Dept. of Mathematics
http://hdl.handle.net/2142/16339
Mon, 18 Jun 2018 08:00:50 GMT2018-06-18T08:00:50ZMetric geometry of the Grushin plane and generalizations
http://hdl.handle.net/2142/99345
Metric geometry of the Grushin plane and generalizations
Romney, Matthew
Given $\alpha>0$, the $\alpha$-Grushin plane is $\mathbb{R}^2$ equipped with the sub-Riemannian metric generated by the vector fields $X = \partial_1$ and $Y = |x_1|^{\alpha} \partial_2$. It is a standard example in sub-Riemannian geometry, as a space which is Riemannian except on a small singular set---here the vertical axis, where the vector field $Y$ vanishes.
The main purpose of this thesis is to study various problems related to the metric geometry of the $\alpha$-Grushin plane and a generalization of it, termed {\it conformal Grushin spaces}. One such problem is the embeddablity of these spaces in some Euclidean space under a bi-Lipschitz or quasisymmetric mapping. Building on work of Seo \cite{Seo:11} and Wu \cite{Wu:15}, we prove a sharp embedding theorem for the $\alpha$-Grushin plane and a general embedding theorem for conformal Grushin spaces under appropriate hypotheses. We also study quasiconformal homeomorphisms of the $\alpha$-Grushin plane.
In the final section, we solve a separate problem regarding quasiconformal mappings in metric spaces. The main result states that if a metric space homeomorphic to $\mathbb{R}^2$ can be quasiconformally parametrized by a domain in $\mathbb{R}^2$, then one can find a mapping which improves the dilatation to within a universal constant. A non-sharp theorem of this type was recently proved by Rajala; our theorem gives the sharp bounds for this problem.
Metric space; Bi-Lipschitz embedding; Sub-Riemannian geometry; Quasiconformal mapping
Sun, 03 Dec 2017 00:00:00 GMThttp://hdl.handle.net/2142/993452017-12-03T00:00:00ZRomney, MatthewOn the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic
http://hdl.handle.net/2142/99314
On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic
Tian, Hongfei
In this thesis we prove the existence of Jordan Decomposition in $D_{G/k}$, the ring of invariant differential operators on a semisimple algebraic group over a field of positive characteristic, and its corollaries. In particular, we define the semisimple center of $D_{G/k}$, denoted by $Z_s(D_{G/k})$, as the set of semisimple elements of its center. Then we show that if $G$ is connected, the semisimple center $Z_s(D_{G/k})$ contains $Z_s(D_{G/k}^{(\nu)})$ for any positive interger $\nu$, where $Z_s(D_{G/k}^{(\nu)})$ is the ring of invariant differential operators on a Frobenius kernel derived from $G$.
Representation theory; Positive characteristic; Invariant differential operators; Semisimple center
Mon, 30 Oct 2017 00:00:00 GMThttp://hdl.handle.net/2142/993142017-10-30T00:00:00ZTian, HongfeiGeneric behaviour of a measure preserving transformation
http://hdl.handle.net/2142/99247
Generic behaviour of a measure preserving transformation
Etedadialiabadi, Mahmood
We study two different problems: generic behavior of a measure preserving transformation and extending partial isometries of a compact metric space. In Chapter $1$, we consider a result of Del Junco--Lema\'nczyk [\ref{DL_B}] which states that a generic measure preserving transformation satisfies a certain orthogonality conditions, and a result of Solecki [\ref{S1_B}] which states that every continuous unitary representations of $L^0(X,\mathbb{T})$ is a direct sum of action by multiplication on measure spaces $(X^{|\kappa|},\lambda_\kappa)$ where $\kappa$ is an increasing finite sequence of non-zero integers. The orthogonality conditions introduced by Del Junco--Lema\'nczyk motivates a condition, which we denote by the DL-condition, on continuous unitary representations of $L^0(X,\mathbb{T})$. We show that the probabilistic (in terms of category) statement of the DL-condition translates to some deterministic orthogonality conditions on the measures $\lambda_\kappa$. Also, we show a certain notion of disjointness for generic functions in $L^0(\mathbb{T})$ and a similar orthogonality conditions to the result of Del Junco--Lema\'nczyk for a generic unitary operator on a Hilbert space $H$.
