $P$. This potential is best understood by augmenting $Q$, forming a bigger multishaded planar algebra $\mathcal{P}$. The isomorphism classes of the algebras $M_{\alpha}$ associated to $\mathcal{P}$ will be computed explicitly. While the isomorphism class of the von Neumann algebras $N_{k}^{\pm}$ associated to $Q$ are still not yet known, they will be shown to be contained in a free group factors and contain a free group factors. This potential is shown to yield a nice free product expression for the law of $\cup$, an element which plays a critical role in understanding algebras that arise from this construction. Many of the ideas in this chapter influenced the work in the Chapter 3.

This dissertation is a collection of four papers involving embedded contact homology (ECH). ECH is a three-manifold invariant recently defined by Michael Hutchings. The dissertation covers both technical results about ECH (for example a proof that the absolute grading in ECH is a topological invariant) and applications of ECH to Reeb dynamics and symplectic embedding problems. All four papers were written while the author was a graduate student at UC Berkeley.

This thesis aims to develop a comprehensive picture of large deviations of affine processes. Affine processes have been studied so extensively, that it would not be foolish to admit that their theory is somewhat complete. Though some questions still remain, the thesis of [cuchiero2011] establishes a comprehensive understanding of affine processes on general convex state spaces. Starting with the basic definition of an affine process as a time-homogeneous Markov process with conditional marginal distributions having a characteristic function with log-affine dependence on the initial state, this paper establishes numerous useful properties of the process. Among these properties include that affine processes have cadlag modifications (paths have left-limits and are right-continuous), and such modifications are themselves jump-diffusions with respect to their natural filtration. This result allowed [keller2015] to prove the transform formula for the real and complex moments, which establishes existence of the process and its finite moments, so long as a system of generalized Riccati differential equations have some solution (which need not be unique).Despite this very comprehensive understanding, large deviation asymptotics of families of affine processes do not have a similar general approach. Seeing as the class of affine processes is quite general, surely there are large deviation principles for certain families of affine processes. The simplest of which include families of state-independent processes like the Brownian motion or Poisson process, whose large deviations are understood asymptotically ε → 0 as time and space are scaled linearly with respective factors 1/ε and ε; this is a simple consequence of the classical Mogulskii's theorem (see [dembo2010]). More general results indeed exist, like [kang2014] which proves a large deviation principle for continuous affine processes on the so-called canonical state space. We are concerned with large deviation principles of similar asymptotics---those which could be interpreted as regularizing the noise of a dynamical system---but on the level of generality that affine processes are already understood.Though our original intent was to prove a large deviation principle for the general class of affine processes on convex state spaces, we discovered other remarkable facts along the way. The following two are of key interest. Firstly, the measure-change techniques one often uses in large deviations are typically split into those of a finite-dimensional (twisting/tilting) and infinite-dimensional (exponential martingale) flavor, but for affine processes one may in fact parameterize the finite-dimensional measure-densities in terms of the infinite-dimensional ones. This was not a fact seen in [kang2014], which---as they mentioned---forced them to choose a proof using exponential martingales. We show how their original finite-dimensional approach was sufficient in establishing an integral form for their rate function. Secondly, seeing as no large deviations result has been made on the general level of affine processes, there has been no attempt at understanding the structure of the associated rate functions. In our work of developing a proof for the principle, we saw the techniques mentioned in [duffy2004], which ultimately inspired us to develop a calculus associated with our rate functions. These results remarkably have no dependence on the affine assumption, and so if a future large deviation principle is to be proved for asymptotic families of jump-diffusions of the same regularization, this thesis will have already established semi-closed forms for the associated rate function.All this said, we do still prove a large deviation principle for the general class of affine processes on convex state spaces. While we require usual light-tail assumptions, we feel the generality is comparable to the treatment of affine processes in [cuchiero2011] and [keller2015]. In fact, we even choose to present affine processes with similar generality up until the point of us needing the assumption. At this juncture, we are able to represent the affine processes as special jump-diffusions, which have representations that are far more familiar to the larger audience that have studied Ito processes. For those that have only seen continuous diffusions, we also offer an Appendix which we feel is a great reference to the calculus associated with jump-diffusions.It goes without saying that our proof is not without the incredible theory of large deviations that already exists. We in particular discuss it at the level of understanding one has from reading [dembo2010], but we for the most part cite [feng2006] and its large deviation results on the Skorokhod space. We also use [puhalskii2001] to use a density argument in developing an integral form for our rate function.

