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Browsing by Author "Shoemaker, Mark, committee member"

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    Defining persistence diagrams for cohomology of a cofiltration indexed over a finite lattice
    (Colorado State University. Libraries, 2022) Rask, Tatum D., author; Patel, Amit, advisor; Shoemaker, Mark, committee member; Tucker, Dustin, committee member
    Persistent homology and cohomology are important tools in topological data analysis, allowing us to track how homological features change as we move through a filtration of a space. Original work in the area focused on filtrations indexed over a totally ordered set, but more recent work has been done to generalize persistent homology. In one avenue of generalization, McCleary and Patel prove functoriality and stability of persistent homology of a filtration indexed over any finite lattice. In this thesis, we show a similar result for persistent cohomology of a cofiltration. That is, for P a finite lattice and F : P → ▽K a cofiltration, the nth persistence diagram is defined as the Möbius inversion of the nth birth-death function. We show that, much like in the setting of persistent homology of a filtration, this composition is functorial and stable with respect to the edit distance. With a general definition of persistent cohomology, we hope to discover whether duality theorems from 1-parameter persistence generalize to more general lattices.
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    Generalized RSK for enumerating projective maps from n-pointed curves
    (Colorado State University. Libraries, 2022) Reimer-Berg, Andrew, author; Gillespie, Maria, advisor; Ghosh, Sudipto, committee member; Hulpke, Alexander, committee member; Shoemaker, Mark, committee member
    Schubert calculus has been studied since the 1800s, ever since the mathematician Hermann Schubert studied the intersections of lines and planes. Since then, it has grown to have a plethora of connections to enumerative geometry and algebraic combinatorics alike. These connections give us a way of using Schubert calculus to translate geometric problems into combinatorial ones, and vice versa. In this thesis, we define several combinatorial objects known as Young tableaux, as well as the well-known RSK correspondence between pairs of tableaux and sequences. We also define the Grassmannian space, as well as the Schubert cells that live inside it. Then, we describe how Schubert calculus and the Littlewood-Richardson rule allow us to turn problems of intersecting geometric spaces into ones of counting Young tableaux with particular characteristics. We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus g, n-marked curve C to Pr, sending the marked points on C to specified general points in Pr, is equal to (r + 1)g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the (r + 1)-ary sequences of length g, and we explore our bijection's combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which r = 1 and several marked points map to the same point in P1, the number of morphisms is still 2g for sufficiently large d.
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    Moduli spaces of rational graphically stable curves
    (Colorado State University. Libraries, 2021) Fry, Andy J., author; Cavalieri, Renzo, advisor; Shoemaker, Mark, committee member; Wilson, James, committee member; Tavani, Daniele, committee member
    We use a graph to define a new stability condition for the algebraic and tropical moduli spaces of rational curves. Tropically, we characterize when the moduli space has the structure of a balanced fan by proving a combinatorial bijection between graphically stable tropical curves and chains of flats of a graphic matroid. Algebraically, we characterize when the tropical compactification of the compact moduli space agrees with the theory of geometric tropicalization. Both characterization results occur only when the graph is complete multipartite.
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    Number of 4-cycles of the genus 2 superspecial isogeny graph
    (Colorado State University. Libraries, 2024) Sworski, Vladimir P., author; Pries, Rachel, advisor; Hulpke, Alexander, committee member; Rajopadhye, Sanjay, committee member; Shoemaker, Mark, committee member
    The genus 2 superspecial degree-2 isogeny graph over a finite field of size p2 is a network graph whose vertices are constructed from genus 2 superspecial curves and whose edges are the degree 2 isogenies between them. Flynn and Ti discovered 4-cycles in the graph, which pose problems for applications in cryptography. Florit and Smith constructed an atlas which describes what the neighborhood of each vertex looks like. We wrote a program in SageMath that can calculate neighborhoods of these graphs for small primes. Much of our work is motivated by these computations. We examine the prevalence of 4-cycles in the graph and, motivated by work of Arpin, et al. in the genus 1 situation, in the subgraph called the spine. We calculate the number of 4-cycles that pass through vertices of 12 of the 14 kinds possible. This also resulted in constructing the neighborhood of all vertices two steps or fewer away for three special types of curves. We also establish conjectures about the number of vertices and cycles in small neighborhoods of the spine.
