An Exposition of Modernism in Geometry and Physics: Isometry and Relativity

Programme

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"Umberto Boccioni. Dynamism of a Soccer Player. 1913". MoMA. Archived from the original on September 21, 2019. Retrieved September 21, 2019

I. Preludio: Preliminaries

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Drawing by Anatoly Fomenko

Manifolds: Just Nice Spaces …

[1] Picture of Six Manifolds (left). George K. Francis. Topological picturebook

[2] Animation of Klein Bottle (right). Rolcott. Imported from Wikipedia webpage

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Klein’s plastic model

[1] Tristan Needham. Visual Complex Analysis. Oxford University Press, 1998

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Ballade for Geometry

1. A character from Gabriel García Márquez ‘s One Hundred Years of Solitude

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[1] John Lee. Introduction to Smooth Manifold. 2nd edition, Springer, 2013.

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[1] Steven G. Krantz. Complex Analysis: The Geometric Viewpoint. 2nd edition, AMS, 2004

[2] https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model

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Curvature

[1] John Lee. Riemannian Manifolds: An Introduction to Curvature. Springer, 1997

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Drawing by Anatoly Fomenko

Uniformization Theorem

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[1] John Lee. Introduction to Topological Manifold. 2nd edition, Springer, 2011.

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Mobius Transformation

[1] https://www.math3ma.com/blog/automorphisms-of-the-riemann-sphere

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Automorphism Groups

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Drawing by Anatoly Fomenko

What is isometry

• Essential Task of Geometry and Klein’s Erlangen program
• Riemann Geometry is the study of invariants under isometries on Riemann manifolds.
• One will feel the same in two isometric spaces.

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[1] Anatole Katok. Lecture on Surfaces. Chapter 1, Fig 1.20

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[1] Steven G. Krantz. Complex Analysis: The Geometric Viewpoint. 2nd edition, AMS, 2004

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Reflection (-)

Glide Reflection (-)

Pictures from [1] Anatole Katok. Lecture on Surfaces. Chapter 1, Fig 1.21 and 1.23

[2] https://kconrad.math.uconn.edu/blurbs/grouptheory/isometryR2.pdf

[3] John Stllwell. Geometry of Surfaces. Springer

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[1] John Stllwell. Geometry of Surfaces. Springer

[2] http://math.uchicago.edu/~may/REU2012/REUPapers/Khan.pdf

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IV. Scherzo: Einstein’s Theory of Relativity

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Drawing by Anatoly Fomenko

Special Relativity

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General Relativity

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V. Finale: Modernism

Geometry is Modernism

-Smooth manifold theory: an alphabet and some grammar…

-Riemann geometry: more about metrics, curvature, and geodesics

-Lie group and analysis on it: Harr measure

-Geometric Topology: Classification of n-manifold

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Anatoly Fomenko

Encore: Spatial Poetics

Poetry is the art of calling the same thing by different names. –Anonymous

Yes, and mathematics is the art of calling different things by the same name. –Poincare [1]

He was told that the student had left mathematics in favor of poetry. “Ah, yes,” said Hilbert, “I always thought he didn’t have enough imagination for Mathematics.”[2]

[1] diverse sources: https://skeptics.stackexchange.com/questions/40302/was-there-a-person-made-the-quote-about-poetry-that-poincar%C3%A9-responded-to

[2]: Aharoni, Ron. "Mathematics, poetry and beauty." Journal of Mathematics and the Arts 8, no. 1-2 (2014): 5-12.

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Bibliography

• [1] Tristan Needham. Visual Complex Analysis. Oxford University Press, 1998.
• [2] John Lee. Introduction to Topological Manifold. 2nd edition, Springer, 2011.
• [3] John Lee. Introduction to Smooth Manifold. 2nd edition, Springer, 2013.
• [4] Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry. Wiley Classics Library Edition, Wiley, 1994.
• [5] Jurgen Jost. Compact Riemann Surfaces: An Introduction to Contemporary Mathematics. 3rd edition, Springer, 2006.
• [6] Steven G. Krantz. Complex Analysis: The Geometric Viewpoint. 2nd edition, AMS, 2004
• [7] John Lee. Riemannian Manifolds: An Introduction to Curvature. Springer, 1997

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And URL links provided on each slide.

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