Motivated by the Calabi-Yau/Landau-Ginzburg correspondence, we study BCOV-type torsion for Landau-Ginzburg B-models. To pave the way for understanding the Landau-Ginzburg model from the index theoretic point of view, we study Witten deformation, which was introduced in one of Witten's extremely influential papers in 1987, on noncompact manifolds. In Part I, we explore the analysis of Witten deformation on noncompact manifolds: we show the asymptotic growth of eigenvalues and the decay of eigenfunctions near infinity as well as the expansion and the estimate of the heat kernel for Schr\"odinger-type operators on noncompact manifolds. With heat kernel expansions of Sch\"ordinger-type operators, we define the Ray-Singer metric (analytic torsion) for the Witten deformation associated with some flat vector bundle and explore several nice properties of it (independence of metrics, Cheeger-Muller/Bismut-Zhang theorem, e.t.c.). Next, we move on to study Landau Ginzburg B-models in Part II. In chapter 6, we investigate the genus-zero theory for Landau-Ginzburg B-models, and establish Calabi-Yau/Landau-Ginzburg correspondence for the $tt^*$ structure and the Weil-Peterson type metric.

The purpose of this paper is to analyze how to predict a business’s operation status during the spread of COVID-19 between the years 2020 to 2022. Exploratory Data Analysis techniques including barplot, geographical visualization, and word cloud are utilized. Three models are chosen to fit and predict business operation status: Logistic Regression, Support Vector Machine, and Bidirectional Long Short Term Memory. After tuning the probability cutoff threshold to 0.0510 with the F1 score, the logistic regression using the solver Newton's Conjugate Gradient generates an accuracy score of around 0.8120. Support Vector Machine algorithm leads to an accuracy score of around 0.9763. By using GloVe for word embedding, bidirectional LSTM results in a loss of 0.1113 and an accuracy score of 0.9758 after 2 epochs. From the performance of the three models, SVM seems to be the most effective and efficient in predicting business operation status through user reviews.

In recent work published by Biswal, Chari, Shereen, and Wand the authors defined a family of symmetric polynomials indexed by pairs of dominant integral weights, G_{\nu, \lambda}(z,q) where z=(z_1, \cdots. z_{n+1})\in\C^{n+1}, and determined that G_{0, \lambda}(z,q) is the graded character of a level two Demazure module for sl_{n+1}[t]. The aim of this thesis is to construct analogues of these polynomials for the generalized Demazure modules for so_{2n}[t] as they are presented by Chari, Davis, and Moruzzi. We do this by constructing modules which interpolate from the presentation provided in that paper and local Weyl modules. We then create short exact sequences between them to relate their graded characters. This allows us to identify coefficients in the corresponding graded characters with the coefficients in G_{\nu,\lambda}(z,q).

