# Browsing by Author "Miller, Ezra"

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Item Open Access Algebraic Data Structure for Decomposing Multipersistence Modules(2020-11-12) Li, JoeySingle-parameter persistent homology techniques in topological data analysis have seen increasing usage in recent years. These techniques have found particular success because of the existence of a complete, discrete, efficiently computable invariant to describe persistence modules in the single-parameter case: the barcode. Attempts to develop an equally robust theory of multiparameter persistent homology, however, have been slow to progress because there is no natural multiparameter analogue to the barcode. Relatively little is known about the structure of decompositions of multiparameter persistence (multipersistence) modules or how to classify their indecomposables. In fact, even for the problem of computing decompositions, there currently is no generalization to multiple parameters of the decomposition algorithm from single-parameter persistent homology. In this paper, we define a new algebraic data structure, the QR code, which was first proposed in https://arxiv.org/abs/1709.08155 but was formulated somewhat erroneously. Additionally, we prove a theorem stating that the QR code recovers all the information of the module it encodes. We suggest that this new data structure, which seeks to encode a module using births and deaths rather than births and relations, may be the correct language in which to solve the problem of decomposing arbitrary finitely generated multipersistence modules.Item Open Access Determinant, Wall Monodromy and Spherical Functor(2015) Wang, KangkangWe apply the definition of determinant in the compactified moduli space as a generalization of the discriminant. We study the relationship between the wall monodromy and the determinant in the GIT wall crossing. The wall monodromy is an EZ-spherical functor in the sense of Horja. By constructing a fibration structure on Z, we obtain a semi-orthogonal decomposition of the derived category of coherent sheaves of Z, hence decompose the EZ-spherical functor into a sequence of its subfunctors. We also show that the intersection multiplicity of the discriminant and the exponent of the discriminant in the determinant both have their correspondences in this decomposition.

Item Open Access Diagrammatics in Categorification and Compositionality(2019) Vagner, DmitryIn the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends—categorification and compositionality—in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment—from a pedagogically novel perspective that introduces almost all notion via diagrammatic language—of the categorical machinery with which we may express the broader no- tions that found the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. Finally, the work culminates in a manifesto on the philosophical role of category theory.

Item Open Access Geometry of Stratified Spaces for the Analysis of Complex Data(2024) Arya, ShreyaTraditionally, statistical data has been in the form of elements of Euclidean space. However, as data complexity increases, it is assumed to lie on lower dimensional non-linear space such as smooth manifolds, in what is known as the manifold hypothesis. Nonetheless, real-world data is not always in the form of smooth manifolds but, in general, can lie on stratified spaces. In this thesis, we explore the geometry of stratified spaces with the overall objective of enabling statistics on these spaces. More specifically, we provide answers to the following two problems.

A fundamental task in object recognition is to identify when two shapes are similar. One approach to rendering this as a precise mathematical problem is to look at the space of all shapes and define a metric on it. This approach has been taken by renowned statisticians and mathematicians like Kendall, Grenander, Mumford, Michor, and others. In this thesis, we provide an algebraic construction of the moduli space of shapes and define metrics on it with the objective of developing a statistical theory on shapes. The construction is far more general than existing constructions, as it doesn't restrict `shapes' to smooth manifolds and includes a broad category of spaces, including many stratified spaces. The foundation of this construction relies on the topological analogue of the Radon transform, building on the work of Schapira who showed that such transforms are injective.

This thesis also provides a starting point for developing a theory of diffusion processes on general stratified spaces. On Euclidean spaces, Brownian motion is constructed by taking scaled limits of random walks. This approach is challenging because stratified spaces are not only non-linear and lack addition but also the tangent spaces of stratified spaces are non-linear, unlike smooth manifolds. So, instead, we define Brownian motion on stratified spaces by taking appropriate limits of Dirichlet forms. Sturm took this approach for general metric measure spaces, where he came up with a measure-theoretic condition required for these Dirichlet forms to converge properly. We prove this is the case for certain compact subanalytic spaces.