In Chapter $2$, we show that for every $\epsilon>0$, every compact metric space $X$ can be extended to another compact metric space, $Y$, such that every partial isometry of $X$ extends to an isometry of $Y$ with $\epsilon-$distortion. Furthermore, we show that the problem of extending partial isometries of a compact metric space, $X$, to isometries of another compact metric space, $X\subseteq Y$, is equivalent to extending partial isometries of $X$ to certain functions in $\operatorname{Homeo}(Y)$ that look like isometries from the point of view of $X$.
Measure preserving transformation; Measurable functions
Fri, 08 Dec 2017 00:00:00 GMThttp://hdl.handle.net/2142/992472017-12-08T00:00:00ZEtedadialiabadi, MahmoodEquivariant E-infinity algebras
http://hdl.handle.net/2142/99207
Equivariant E-infinity algebras
Smith, Mychael
The equivariant 𝔼∞G operad has the property that 𝔼∞G(n) is the total space for the G-equivariant universal principal Σn bundle. There is a forgetful functor from 𝔼∞G-algebras to 𝔼∞-algebras, where 𝔼∞ is the classic 𝔼∞ operad. This functor admits a homotopical right adjoint R. The goal of this thesis is to understand R by expressing the free 𝔼∞G-algebra on a G-space X as a homotopy colimit of classic 𝔼∞ algebras.
Equivariant; Homotopy theory; E-infinity algebra
Tue, 28 Nov 2017 00:00:00 GMThttp://hdl.handle.net/2142/992072017-11-28T00:00:00ZSmith, MychaelOn intrinsic ultracontractivity of perturbed Levy processes and applications of Levy processes in actuarial mathematics
http://hdl.handle.net/2142/99190
On intrinsic ultracontractivity of perturbed Levy processes and applications of Levy processes in actuarial mathematics
Yi, Bingji
In this thesis, we study certain aspects of Levy processes and their applications. In the first part of this thesis, we study the applications of Levy processes in actuarial mathematics. Our topics are closely related to the generalized Ornstein-Uhlenbeck processes. We investigate their intimate relationships with the exponential functionals of Levy processes, which enable us to develop efficient semi-analytical algorithms to solve the pricing and risk management problem of certain exotic variable annuity products. In particular, we consider two variable annuity products with guaranteed benefits, the Guaranteed Minimum Accumulation Benefit (GMAB) and the Guaranteed Minimum Withdrawal Benefit (GMWB). For the first one, we develop efficient semi-analytical algorithms to compute its risk measures and hedging costs to solve the risk management problem of the rider. For the other one, we consider pricing the rider. We identify the Laplace transforms of the GMWB rider's risk-neutral values analytically, which leads to efficient solutions to its pricing problem.
In the second part, we consider the intrinsic ultracontractivity of certain Levy processes under nonlocal perturbations. More precisely, we establish the intrinsic ultracontractivity of the Laplacian (corresponding to Brownian motions) and the fractional Laplacian (corresponding to symmetric $\alpha$-stable processes) perturbed by a class of nonlocal operators. Conditions on the nonlocal perturbations are given in order to guarantee that the perturbed operators are intrinsically ultracontractive in general bonded open sets. The methods we use are probabilistic. Essentially, the methods rely on the heat kernel estimates of the fundamental solutions of the operators as well as the Levy systems of the corresponding processes.
Levy processes; Intrinsic ultracontractivity; Variable annuity guaranteed benefits
Mon, 23 Oct 2017 00:00:00 GMThttp://hdl.handle.net/2142/991902017-10-23T00:00:00ZYi, BingjiEssays on the relationship between public transit usage and obesity
http://hdl.handle.net/2142/99128
Essays on the relationship between public transit usage and obesity
She, Zhaowei
My dissertation consists of two essays which analyze the impact of public transit usage on obesity.
Chapter 1 introduces the backgrounds of this field and layout the general framework of this thesis work.
Chapter 2 conducts a cross sectional study on the impact of county population level public transit usage on obesity rates. Since the obese population may have different commuting preference in comparison to non-obese population, one can over or under estimate this effect if these preference differences are not properly controlled. This study adopts an instrumental regression approach to implicitly control for the possible selection bias due to different commuting preferences among different populations. The 2009 health data from the Behavioral Risk Factor Surveillance System (BRFSS) and transportation data from the 2009 National Household Travel Survey (NHTS) are aggregated and matched at the county level. Measures of county level public transit accessibility and vehicle ownership rates are chosen as instrumental variables to implicitly control for unobservable commuting preferences. The model suggests that a one percent increase in county population usage of public transit is associated with a 0.287 percent decrease in county population obesity rate at the alpha=0.01 statistical significance level, when commuting preferences, amount of non-travel physical activity, health resource and distribution of income are fixed. This study provides empirical support for the effectiveness of encouraging public transit usage as an intervention strategy for obesity.