In this dissertation, we determine best-to-date upper bounds on the resolvent degree of solving general polynomials. Chapters 1 and 2 provide the necessary background information and Chapters 3 and 4 establish new results.In Chapter 1, we provide a high-level introduction of the dissertation and provide a history of the literature on resolvent degree. We also establish standard mathematical notation and terminology.In Chapter 2, Section 1, we recall the definitions of essential dimension and resolvent degree, as well as some standard results. In Section 2, we give an introduction to the theory of Tschirnhaus transformations and explain how we will use this theory to obtain upper bounds on resolvent degree by determining special points on Tschirnhaus complete intersections.In Chapter 3, Section 1, we recover the classical notion of the polars of a hypersurface at a point, so that we can introduce the polar cone of a hypersurface at a point and the connection between the polar cone at a point and lines on the hypersurface through that point. We then extend these notions to intersections of hypersurfaces and introduce iterated polar cones and their connections to k-planes in Section 2. Finally, we recover the obliteration algorithm of Sylvester and present it in a modern geometric context in Section 3.In Chapter 4, we establish our bounds on resolvent degree and Sections 1, 2, and 3 each highlight different constructions. In Section 4.4, we indicate obstructions to further bounds on resolvent degree via iterated polar cone constructions. We then proceed to establish approximations via elementary functions and compare the bounds we obtain with previous bounds in Sections 5 and 6. Finally, we posit several questions for future research in Section 7.Appendix A contains numerical information regarding our bounding function G'(m) and the previous best bounding function F(m), as well as implementations of the geometric obliteration algorithm. Appendix B contains three translations [Sut2019, Sut2021A, Sut2021B] of papers in the classical resolvent degree literature [Che1954, Kle1905, Wim1927].

We explore methods that allow us to change an existing network's structure to achieve a desired PageRank. Being able to change the ranking of the nodes is desirable for any network for which the PageRank determines the importance of the nodes. We search for a perturbation to the original network that when added will give the desired ranking. We derive multiple algorithms to solving this Inverse PageRank problem using both constrained optimization techniques and root finding approaches. In these approaches, we ensure that the answer we find satisfies the conditions for an adjacency matrix of an undirected graph, and we explore techniques that promote sparsity of the final solution. We want sparse solutions so that we only have to change a fraction of the existing edges. We develop a general approach that can be applied to any network. We describe C++ implementations for a set of algorithms, the best of which is scalable to thousands of vertices and much faster than Matlab-based approaches we have tried. This same set of algorithms is subjected to a number of tests designed to study the relationship between the initial guess and the final solution. Our ultimate goal is to solve the inverse PageRank problem on a social network of over 6000 venture capitalists to try and reproduce the top 10 list from a previously published set of rankings. Using a root-finding algorithm where nonnegativity constraints are handled implicitly, we are able to find a highly sparse solution to this inverse problem. This thesis consists of a draft of a research paper, co-authored by my advisor and me, that we plan to submit to a peer-reviewed journal of applied/computational mathematics.

