The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from K\"ahler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases of interest and give a formula for homotopy equivalence in one particular 7-dimensional case. We also show that our construction gives at least a two dimensional cone of both Sasaki-Ricci solitons and extremal Sasaki metrics.Comment: 38 pages, paragraph added to introduction and Proposition 4.1 added, Proposition 4.15 corrected, Remark 5.5 added, and explanation for irregular Sasaki-Einstein structures expanded. Reference adde

The Sasaki Join, Hamiltonian 2-forms, and Constant Scalar Curvature

We describe a general procedure for constructing new Sasaki metrics of constant scalar curvature from old ones. Explicitly, we begin with a regular Sasaki metric of constant scalar curvature on a 2n+1-dimensional compact manifold M and construct a sequence, depending on four integer parameters, of rays of constant scalar curvature (CSC) Sasaki metrics on a compact Sasaki manifold of dimension $2n+3$. We also give examples which show that the CSC rays are often not unique on a fixed strictly pseudoconvex CR manifold or a fixed contact manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two dimensional subcone of Sasaki Ricci solitons in the Sasaki cone, and a unique Sasaki-Einstein metric in each of the two dimensional sub cones.Comment: 32 pages. A gap in the argument of applying the admissibility conditions to irregular Sasakian structures is filled. Some minor corrections and additions are also made. This is the final version which will appear in the Journal of Geometric Analysis. It also encorporates much from our paper arXiv:1309.706

The Frequency Interpretation in Probability

AbstractWe formulate and discuss the notion of generic sequence (and random sequence) associated with a sequence of random variables. These are substantial generalizations of the notion of “collective” introduced by R. von Mises as an attempt to give an operational meaning to various probabilistic ideas. The definition of collectives lacked a precise formulation, and A. Church and others attempted to give a rigorous meaning to the theory by utilizing ideas from the theory of computability. Nevertheless, the theory had major flaws. Even though some important contributions to special cases were made by P. Martin-Löf, the whole circle of ideas has languished due to lack of generality. The theory presented in the present article resolves many of these problems, and provides a coherent framework for the relevant ideas

Educating Health Professionals in the Era of Ubiquitous Information

Dr. Charles Chuck Friedman, an internationally known biomedical informatics scholar and health IT expert, will give a talk as part of the Chancellor\u27s Leadership Forum on Monday, December 17. This is a reprise of a highly acclaimed talk he gave at a recent national meeting of the Association of American Medical Colleges (AAMC). Dr. Friedman will discuss the following: • How the ubiquity of information available from the Internet will impact medical student learning • The implications of medical education we can reasonably expect by 2020\u2

Computable knowledge: An imperative for Learning Health Systems

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151989/1/lrh210203.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151989/2/lrh210203_am.pd