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Book Review: The Poincare Conjecture: In Search of the Shape of the Universe by Donal O’Shea

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Donal O’Shea’s Poincare Conjecture: In Search for the Shape of the Universe is an excellent rendition on the classical theories of shape for the planet earth and the universe itself utilizing Pythagoras, Euclidean Geometry, Einstein and modern mathematicians. Fractal geometry is another possible application because the earth’s spherical presentation is non-linear in many places.

A basic assumption is that we do not know the shape of earth for sure. In fact, the shape of the planet is not constant. Instead, there are finite and not so finite changes in topography due to volcano activity, earthquakes, and other significant disturbances that literally change the face of the planet on a continuing basis.

Continuous space has infinite dimensions. The essence of the Poincare Conjecture is that there is no boundary for earth in the classic sense of a beginning and an end. Every loop on a sphere shrinks to a point. In addition, there are no two parallel lines on a sphere because any two lines intersect at some point.

The Poincare Conjecture also asserts that three compact manifolds on which a closed path shrinks to a point is the exact topology as its 3-sphere. An equivalent form of the conjecture involves a homotopy equivalence.

In mathematical topology, two continuous functions from one topological space to another are called homotopic when one can be “continuously deformed” into the other. If a 3-manifold is homotopy equivalent to the 3-sphere, then it is homeomorphic to it. A homeomorphism or topological isomorphism or bi-continuous function is a continuous function between topological spaces that have a continuous inverse function.

Grigori Perelman proved the full geometry conjecture in 2003 by employing the Ricci flow. In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold by smoothing out irregularities . A Riemannian metric opens the possibility to define various geometric notions on a Riemannian manifold such as areas, angles, and curve lengths.

Poincare Conjecture: In Search for the Shape of the Universe is a perfect acquisition for physicists, mathematicians, logicians, and an audience of professionals in the allied areas of mathematics and computer science. O’Shea’s presentation is strong in many spots and difficult to understand in others.

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About Dr Joseph S Maresca

I've taught approx. 34 sections of collegiate courses including computer applications, college algebra, collegiate statistics, law, accounting, finance and economics. The experience includes service as a Board Director on the CPA Journal and Editor of the CPA Candidates Inc. Newsletter. In college, I worked as a statistics lab assistant. Manhattan College awarded a BS in an allied area of operations research. The program included courses in calculus, ordinary differential equations, probability, statistical inference, linear algebra , the more advanced operations research, price analysis and econometrics. Membership in the Delta Mu Delta National Honor Society was granted together with the degree. My experience includes both private account and industry. In addition, I've worked extensively in the Examinations Division of the AICPA from time to time. Recently, I passed the Engineering in Training Exam which consisted of 9 hours of examination in chemistry, physics, calculus, differential equations, linear algebra, probability/ statistics, fluids, electronics, materials science/structure of matter, mechanics, statics, thermodynamics, computer science, dynamics and a host of minor subject areas like engineering economics. A very small percentage of engineers actually take and pass the EIT exam. The number has hovered at circa 5%. Several decades ago, I passed the CPA examination and obtained another license in Computer Information Systems Auditing. A CISA must have knowledge in the areas of data center review, systems applications, the operating system of the computer, disaster recovery, contingency planning, developmental systems, the standards which govern facility reviews and a host of other areas. An MBA in Accounting with an Advanced Professional Certificate in Computer Applications/ Information Systems , an Advanced Professional Certificate in Finance and an Advanced Professional Certificate in Organizational Design were earned at New York University-Graduate School of Business (Stern ). In December of 2005, an earned PhD in Accounting was granted by the Ross College. The program entrance requires a previous Masters Degree for admittance together with a host of other criteria. The REGISTRAR of Ross College contact is: Tel . US 202-318-4454 FAX [records for Dr. Joseph S. Maresca Box 646 Bronxville NY 10708-3602] The clinical experience included the teaching of approximately 34 sections of college accounting, economics, statistics, college algebra, law, thesis project coursework and the professional grading of approx. 50,000 CPA examination essays with the American Institute of Certified Public Accountants. Additionally, membership is held in the Sigma Beta Delta International Honor Society chartered in 1994. Significant writings include over 10 copyrights in the name of the author (Joseph S. Maresca) and a patent in the earthquake sciences.
  • Neil Black Douglas

    Are the Tropics of Capricorn and
    Cancer not // ?

  • bliffle

    The Poincare conjecture is a little esoteric, but there are homelier examples of indeterminancy, such as “what is the length of the English coastline?”.

    Oh, and the Tropics are not Great Circles, which are the manifestation of straight lines on a sphere.

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