Boij-Soederberg theory represents a breakthrough in our understanding of free resolutions. Boij and Soederberg proposed the radical perspective that the numerics of graded free resolutions are best understood "up to scalar multiplication." The conjectures of Boij and Soederberg were then proven in a series of papers. Further, these papers describes the cone of cohomology tables of vector bundles on projective n-space by illustrating a duality with the cone of free resolutions. We use the phrase Boij-Soederberg theory to refer to the study of these two cones and the corresponding decomposition theorems.Applications of graded free resolutions appear throughout algebraic geometry, commutative algebra, topology, combinatorics, and more; since Boij-Soederberg theory provides a structure theorem about the shapes of graded free resolutions, we might hope to apply the theory widely. One such application arose instantly, leading to a proof of the Herzog-Huneke-Srinivasan Multiplicity Conjecture, which had been open for decades.However, Boij-Soederberg theory is such a radical departure from usual approaches to free resolutions that it is not immediately applicable to many of the situations where free resolutions arise. Boij-Soederberg theory is almost orthogonal to all previous approaches to understanding graded free resolutions, as the theory uses the combinatorial structure of Betti diagrams to group modules into families, whereas more traditional approaches use flatness to understand families of modules. The overarching goal of this thesis is thus to connect Boij-Soederberg theory with some of the previous avenues of research where graded free resolutions have arisen, and we pursue this theme in several directions.Chapter 2 builds a framework for overcoming the limitation of working "up to scalar multiplication'". Namely, we apply Boij-Soederberg theoretic results about the cone of Betti diagrams in order to investigate the integral structure of the semigroup of Betti diagrams. Our main results show that this semigroup is locally finitely generated, but that it can otherwise be quite pathological. In addition, we construct a number of nontrivial obstructions which prevent a diagram in the cone of Betti diagrams from belonging to the semigroup of Betti diagrams.In Chapter 3, we consider the question of whether the Boij-Soederberg decomposition of a Betti diagram of a module is related to a flat deformation of the module. This is an essential mystery raised by Boij-Soederberg theory, and we provide the first results in this direction by producing large families where the Boij-Soederberg decomposition of a Betti diagram closely reflects a special filtration of the module. These results suggest that Boij-Soederberg theory might have deeper, as yet undiscovered, consequences for free resolutions. In addition, we provide applications to the classification of very singular spaces of matrices. This application is related to questions about classifying vector spaces of low rank matrices and torsion-free sheaves on projective spaces. This chapter is based on joint work with David Eisenbud and Frank-Olaf Schreyer.In Chapter 4, we apply Boij-Soederberg theory to prove a special case of a famous conjecture in commutative algebra: the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. Whereas the Multiplicity Conjecture is about the possible shapes of graded free resolutions, the Buchsbaum-Eisenbud-Horrocks Rank Conjecture is about the possible sizes of graded free resolutions. The conjecture is closely related to a topological conjecture of Carlsson about certain finite group actions on products of spheres. We prove a broad new case of the rank conjecture, and our method of proof--which involves a combination of Boij-Soederberg theory and optimization--is essentially unrelated to any previous work on the conjecture.Finally, in Chapter 5, we apply Boij-Soederberg theory to the study of the asymptotics of free resolutions. Namely, we provide a lower bound for the Betti numbers of I^t when I is an ideal generated in a single degree. This builds on recent studies of asymptotic Castelnuovo-Mumford regularity, by Cutkosky, Herzog, Kodiyalam, Trung, and Wang.

First, we show that a Ricci flow can be started from a non-compact complete manifold, if the manifold is non-collapsed and satisfies a lower bound for many known curvature conditions.In this theorem, we do not need the manifold to have bounded curvature, which was assumed inan earlier work by Bamler-Cabezas-Rivas-Wilking.Second, we show that a Ricci flow can be started from a 3d complete manifold with non-negative Ricci curvature.This gave a partial affirmative answer to a conjecture by Topping.we prove it by generalizing the concept of singular Ricci flow by Kleiner and Lott to non-compact initial conditions.Third, we find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies Hamilton's flying wing conjecture. We also show that the scalar curvature does not vanish at infinity in a 3d flying wing. For dimension $n\ge 4$, we find a family of non-collapsed, $\mathbb{Z}_2\times O(n-1)$-symmetric, but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. This thesis is a composition of the following three papers of the author: "Ricci flow under local almost non-negative curvature conditions", "Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows", "A family of 3d steady gradient solitons that are flying wings" \cite{Lai2019,Lai2020,Lai2020a}.

This dissertation consists of several papers. First, we start off introducing domain adaptation theory and briefly introduce optimal transport. Such an introduction allows the reader to understand why studying problems in optimal transport theory is so valuable. Our first key result establishes bounds between regularized and unregularized optimal transport. Instead of using an entropic regularization, which is used in the Sinkhorn divergence, we regularize using dual potentials in a reproducing kernel Hilbert space. After this, we derive sample complexity bounds for the regularized optimal transport problem, and we show this is a substantial improvement over unregularized optimal transport. With these two results, one can approximate the theoretical optimal transport distance. Next, we prove the first and second moments of the source and target distributions are enough to determine explicitly the optimal transport map and also that this is a linear mapping. Furthermore, we propose an alternative regularization for the transport map between two distributions. After this, we briefly diverge from optimal transport theory and introduce work on prior elicitation. In particular, we extend a result from \cite{Pinelis2017} on non-asymptotic bounds for maximum likelihood estimators to that for M-estimators. Crucially, we show sufficient assumptions for these to hold and use these to theoretically justify our prior elicitation objective. Last, we return to optimal transport and introduce a variant to compare multiple probability measures, which we call sliced multi-marginal optimal transport. There, we propose a paradigm based on random one-dimensional projections.

