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Browsing Department of Mathematics by Author "Achter, Jeff, committee member"
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Item Open Access Arithmetic in group extensions using a partial automation(Colorado State University. Libraries, 2010) Ziliak, Ellen, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Betten, Anton, committee member; McConnell, Ross, committee memberThe purpose of this paper is to describe the structure of an extension group G which has a normal subgroup K and a quotient group Q = G/K . To describe the structure of G concretely, we want to be able to do arithmetic in G based on the arithmetic done in both the normal subgroup K and the quotient group Q. We will begin by looking at the 2-cohomology group which is the standard approach for working on this problem. This will lead us to questions concerning storage which we would like to reduce. Therefore we will consider the case where our groups arc finitely presented and see how storage may be reduced. During this reduction we will see that it will be necessary to be able to rewrite words in a free group as a product of generators of the normal subgroup K. We begin by looking at current approaches to this problem, which requires computing an (augmented) coset table. If we will let Q be a finite group for which we also have a presentation < S\R >, (i.c. Q = F/N with F —< S > and N the normal closure of R in F). We assume that Q does not have a confluent rewriting system. We want to rewrite a word in S, representing the identity in Q as a product of conjugates in R. Such rewriting can be done using an (augmented) coset table for N in F which can be visualized in a graph by a coset automaton, also called the full Cayley Graph. Tracing in the graph through words in F will allow us to rewrite these words as a product of generators of N. The difficulty that arises in this approach lies in storing and constructing the augmented coset table. Instead we will construct an object called a partial automaton which is a subgraph of the coset automaton. The partial automaton will have the property that it contains a loop for every relator in R. We will first show that this graph can be used to reconstruct the coset automaton, which means it contains the same information as the coset automaton even though it is much smaller. Our next step will be to use the partial automaton to rewrite words in N as a product of conjugates in R. Since the partial automaton is much smaller than the coset automaton, and it does not contain doubly labeled edges as an augmented coset automaton would it require substantially less memory to store. A word in N is represented by a loop in the coset automaton, therefore if we wish to rewrite this word as a product of conjugates of relators, we essentially want to describe this larger loop as a product of smaller loops. Where we will restrict our smaller loops to be loops in the partial automaton. To do this rewriting we place the partial automaton locally at different states in the coset automaton until we cover the entire loop. By placing the partial automaton at different states in the graph we will then the conjugate of relators. Unfortunately we cannot just place the partial automaton arbitrarily at different states, because we would have many different choices of the conjugates of relators we could choose. Instead we must use one further tool, which is the fact that our normal subgroup N is itself a free group. Therefore N has a free generating set, where the generators of N are conjugates of relators. With this generating set we can rewrite words in N uniquely as a product of the generators. We will therefore, use the partial automaton to compute the generators of the free generating set for N and then use these generators to rewrite our word in N as a product of conjugate of relators. By using the partial automaton to do this rewriting we can quickly do rewriting in much larger examples. This algorithm has been implemented in GAP and to suggest the improvement we rewrote several words in the group PSp^ which is a group of order 1,451, 520. The partial automaton had 145 states and after some initial set up which will be described in the paper the run time for this rewriting took less than a half a second per word.Item Open Access Bridgeland stability of line bundles on smooth projective surfaces(Colorado State University. Libraries, 2014) Miles, Eric W., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Peterson, Chris, committee member; Prasad, Ashok, committee member; Pries, Rachel, committee memberBridgeland Stability Conditions can be thought of as tools for creating and varying moduli spaces parameterizing objects in the derived category of a variety X. Line bundles on the variety are fundamental objects in its derived category, and we characterize the Bridgeland stability of line bundles on certain surfaces. Evidence is provided for an analogous characterization in the general case. We find stability conditions for P1 × P1 which can be seen as giving the stability of representations of quivers, and we deduce projective structure on the Bridgeland moduli spaces in this situation. Finally, we prove a number of results on objects and a construction related to the quivers mentioned above.Item Open Access Counting Artin-Schreier curves over finite fields(Colorado State University. Libraries, 2015) Ho, Anne M., author; Pries, Rachel, advisor; Achter, Jeff, committee member; Lee, Myung Hee, committee member; Penttila, Tim, committee memberSeveral authors have considered the weighted sum of various types of curves of a certain genus g over a finite field k := Fq of characteristic p where p is a prime and q = pm for some positive integer m. These include elliptic curves (Howe), hyperelliptic curves (Brock and Granville), supersingular curves when p = 2 and g = 2 (Van der Geer and Van der Vlught), and hyperelliptic curves of low genus when p = 2 (Cardona, Nart, Pujolàs, Sadornil). We denote this weighted sum ∑[C] 1/|Autk(C)|' where the sum is over k-isomorphism classes of the curves and Autk(C) is the automorphism group of C over k. Many of these curves mentioned above are Artin-Schreier curves, so we focus on these in this dissertation. We consider Artin-Schreier curves C of genus g = d(p - 1)/2 for 1 ≤ d ≤ 5 over finite fields k of any characteristic p. We also determine a weighted sum for an arbitrary genus g in one-, two-, three-, and four-branch point cases. In our cases, we must consider a related weighted sum ∑/[C] 1/|CentAutk(C)‹t›|' where CentAutk(C) ‹t› is the centralizer of ‹t› in Autk(C). We discuss our methods of counting, our results, applications, as well as geometric connections to the moduli space of Artin-Schreier covers.Item Open Access Hyperovals, Laguerre planes and hemisystems - an approach via symmetry(Colorado State University. Libraries, 2013) Bayens, Luke, author; Penttila, Tim, advisor; Achter, Jeff, committee member; Bohm, Willem, committee member; Peterson, Chris, committee memberIn 