Mathematics
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Item Towards a Classification of Almost Complex and Spin^h Manifolds(2024) Mills, Keith; Rosenberg, Jonathan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We show that all homotopy CP^ns, smooth closed manifolds with the oriented homotopy type of CP^n, admit almost complex structures for 3 ≤ n ≤ 6, and classify these structures by their Chern classes for n=4, 6. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP^4s. We also show that all homotopy RP^(2k+1)s admit stably almost complex structures. Spin^h manifolds are the quaternionic analogue to spin^c manifolds. At the prime 2 we compute the spin^h bordism groups by proving a structure theorem for the cohomology of the spin^h bordism spectrum MSpin^h as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of MSpin^h as a wedge sum of familiar spectra. We also compute the decomposition of H^*(MSpin^h; Z/2Z) explicitly in degrees up through 30 via a counting process.Item On the Difficulty of Breaking Substitution Ciphers(2021) Wertheimer, Phil; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We analyze different methods of attacking substitution ciphers using $m$-gram frequency analysis. For $m=1$ this amounts to studying symbol counts in random strings, and for $m\geq 2$ we use the Markov Chain Monte Carlo method introduced by Diaconis \cite{mcmcr}. Our study includes both numerical simulations of the English language and theoretical analysis of random alphabets, which are probabilistic constructions for studying the distribution of $m$-grams in random strings. We present several results in the direction of explaining why the $2$-gram method performs the best in breaking the substitution ciphers.Item Cluster Algebras and Polylogarithm Relations(2021) Greenberg, Zachary; Zickert, Christian; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We seek to illuminate the connection between multiple polylogarithm relations and cluster algebras in two ways. First, we give a uniform description of the cluster modular group of affine and doubly extended cluster algebras. This will be critical for the future work of extracting polylogarithm relations from infinite type cluster algebras. Second, we introduce a differential one form, ωn, associated to each multiple polylogarithm, which can be used to compute multiple polylogarithm relations. This form satisfies a clean recurrence relation, mirroring the inductive definition of multiple polylogarithms. We are able to use this recurrence to find several families of “small” polylogarithm relations that hold in any weight. Finally for small values of n, we extract polylogarithm relations from type An and Dn cluster algebras.Item Class groups of characteristic-p function field analogues of Q(n^(1/p))(2021) Reich, Steven; Washington, Lawrence; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In the theory of cyclotomic function fields, the Carlitz module $\Lambda_M$ associated to a polynomial $M$ in a global function field of characteristic $p$ provides a strong analogy to the roots of unity $\mu_p$ in a number field. In this work, we consider a natural extension of this theory to give a compatible analogue of the $p$-th root of an integer $n$. The most fundamental case, and the one which most closely mimics the number field situation, is when the Carlitz module is defined by a linear polynomial (which can be assumed to be $T$) in $k={\mathbb F}_q(T)$. The Carlitz module $\Lambda_T$ generates a degree-$(q-1)$ extension $k(\Lambda_T)$ which shares many properties with the field ${\mathbb Q}(\mu_p)$, where $\mu_p$ is the module of $p$-th roots of unity. To form the analogue of ${\mathbb Q}(\sqrt[p]{n})$, we define a degree-$q$ extension $F/k$ associated to a polynomial $P(T) \in k$, for which the normal closure is formed by adjoining $\Lambda_T$. In the introduction, we describe in detail the parallels between this construction and that in the number field setting. We then compute the class number $h_F$ for a large number of such fields. The remainder of the work is concerned with proving results about the class groups and class numbers of this family of fields. These are:\begin{itemize} \item a formula relating the class number of $F$ to that of its normal closure, along with a theorem about the structure of the class group of the normal closure \item a formula relating the class number of a compositum of such $F$ to the class numbers of the constituent fields \item conditions on $P(T)$ for when the characteristic, $p$, of $F$ divides its class number, along with bounds on the rank of the $p$-part of the class group. \end{itemize}Item Generalizations of Schottky groups(2017) Burelle, Jean-Philippe; Goldman, William M; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Schottky groups are classical examples of free groups acting properly discontinuously on the complex