This thesis presents multiple fundamental mathematical contributions to Generalized Linear Models (GLMs) ubiquitous to the sciences. The methodologies considered are shown to overcome biased estimates for parameters of interest in the sciences through new mathematical results and their applications in both nonparametric and parametric settings. The results are shown to be uniformly better in comparison to existing widely used methods in the sciences. In extensive simulation studies the methodologies outperform existing Artificial Intelligence (AI) and Machine Learning (ML) methods in the sciences for all around better Model fits, Inference and Prediction (MIP) results without losing interpretability of the parameter estimates. This is because the mathematical construction and their accompanying mathematical foundations ensure that the estimation procedure strongly converges to the parameters of interest. In the first application, I present a parametric version of the methodology (© Elsevier and Journal of Informetrics) titled “Functional analysis of generalized linear models under non-linear constraints with applications to identifying highly-cited papers.” In the second application, I extend this methodology in an entirely nonparametric setting which gives equivalent results to the parametric formulation under various circumstances, but may outperform it as well in others, especially if the underlying Data Generating Process (DGP) is asymmetric. Furthermore, I show that the categorical data models on which the methodologies are applied can be extended to any GLM, continuous or otherwise, while maintaining model interpretability and convergence results. In addition, I present a new prediction performance diagnostic statistic, called Adjusted ROC Statistic (ARS), which allows us to compare whether the prediction performance of various models fitted are statistically different. The nonparametric methodology is then further extended to give a new formulation of the binary regression framework widely used in the sciences. Through extensive simulation studies I show that this version of the methodology is more robust than the previous versions discussed. This general framework is then extended to various AI and ML applications widely used in the sciences. The entirety of the work also has some important consequences for our continued discussion on “statistical significance” vs. “scientific significance.” This includes the need for us to consider the strength of convergence of our methodology in addition to the subtle connections between Topological Spaces and Measure Spaces. Each of which are crucial to ensure almost sure convergence of the parameter estimates through the estimation algorithm presented termed, Latent Adaptive Hierarchical EM Like algorithm or LAHEML. As such, the results present a significantly expanded and more accurate toolset for Mathematicians, Statisticians, Scientists and Decision Makers at all levels for better model fit, inference and prediction outcomes.

Toric manifolds and toric varieties have played an important and particularly illuminating role in algebraic, symplectic, and K\"ahler geometry going back at least to the 1970's. The high degree of symmetry in many cases allows one to reduce the complexity of a given geometric question, giving the impression that we can really ``see'' the structure. A notable example of this in the K\"ahler setting is Donaldson's classic paper [27]. Until relatively recently, this has been mostly confined to the setting of compact manifolds (complete varieties). Both the algebraic and the symplectic perspectives have come to be fairly well understood individually in the non-compact case, but there has been little application of the intersection of the two theories, which naturally comprises toric K\"ahler geometry. The notable exception to this rule is the case of toric K\"ahler cones [56, 55, 39, 17, 16]. These non-compact manifolds inherit a great deal of structure from their compact \emph{link}, which are interesting spaces in their own right. However, these manifolds are (typically) highly singular at precisely one point.In this dissertation, we describe a procedure for the study of K\"ahler geometry on smooth non-compact toric manifolds. This involves tying together the algebraic and symplectic perspectives. We study a class of toric manifolds and show that here we can apply a non-compact version of the classical Delzant classification together with its application to K\"ahler geometry [4, 41, 24, 42]. We apply this to non-compact shrinking K\"ahler-Ricci solitons, where we propose a useful class of complete K\"ahler metrics, and ultimately prove a general uniqueness theorem for shrinking K\"ahler-Ricci solitons on non-compact toric manifolds.