Parts of the thesis are based on joint work with Justin Curry and Sayan Mukherjee.

Item Open Access Implementing Non-Canonical Sylvan Resolutions(2021-04-19) Klett, PhoebeAn implementation in Macaulay2 of the Non-Canonical Sylvan Resolutions explicitly defined in Minimal resolutions of monomial ideals. Intuition, explanation is given for the theoretical as well as the applied math, and any choices made are justified. A practical user's guide to the software is provided, and by-hand examples help to give a conceptual underpinning of the work done.Item Open Access Invariants and Metrics for Multiparameter Persistent Homology(2019) Thomas, AshleighThis dissertation is about building fundamental techniques for comparing data via a geometric and topological data analysis method called multiparameter persistent homology. The techniques used are largely algebraic. A new summary statistic, called the multirank function, is introduced as a measure of persistence output that detects relationships between important features of the data being analyzed. Also introduced is a technique for modifying existing metrics on the space of persistence outputs. Existing metrics can return infinite distances, which do not give as much information as a finite distance; the proposed modification gives fewer such situations. The final chapter of this dissertation details work in a long-term biology research project. Persistence is used to study the relationship between continuous morphological variation and rates of topologically abnormal morphologies in populations of fruit flies. Some preliminary computations showing proof of concept are included. Future plans involve using theoretical contributions from this dissertation for final analysis of the fly data.

The distance modification is joint work with Ezra Miller and the biology application is joint with Surabhi Beriwal, Ezra Miller, and biologists at the Houle Lab at Florida State University.

Item Open Access Lattice point methods for combinatorial games(2011) Guo, AlanWe encode arbitrary finite impartial combinatorial games in terms oflattice points in rational convex polyhedra. Encodings provided by theselatticegamescan be made particularly efficient for octal games, which we generalize tosquarefree games. These encompass all heap games in a natural setting where theSprague–Grundy theorem for normal play manifests itself geometrically. We providepolynomial time algorithms for computing strategies for lattice games provided thatthey have a certain algebraic structure, called anaffine stratification.Item Open Access Minimal Resolutions of Monomial Ideals(2020) Ordog, Erika AnastasiaIt has been an open problem since the early 1960s to construct free resolutions of monomial ideals. Beginning with the 1966 Taylor resolution, the first resolution for arbitrary monomial ideals, there have since been many constructions of free resolutions of monomial ideals, satisfying some of the following properties: canonical, minimal, universal, closed form, and combinatorial. The goal is for such a construction to satisfy all of the desired properties. The constructions given so far each satisfy some subset of these properties. This dissertation gives a full solution to the problem,satisfying all of the desired properties, over a field of characteristic 0 and most positive characteristics, with these positive characteristics depending on the ideal. The differential is a weighted sum over lattice paths in $\mathbb{Z}^n$ that come from analogues of spanning trees in simplicial complexes that are indexed by the lattice. Over a field of any characteristic, noncanonical sylvan resolutions are defined. The noncanonical resolutions are minimal, universal, closed form, and combinatorial. The differentials sum over choices of these generalized spanning trees at points along the lattice paths. Finally, a combinatorial description of the canonical three-variable case is given, and noncanonical sylvan resolutions are used to produce planar graph resolutions in the three-variable case and a minimal resolution of the Stanley–Reisner ideal of the minimal triangulation of $\mathbb{RP}^2$ in characteristic 2. Substantial portions of the results are based on joint work with John Eagon and Ezra Miller.

Item Open Access Monoid Congruences, Binomial Ideals, and Their Decompositions(2014) ONeill, Christopher DavidThis dissertation refines and extends the theory of mesoprimary decomposition, as introduced by Kahle and Miller. We begin with an overview of the existing theory of mesoprimary decomposition

in both the combinatorial setting of monoid congruences and the arithmetic setting of binomial ideals. We state all definitions and results that are relevant for subsequent chapters.

