Browsing by Author "Miller, Ezra"

We 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.

In 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.

Traditionally, 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.

This 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.

It 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.

This 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.

This 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.