Mathematics
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Item HOPF ALGEBRA OF MULTIPLE POLYLOGARITHMS AND ASSOCIATED MIXED HODGE STRUCTURES(2024) Li, Haoran; Zickert, Christian K; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This thesis constructs a variation of mixed Hodge structures based on multiple polylogarithms, and attempts to build candidate complexes for computing motivic cohomology. Firstly, we consider Hopf algebras with generators representing multiple polylogarithms. By quotienting products and functional relations, we get Lie coalgebras whose Chevalley-Eilenberg complexes are conjectured to compute rational and integral motivic cohomologies. We also associate one-forms to multiple polylogarithms, which exhibit combinatorial properties that are easy to work with. Next, we introduce a variation matrix which describes a variation of mixed Hodge structures encoded by multiple polylogarithms. Its corresponding connection form is composed of the one-forms associated to the multiple polylogarithms. Lastly, to ensure the well-definedness of the Hodge structures, we must compute the monodromies of multiple polylogarithms, for which we provide an explicit formula, extending the previous work done for multiple logarithms, a subfamily of multiple polylogarithms.Item Eventually Stable Quadratic Polynomials over Q(i)(2024) McDermott, Jermain; Washington, Lawrence C; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Let $f$ be a polynomial or a rational function over a field $K$. Arithmetic dynamics studies the algebraic and number-theoretic properties of its iterates $f^n:=f \circ f \circ ... \circ f$.\\ A basic question is, if $f$ is a polynomial, are these iterates irreducible or not? We wish to know what can happen when considering iterates of a quadratic $f= x^2+r\in K[x]$. The most interesting case is when $r=\frac{1}{c}$, which we will focus on, and discuss criteria for irreducibility, i.e. \emph{stability} of all iterates. We also wish to prove that if 0 is not periodic under $f$, then the number of factors of $f^n$ is bounded by a constant independent of $n$, i.e. $f$ is \emph{eventually stable}. This thesis is an extension to $\Qi$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. This thesis involves a mixture of ideas from number theory and arithmetic geometry. We also show how eventual stability of iterates ties into the density of prime divisors of sequences.Item Langlands-Kottwitz Method on Moduli Spaces of Global Shtukas(2024) Song, Shin Eui; Haines, Thomas; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We apply the approach of Scholze to compute the trace of Hecke operator twisted by some power of Frobenius on the cohomology of the moduli spaces of global shtukas in the case of bad reduction. We find a formula that involves orbital integrals and twisted orbital integrals which can be compared with the Arthur-Selberg trace formula. This extends the results of Ngo and Ngo Dac on counting points of moduli spaces of global shtukas over finite fields. The main problem lies in finding a suitable compactly supported locally constant function that will be plugged into the twisted orbital integrals. Following Scholze, we construct locally constant functions called the test functions by using deformation spaces of bounded local shtukas. Then we establish certain local-global compatibility to express the trace on the nearby cycle sheaves on the moduli space of global shtukas to the trace on the deformation spaces.Item Metastable Distributions for Semi-Markov Processes(2024) Mohammed Imtiyas, Ishfaaq Ahamed; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this work, we consider semi-Markov processes whose transition times and transitionprobabilities depend on a small parameter ε. Understanding the asymptotic behavior of such processes is needed in order to study the asymptotics of various randomly perturbed dynamical and stochastic systems. The long-time behavior of a semi-Markov process Xε t depends on how the point (1/ε, t(ε)) approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale t(ε). The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.Item Polynomials with Equal Images over Number Fields(2024) Hirsh, Jordan; Washington, Lawrence C; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Chapman and Ponomarenko [1] characterized when two polynomials f, g ∈ Q[x] have thesame image f(Z) = f(Z). We extend this result to rings of integers in number fields. In particular, if K is a finite extension of Q and O is the ring of algebraic integers in K, we characterize when polynomials f, g ∈ K[x] satisfy f(O) = g(O). As part of our proof, we give a variant of Hilbert’s irreducibility theorem.Item GOOD POSITION BRAIDS, TRANSVERSAL SLICES AND AFFINE SPRINGER FIBERS(2024) Duan, Chengze; Haines, Thomas TH; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In the study of Iwahori-Hecke algebras, Geck and Pfeiffer introduced good elements inCoxeter groups. These elements played a crucial role in the work of He and Lusztig on generalizing Steinberg’s cross-sections and Steinberg slices. This work yields the transversal slices for basic unipotent conjugacy classes in a reductive group G. We improve this result by introducing some more general braid elements called good position braids. We use them to construct transversal slices for any unipotent conjugacy classes in G. On the other hand, these good position braids also correspond to affine Springer fibers via root valuation strata. The correspondence leads to a reformulation of the dimension formula of affine Springer fibers. We also expect these braid elements to help with a conjecture of Goresky, Kottwitz and MacPherson on the cohomology of affine Springer fibers.Item CYCLOTOMIC Z2-EXTENSION OF REAL QUADRATIC FIELDS WITH CYCLIC IWASAWA MODULE(2024) Avila Artavia, Josue David; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)For a number field K and a prime p, let K∞ denote the cyclotomic Zp-extension of K, andAn denote the p-primary part of the class group of its n-th layer Kn. Greenberg conjectured that for a totally real field, the order of An becomes constant for sufficiently large n. Motivated by the work of Mouhib and Movahhedi, we focus on the case where p = 2 and K is a real quadratic field such that the Iwasawa module