We investigate continued fractions and some aspects of Diophantine approximation in the following settings: - approximation of vectors in the product of the Archimedean completions of a number field by elements of the field (embedded diagonally), - approximation of vectors in Euclidean space by elements of definite Clifford algebras. In the first setting, when the number field is one of the Euclidean imaginary quadratic fields, we more thoroughly investigate two algorithms: nearest integer continued fractions (introduced by A. Hurwitz) and continued fraction algorithms associated to a class of right-angled hyperbolic Coxeter groups (variations on the continued fraction algorithms of A. L. Schmidt). In the second setting, we go into slightly more detail concerning approximation in three-dimensional Euclidean space by quotients of Lipschitz quaternions. In both settings we give explicit examples of badly approximable numbers via: the Dani correspondence and bounded geodesic trajectories in the appropriate locally symmetric spaces associated to the special linear group of degree two, boundedness of partial quotients in various continued fraction algorithms, and elementary arguments in the vein of Liouville's theorem concerning rational approximation to algebraic numbers. In addition, some properties of the discrete Markoff spectrum are explored, namely we demonstrate the transcendence of limits of roots of Markoff forms and evaluate a sums over Markoff numbers. The former result follows from work of Bugeaud and Adamczewski (an application of the Subspace Theorem) and the latter is a special case of Mcshane's identity (for a sum over lengths of simple closed geodesics on a once-punctured hyperbolic torus).
I will discuss index theory in the context of both manifolds with Baas-Sulivan singularities and their generalization to fibered singularities. A basic but informative class of examples are Z/k-manifolds. As will be developed in detail in the talk, a Z/k-manifold is obtained from a manifold with boundary whose boundary decomposes into k-copies of a single closed manifold. The fibered generalization in this special case is obtained from a manifold with boundary whose boundary is the total space of a k-fold covering space. Much of what will be discussed is based on "Positive scalar curvature on manifolds with fibered singularities" by Botvinnik and Rosenberg (https://arxiv.org/abs/1808.06007).
Index theory for manifolds with fibered Baas-Sulivan singularities
Apr. 04, 2019 3pm (MATH 350)
Probability
Zhenhua Wang (CU Boulder)
X
Let S be the random walk obtained from "coin turning" with some sequence {p_n}_{n>=1}. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics.
We prove that an invariance principle, albeit with a non-classical scaling, holds for "not too small" sequences, the order const n^{-1} (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold.
In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed.
We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk.
