On the vanishing of homology in random Čech complexes

We compute the homology of random ?ech complexes over a homogeneous Poisson process on the d‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd?s ‐Rényi phase transition, where the ?ech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017     

Modular Symbols and the Topological Nonrigidity of Arithmetic Manifolds

In this paper we address the existence of smooth manifolds proper homotopy equivalent to nonuniform arithmetic manifolds M = Γ\G/K that are not homeomorphic to it. While the manifolds M are properly rigid if rank_ℚ(Γ) ≤ 2, we show that the so‐called virtual structure group has infinite rank as a ℚ‐vector space if rank_ℚ(Γ) ≥ 4.© 2015 Wiley Periodicals, Inc.     

Trinuclear Gold Clusters Supported by Cyclic (alkyl)(amino)carbene Ligands: Mimics for Gold Heterogeneous Catalysts

The synthesis of air‐ and moisture‐stable trinuclear mixed‐valence gold(I)/gold(0) clusters is described. They promote the catalytic carbonylation of amines under relatively mild conditions. The synthetic route leading to the trinuclear clusters involves a simple ligand exchange from the readily available μ³‐oxo‐[(Ph₃PAu)₃O]⁺ complex. This synthetic method paves the way for the preparation of a variety of mixed‐valence gold(I)/gold(0) polynuclear clusters. Moreover, the well‐defined nature of the complexes demonstrates that the catalytic process involves a rare example of a definite change of oxidation state of gold from Au⁰₂Au^I to Au^I₃.     

Conical parametric scattering of a single extraordinary beam in BaTiO3 crystal

In this paper, the parametric scattering of a single extraordinary polarized beam of laser in BaTiO₃ photorefractive crystal has been investigated experimentally and theoretically. The resulting pattern consists of beam fanning, isotropic ring, and anisotropic one. Among all parts of scattering pattern, isotropic ring has not been studied as much as beam fanning and anisotropic ring, and there still are some differences in reports about it. Therefore, the study has mainly focused on this part. In this experimental configuration, isotropic ring is just visible in positive angles although the other parts of parametric scattering pattern can be visible from behind and in front of the crystal. In addition to steady state pattern in forward and backward directions, its transient behavior with the rotation of crystal has been studied. The results of experiments have been analyzed carefully, and their theoretical explanations have been presented based on the standard theory of parametric scattering in photorefractive crystals. It has been shown that this configuration corresponds to the so called parametric B-process scattering.     

Interpolation,the Rudimentary Geometry of Spaces of Lipschitz Functions,and Geometric Complexity

Foundations of Computational Mathematics - We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring...     

Finding the Homology of Submanifolds with High Confidence from Random Samples

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question. The main results of this paper were first presented at a conference in honor of John Franks and Clark Robinson at Northwestern University in April 2003. These results were formally written as Technical Report No. TR-2004-08, Department of Computer Science, University of Chicago.