X∞ = lim←An is cyclic. They determined all such fields and proved that Greenberg’s conjecture holds for some cases. In this dissertation, we provide new examples of infinite families of real quadratic fields satisfying Greenberg’s conjecture which were not covered completely in the work of Mouhib and Movahhedi. To achieve this, we use the theory of binary quadratic forms and biquadratic extensions to determine a fundamental system of units and the class number of the first few layers of the cyclotomic Z2-extension. Additionally, in certain cases, we can determine the size of the module X∞ and the level of the cyclotomic tower where the size of An becomes constant.Item Motivic Homotopy Theory and Synthetic Spectra(2024) Dziedzic, Charles Richard; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Motivic homotopy studies the application of techniques from homotopy theory to algebraic geometry, using A1 as the analogue of the unit interval. Voevodsky found early success in using his constructions to prove Milnor’s conjecture and the Bloch-Kato conjecture. An interesting and deep theory arises when constructing the unstable and stable motivic categories, and it has developed into its own field of study. We begin with a survey of these constructions, detailing the equivalences between the different model used in the construction of H(S) and SH(S). Here, we draw connections between all the constructions one might encounter across the literature, and provide explicit statements on their equivalence. Stable homotopy theorists have also found utility in motivic homotopy, using the stable mo- tivic homotopy category SH to advance computations of the stable homotopy groups of spheres, such as in the work of Isaksen-Wang-Xu. Other work by Bachmann-Kong-Wang-Xu has made great progress in our understanding of motivic homotopy theory. Synthetic spectra are a construction of Pstragrowski which represent a ‘return to form’ ofsome sort, as they are constructed entirely in the ∞−category of spectra. However, they give rise to a natural bigrading and a strong connection to motivic homotopy; one of the main results is an equivalence of ∞−categories with cellular motivic spaces over C, SpCcell. We build up enough of the general theory to establish the connection with motivic homotopy and comment on recent applications.Item G-INVARIANT REPRESENTATIONS USING COORBITS(2024) Tsoukanis, Efstratios; Balan, Radu V; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Consider a finite-dimensional real vector space and a finite group acting unitarily on it. We investigate the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding relies on subsets of sorted coorbits with respect to chosen window vectors.Our main injectivity results examine the conditions under which such embeddings are injective. We establish these results using semialgebraic techniques. Furthermore, our main stability result states and demonstrates that any embedding based on sorted coorbits is automatically bi-Lipschitz when injective. We establish this result using geometric function techniques. Our work has applications in data science, where certain systems exhibit intrinsic invariance to group actions. For instance, in graph deep learning, graph-level regression and classification models must be invariant to node labeling.Item Long-term behavior of randomly perturbed Hamiltonian systems: large deviations and averaging(2024) Yan, Shuo; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation concerns various asymptotic problems related to the long-term macroscopic behavior of randomly perturbed Hamiltonian systems, with different types of perturbations and on different time scales. Since the perturbations of the systems are assumed to be small while the systems are observed at large times, non-trivial phenomena can arise due to the interplay between the perturbation size and the temporal and spatial scales, and the systems often demonstrate qualitatively distinct types of behavior depending on the subtle quantitative relation between asymptotic parameters. More specifically, on a natural time scale, i.e., in time required for the dynamics to move distance of order one, we investigate the dynamical systems with fast-oscillating perturbations and obtain precise estimates on the distribution. In particular, we calculate the exact asymptotics of the distribution in the case of linear dynamical systems. This problem also inspires the study of the local limit theorem for time-inhomogeneous functions of Markov processes. The local limit theorem is a significant and widely used tool in problems of pure and applied mathematics as well as statistics. This result has been included in [1] and submitted for publication. On the time scale that is inversely proportional to the effective size of the perturbation, we prove that the evolution of the first integral of the Hamiltonian system with fast-oscillating perturbations converges to a Markov process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices. The result is parallel to the celebrated Freildin-Wentzell theory on the averaging principle of additive white-noise perturbations of Hamiltonian systems, and provides a description of the long-term behavior of a system when adopting an alternative approach to modeling random noise. Moreover, the current result provides the first scenario where the motion on a graph and the corresponding gluing conditions appear due to the averaging of a slow-fast system. It allows one to consider, for instance, the long-time diffusion approximation for an oscillator with a potential with more than one well. This result has been submitted for publication [2]. In the last part of the dissertation, we return to the more classical case of additive diffusion-type perturbations, combine the ideas of large deviations and averaging, and establish a large deviation principle for the first integral of the Hamiltonian system on intermediate time scales. Besides representing a new step in large deviations and averaging, this result will have important applications to reaction-diffusion equations and branching diffusions. The latter two concepts concern the evolution of various populations (e.g., in biology or chemical reactions). This result has been published in the journal Stochastics and Dynamics [3].