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    Properties of tautological classes and their intersections
    (Colorado State University. Libraries, 2019) Blankers, Vance T., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Shoemaker, Mark, committee member; Heineman, Kristin, committee member
    The tautological ring of the moduli space of curves is an object of interest to algebraic geometers in Gromov-Witten theory and enumerative geometry more broadly. The intersection theory of this ring has a highly combinatorial structure, and we develop and exploit this structure for several ends. First, in Chapter 2 we show that hyperelliptic loci are rigid and extremal in the cone of effective classes on the moduli space of curves in genus two, while establishing the skeleton for similar results in higher genus. In Chapter 3 we connect the intersection theory of three families of important tautological classes (Ψ-, ω-, and κ-classes) at both the cycle and numerical level. We also show Witten's conjecture holds for κ-classes and reformulate the Virasoro operators in terms of κ-classes, allowing us to effectively compute relations in the κ-class subring. Finally, in Chapter 4 we generalize the results of the previous chapter to weighted Ψ-classes on Hassett spaces.
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    Relative oriented class groups of quadratic extensions
    (Colorado State University. Libraries, 2024) O'Connor, Kelly A., author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Shoemaker, Mark, committee member; Rugenstein, Maria, committee member
    In 2018 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. This work generalized the classical correspondence between ideal classes of quadratic orders and classes of integral binary quadratic forms to any base number field of narrow class number 1. Zemková explicitly computed these relative oriented class groups for quadratic extensions of the rationals. We consider extended versions of this work and develop general strategies to compute relative oriented class groups for quadratic extensions of higher degree number fields by way of the action of Gal(K/Q) on the set of real embeddings of K. We also investigate the binary quadratic forms side of Zemková's bijection and determine conditions for representability of elements of K. Another project comprising work done jointly with Lian Duan, Ning Ma, and Xiyuan Wang is included in this thesis. Our project investigates a principal version of the Chebotarev density theorem, a famous theorem in algebraic number theory which describes the splitting of primes in number field extensions. We provide an overview of the formulation of the principal density and describe its connection to the splitting behavior of the Hilbert exact sequence.
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    Site-selective pyridine functionalization via nucleophilic additions to activated pyridiniums
    (Colorado State University. Libraries, 2024) Nguyen, Hillary M. H., author; McNally, Andrew, advisor; Bandar, Jeff, committee member; Chung, Jean, committee member; Shoemaker, Mark, committee member
    Pyridines and diazines are important heterocycles commonly found in pharmaceuticals, agrochemicals, ligands, and various other organic molecules. Pyridines existing in these molecules usually have multiple bonds connected to them that contribute to their reactivity and characteristics. Therefore, there are ongoing efforts l to find new methods to functionalize these heterocycles. Our lab has contributed to this field by developing methods to functionalize pyridines directly from the C–H bond through phosphonium salts or Zincke imines. Chapter One gives an overview of the current methods for pyridine functionalization and their limitations. Chapter Two describes the synthesis of N-Tf Zincke imines and their use for regioselective 3-position pyridine functionalization. Bipyridines and pyridine-piperidine coupled products are accessed through this method. Chapter Three discusses using N-Tf Zincke imines to form 15N pyridines and coupled with deuteration forms higher mass isotopologues. Chapter Four describes the formation of N-alkyl pyridinium salts from N-Tf Zincke imines. This chapter focuses on optimizing the ring-opening of 2-ester pyridines and ring-closing them with amino esters to access pipecolic esters for macrocyclization. Chapter Five highlights direct nucleophile additions to the 4-position of N-Tf pyridinium salts for pyridine functionalization. 4-aminated pyridines are formed with both aliphatic amines and anilines from the C–H bond. The regioselectivity of this amination is controlled by the basicity of the reaction. In addition, 4-NH2 pyridines are achieved through this method by adding benzophenone imine, an ammonia surrogate. This reaction extends to adding in heteroatom nucleophiles including alcohols, thioesters, amides, and sulfonamides.