Differential Geometry Intro

Catalog Description MAT 4140 – 3 hours [On-Demand] Differential Geometry Bulletin Description: This is an introductory course in the differential geometry of curves and surfaces in space, presenting both theoretical and computational components, intrinsic and extrinsic viewpoints, and numerous applications. The geometry of space-time will also be considered. Prerequisite: MAT 2130.

Background Geometry: Riemannian Geometry PhD Physics: Special relativity with Ralph Alpher, one of the creators of the big bang. Research: Geometry of Orbifolds, Pop Culture & Math, Women & Minorities in Math

Course Goals To develop geometric problem solving skills and 3-D spatial visualization skills. To develop a greater appreciation for connections between various disciplines of mathematics, including geometry, linear algebra, complex analysis, and differential equations, along with an introduction to these subjects as they apply to differential geometry. To understand the importance of differential geometry in various scientific fields, including physics.

Topics Geometry of curves in space, including Frenet formulas Theory of surfaces, including curvature, geodesics, and metrics Geometry of space-time and applications to general relativity (as time allows)

Culmination of Early Ideas At a very early age, children develop a very rich `visual intelligence' in terms of perception and experiences. They have questions and lots of these questions and explorations can be connected to geometry if we use the right types of physical and visual presentations. They have developed advanced skills for which the precise vocabulary is 3-D differential geometry and differential topology [28 (Hoffmann, 1998), 36 (Koenderink, 1990)]...we should connect with these abilities. (Whiteley, 1999)

Rich History and New Relevance Machinery Design Classification of Spaces Fundamental Forces in the Universe Study of DNA Medical Imaging Computer Graphics

Approaches Intuitive Calculable Useful Interdisciplinary: unification of many topics including geometry, spatial visualization, calculus, linear algebra, differential equations, and complex variables along with various topics from the sciences, including physics

Bee Dance All Things Considered, December 5, 1997 In mapping a six-dimensional figure onto two-dimensions, mathematician Barbara Shipman recognized the pattern as that of the honeybee's ritual dance. To her, this implies that bees can sense the quantum world, since it is in that realm that six-dimensional geometry has real meaning. The bees use the dance to communicate to others in the hive the location and distance of a pollen source.

Smokestack Problem Frank Morgan got a call from a company constructing a huge smokestack, which required the attachment of a spiraling strip or stake for structural support.

Strake cut out of flat metal pieces What inner radius would make it fit best?

Inconsistency in Mercury’s Orbit Newton used solar gravitational attraction and calculus to explain Kepler’s elliptical planetary orbits. The orbit rotated (precessed) at an unexpected rate General relativity and the geometry of space-time

Map Projections Coordinate charts for surfaces are named for cartographic maps. NASA Mercator map of Saturn’s moon Phoebe taken by the Cassini spacecraft in June 2004

Soap Bubbles Pressure versus surface tension What shape does it take on, and why?

Curves Euclid, Archimedes, and Appolonius Newton, Leibniz, the Bernoullis, and Euler Some of the earliest results were motivated by the desire for more accurate clocks: I have been occupied with a new discovery... In order to make my clock even more exact.... What, however, I never had expected I would discover, I have now hit upon, the undoubtedly true shape of curves... I determined it by geometric reasoning. (C. Huygens December 1659)