projective line. We discuss two different applications of similar constructions. The first gives examples of 3-dimensional Lorentzian Kleinian groups which act properly discontinuously on an open dense subset of the Einstein universe. The second gives a large class of examples of free subgroups of automorphisms groups of partially cyclically ordered spaces. We show that for a certain cyclic order on the Shilov boundary of a Hermitian symmetric space, this construction corresponds exactly to representations of fundamental groups of surfaces with boundary which have maximal Toledo invariant.Item Algorithms and Generalizations for the Lovasz Local Lemma(2015) Harris, David; Srinivasan, Aravind; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The Lovasz Local Lemma (LLL) is a cornerstone principle of the probabilistic method for combinatorics. This shows that one can avoid a large of set of “bad-events” (forbidden configurations of variables), provided the local conditions are satisfied. The original probabilistic formulation of this principle did not give efficient algorithms. A breakthrough result of Moser & Tardos led to an framework based on resampling variables which turns nearly all applications of the LLL into efficient algorithms. We extend and generalize the algorithm of Moser & Tardos in a variety of ways. We show tighter bounds on the complexity of the Moser-Tardos algorithm, particularly its parallel form. We also give a new, faster parallel algorithm for the LLL. We show that in some cases, the Moser-Tardos algorithm can converge even thoughthe LLL itself does not apply; we give a new criterion (comparable to the LLL) for determining when this occurs. This leads to improved bounds for k-SAT and hypergraph coloring among other applications. We describe an extension of the Moser-Tardos algorithm based on partial resampling, and use this to obtain better bounds for problems involving sums of independent random variables, such as column-sparse packing and packet-routing. We describe a variant of the partial resampling algorithm specialized to approximating column-sparse covering integer programs, a generalization of set-cover. We also give hardness reductions and integrality gaps, showing that our partial resampling based algorithm obtains nearly optimal approximation factors. We give a variant of the Moser-Tardos algorithm for random permutations, one of the few cases of the LLL not covered by the original algorithm of Moser & Tardos. We use this to develop the first constructive algorithms for Latin transversals and hypergraph packing, including parallel algorithms. We analyze the distribution of variables induced by the Moser-Tardos algorithm. We show it has a random-like structure, which can be used to accelerate the Moser-Tardos algorithm itself as well as to cover problems such as MAX k-SAT in which we only partially avoid bad-events.Item Investigations of Highly Irregular Primes and Associated Ray Class Fields(2014) Stern, Morgan Benjamin; Washington, Lawrence; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We investigate properties of the class number of certain ray class fields of prime conductor lying above imaginary quadratic fields. While most previous work in this area restricted to the case of imaginary quadratic fields of class number 1, we deal almost exclusively with class number 2. Our main results include finding 5 counterexamples to a generalization of the famous conjecture of Vandiver that the class number of the pth real cyclotomic field is never divisible by p. We give these counterexamples the name highly irregular primes due to the fact that any counterexample of classical Vandiver is an irregular prime. In addition we explore whether several consequences of Vandiver's conjecture still hold for these highly irregular primes, including the cyclicity of certain class groups.Item Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps(2014) Contreras, Fabian Elias; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation consists of two parts. In the first part, we consider a piecewise expanding unimodal map (PEUM) $f:[0,1] \to [0,1]$ with $\mu=\rho dx$ the (unique) SRB measure associated to it and we show that $\rho$ has a Taylor expansion in the Whitney sense. Moreover, we prove that the set of points where $\rho$ is not differentiable is uncountable and has Hausdorff dimension equal to zero. In the second part, we consider a family $f_t:[0,1] \to [0,1]$ of PEUMs with $\mu_t$ the correspoding SRB measure and we present a new proof of \cite{BS1} when considering the observables in $C^1[0,1]$ . That is, $\Gamma(t)=\int \phi d\mu_t$ is differentiable at $t=0$, with $\phi \in C^1[0,1]$, when assuming $J(c)=\sum_{k=0}^{\infty} \frac{v(f^k(c))}{Df^k(f(c))}$ is zero. Furthermore, we show that in fact $\Gamma(t)$ is never differentiable when $J(c)$ is not zero and we give the exact modulus of continuity.