We classify redundant mesoprimary components in both the combinatorial and arithmetic settings. Kahle and Miller give a class of redundant components in each setting that are redundant in every mesoprimary decomposition. After identifying a further class of redundant components at the level of congruences, we give a condition on the associated monoid primes that guarantees the existence of unique irredundant mesoprimary decompositions in both settings.

We introduce soccular congruences as combinatorial approximations of irreducible binomial quotients and use the theory of mesoprimary decomposition to give a combinatorial method of constructing irreducible decompositions of binomial ideals. We also demonstrate a binomial ideal which does not admit a binomial irreducible decomposition, answering a long-standing problem of Eisenbud and Sturmfels.

We extend mesoprimary decomposition of monoid congruences to congruences on monoid modules. Much of the theory for monoid congruences extends to this new setting, including a characterization of mesoprimary monoid module congruences in terms of associated prime monoid congruences and a method for constructing coprincipal decompositions of monoid module congruences using key witnesses.

We conclude with a collection of open problems for future study.

Item Open Access Sampling From Stratified Spaces(2020) Tran, Do VanThis dissertation studies Central Limit Theorems (CLTs) of Frechet means on stratified spaces. The broad goal of this work is to answer the following question: What information one should expect to get by sampling from a stratified space? In particular, this work explores relationships between geometry and different forms of CLT, namely classic, smeary and sticky. The work starts with explicit forms of CLTs for spaces of constant sectional curvature. As a consequence, we explain the effect of sectional curvature on the behaviors of Frechet means. We then give a sufficient condition for a smeary CLT to occur on spheres. In the later part, we propose a general form of CLT for star shaped Riemannian stratified spaces. The general CLT we propose is universal in the sense that it contains all of the different forms of aforementioned CLTs. The proposed CLT is verified on manifolds and on any flat 2-dimensional spaces with an isolated singularity.

Item Open Access Statistical analysis of fruit fly wing vein topology(2018-04) Beriwal, SurabhiThe fruit fly Drosophila melanogaster is a commonly used model organism for evolution given that the species showcases interesting behaviors and is easy to modify and rear. Among other things, the Drosophila wings are studied because their structure is tractable, consistent, and traceable developmentally. Along with Dr. Ezra Miller and Ashleigh Thomas, I studied evolutionary changes to Drosophila melanogaster wings using persistent homology. The biological hypothesis posits that selecting for continuous wing deformation leads to higher rates of topological novelty. We are interested in understanding whether selection on a continuous trait can itself cause higher rates of variation of a (separate) discrete trait. We work joint with Dr. David Houle at Florida State University.Item Open Access Sticky central limit theorems at isolated hyperbolic planar singularities(Electronic Journal of Probability, 2015-01-01) Huckemann, Stephan; Mattingly, Jonathan C; Miller, Ezra; Nolen, James© 2015, University of Washington. Akll rights reserved.We derive the limiting distribution of the barycenter bn of an i.i.d. sample of n random points on a planar cone with angular spread larger than 2π. There are three mutually exclusive possibilities: (i) (fully sticky case) after a finite random time the barycenter is almost surely at the origin; (ii) (partly sticky case) the limiting distribution of √nbn comprises a point mass at the origin, an open sector of a Gaussian, and the projection of a Gaussian to the sector’s bounding rays; or (iii) (nonsticky case) the barycenter stays away from the origin and the renormalized fluctuations have a fully supported limit distribution—usually Gaussian but not always. We conclude with an alternative, topological definition of stickiness that generalizes readily to measures on general metric spaces.Item Open Access Sticky central limit theorems on open books(Annals of Applied Probability, 2013-12-01) Hotz, Thomas; Huckemann, Stephan; Le, Huiling; Marron, JS; Mattingly, Jonathan C; Miller, Ezra; Nolen, James; Owen, Megan; Patrangenaru, Vic; Skwerer, SeanGiven a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine.We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky). © Institute of Mathematical Statistics, 2013.