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries -- we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups -- indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting transitively on flags, with an additional hypothesis -- a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order. We also prove a correspondence between translation ovoids of translation generalized quadrangles arising from a pseudo-oval O and translation flocks of the elation Laguerre plane arising from the dual pseudo-oval O*. The last topic we consider is the existence of hemisystems in finite hermitian spaces. Hemisystems were introduced by Segre in 1965 -- he constructed a hemisystem of H(3,32) and rasied the question of their existence in other spaces. Much of the interest in hemisystems is due to their connection to other combinatorial structures, such as strongly regular graphs, partial quadrangles, and association schemes. In 2005, Cossidente and Penttila constructed a family of hemisystems in H(3,q2), q odd, and in 2009, the same authors constructed a family of hemisystem in H(5,q2), q odd. We develop a new approach that generalizes the previous constructions of hemisystems to H(2r - 1,q2), r > 2, q odd.Item Open Access Intersections of ψ classes on Hassett spaces of rational curves(Colorado State University. Libraries, 2018) Sharma, Nand, author; Cavalieri, Renzo, advisor; Peterson, Chris, committee member; Achter, Jeff, committee member; Prasad, Ashok, committee memberHassett spaces are moduli spaces of weighted stable pointed curves. In this work, we consider such spaces of curves of genus 0 with weights all 1/q , q being a positive integer greater than or equal to 2. These spaces are interesting as they have different universal families and different intersection theory when compared with classical moduli spaces of pointed stable rational curves. We develop closed formulas for intersections of ψ-classes on such spaces. In our main result, we encode the formula for top intersections in a generating function obtained by applying an exponential differential operator to the Witten-potential.Item Open Access Modular group and modular forms(Colorado State University. Libraries, 2010) Schmidt, Eric, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Cavalieri, Renzo, committee member; Harton, John, committee memberWe prove some results about the structure of SL2(Z) and related groups. We define modular forms for this group and develop the basic theory. We then use the theory of lattices to construct examples of modular forms.Item Open Access Molecular configurations and persistence: branched alkanes and additive energies(Colorado State University. Libraries, 2022) Story, Brittany M., author; Adams, Henry, advisor; Shipman, Patrick, committee member; Achter, Jeff, committee member; Fremstad, Anders, committee memberEnergy landscapes are high-dimensional functions that encapsulate how certain molecular properties affect the energy of a molecule. Chemists use disconnectivity graphs to find transition paths, the lowest amount of energy needed to transfer from one energy minimum to another. But disconnectivity graphs fail to show not only some lower-dimensional features, such as transition paths with an energy value only slightly higher than the minimum transition path, but also all higher-dimensional features. Sublevelset persistent homology is a tool that can be used to capture other relevant features, including all transition paths. In this paper, we will use sublevelset persistent homology to find the structure of the energy landscapes of branched alkanes: tree-like molecules consisting of only carbons and hydrogens. We derive complete characterizations of the sublevelset persistent homology of the OPLS-UA energy function on two different families of branched alkanes. More generally, we explain how the sublevelset persistent homology of any additive energy landscape can be computed from the individual terms comprising that landscape.Item Open Access Open and closed Gromov-Witten theory of three-dimensional toric Calabi-Yau orbifolds(Colorado State University. Libraries, 2013) Ross, Dustin J., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Lunkenheimer, Erika, committee member; Peterson, Chris, committee memberWe develop the orbifold topological vertex, an algorithm for computing the all-genus, open and closed Gromov-Witten theory of three-dimensional toric Calabi-Yau orbifolds. We use this algorithm to study Ruan's crepant resolutions conjecture and the orbifold Gromov-Witten/Donaldson-Thomas correspondence.Item Open Access 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 memberThe 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.Item Open Access Some topics in combinatorial phylogenetics(Colorado State University. Libraries, 2010) McBee, Cayla D., author; Penttila, Tim, advisor; Achter, Jeff, committee member; Hulpke, Alexander, committee member; Mueller, Rachel, committee memberThis thesis is in combinatorial phylogenetics and is focused on a study of Hadamard conjugation. It examines the question of whether the presence of an abelian permutation group acting regularly on the states is necessary for the application of this technique. New connections between phylogenetics and algebraic combinatorics are suggested, especially with (commutative) association schemes.Item Open Access The conjugacy extension problem(Colorado State University. Libraries, 2021) Afandi, Rebecca, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Rajopadhye, Sanjay, committee memberIn this dissertation, we consider R-conjugacy of integral matrices for various commutative rings R. An existence theorem of Guralnick states that integral matrices which are Zp-conjugate for every prime p are conjugate over some algebraic extension of Z. We refer to the problem of determining this algebraic extension as the conjugacy extension problem. We will describe our contributions to solving this problem. We discuss how a correspondence between Z-conjugacy classes of matrices and certain fractional ideal classes can be extended to the context of R-conjugacy for R an integral domain. In the case of integral matrices with a fixed irreducible characteristic polynomial, this theory allows us to implement an algorithm which tests for conjugacy of these matrices over the ring of integers of a specified number field. We also describe how class fields can be used to solve the conjugacy extension problem in some examples.Item Open Access The Möbius number of the symmetric group(Colorado State University. Libraries, 2012) Monks, Kenneth M., author; Hulpke, Alexander, advisor; Penttila, Tim, committee member; Achter, Jeff, committee member; Toki, Walter, committee memberThe Möbius number of a finite group is its most important nontrivial combinatorial invariant. In this paper, we compute the Möbius numbers of many partially-ordered sets, including the odd-partition posets and the subgroup lattices of many infinite families of groups. This is done with an eye towards computing the Möbius number of the symmetric group on 18 points.