Chapter 3 presents a longitudinal study on this topic. Annual health data from the Behavioral Risk Factor Surveillance System (BRFSS) and transportation data from the National Household Travel Survey (NHTS) were aggregated and matched at the county level, to create a panel data set with 229 counties (from 45 states) across two time periods, 2001 and 2009. Possible confounding variables such as amount of leisure time physical activity, health care coverage and distribution of income are explicitly controlled. All time-invariant county level heterogeneities are implicitly controlled using first difference estimators. This study shows that making frequent public transit commuting possible in a county can effectively decrease the county obesity rate. Specifically, a one percent emergence of frequent public transit riders in a county population is estimated to decrease the county population obesity rate by 0.18% or more. This result supports findings in previous research that the extra amount of physical activities involved in public transit usage can have a statistically significant impact on obesity. In addition, this study also provides empirical evidence for the effectiveness of encouraging public transit usage as a public health intervention for obesity.
Chapter 4 concludes this thesis work as well as postulates directions for future study.
Obesity; Public transit usage
Wed, 19 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/991282017-07-19T00:00:00ZShe, ZhaoweiCohomology of associative algebras and spectral sequences
http://hdl.handle.net/2142/98983
Cohomology of associative algebras and spectral sequences
Shih, Kung-Sing
Algebras, Abstract; Mathematics
Mon, 11 May 1953 00:00:00 GMThttp://hdl.handle.net/2142/989831953-05-11T00:00:00ZShih, Kung-SingStructures and dynamics
http://hdl.handle.net/2142/98378
Structures and dynamics
Panagiotopoulos, Aristotelis
Our results are divided in three independent chapters.
In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: "For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M)." From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15].
In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16].
In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16].
Polish groups; Fraisse; Turbulence; Hjorth; Left invariant; Becker; Projective Fraisse; Infinite games; Borel complexity
Thu, 13 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983782017-07-13T00:00:00ZPanagiotopoulos, AristotelisAsymptotically optimal shapes for counting lattice points and eigenvalues
http://hdl.handle.net/2142/98364
Asymptotically optimal shapes for counting lattice points and eigenvalues
Liu, Shiya
In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the "radius" approaches infinity. In particular, the result implies when 1 < p < ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity.
The case p = 2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0 < p < 1 by building on our results here.
The case p = 1 remains open: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity?
In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number of shifted lattice points in the same family of reciprocal stretching curves as in Part I.
The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates.
Our results handle the p-circle when p > 1 (concave) and also when 0 < p < 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.
Lattice points; Planar convex domain; P-ellipse; Spectral optimization; Laplacian; Dirichlet eigenvalues; Neumann eigenvalues; Translated lattice; Schrödinger eigenvalues; Harmonic oscillator
Tue, 11 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983642017-07-11T00:00:00ZLiu, ShiyaColoring and covering problems on graphs
http://hdl.handle.net/2142/98358
Coloring and covering problems on graphs
Loeb, Sarah Jane
The \emph{separation dimension} of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times.
In contrast to separation dimension, we show fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt{2}$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. We conjecture that when $n=3m$ the $K_4$-free $n$-vertex graph maximizing $\pi_f(G)$ is $K_{m,m,m}$.
We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs, and let $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$.
A \emph{star $k$-coloring} is a proper $k$-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring.
A proper vertex coloring of a graph $G$ is \emph{$r$-dynamic} if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. We investigate $3$-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs.
Given a proper total $k$-coloring $c$ of a graph $G$, we define the \emph{sum value} of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} $\chi''_{\Sigma}(G)$. Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that $\chi''_{\Sigma}(G)\leq \Delta(G)+3$ for any simple graph with maximum degree $\Delta(G)$. We prove this bound to be asymptotically correct by showing that $\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1))$. The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring.
Graph coloring; Graph covering
Mon, 10 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983582017-07-10T00:00:00ZLoeb, Sarah Jane