2007 March 19 / E-Mail Me
In lieu of a final exam our class will undertake final projects in groups of three. Each group will produce a report and a short talk (5 minutes per person, say) to be delivered to the class in the final week of the semester. I will meet with each group several times to suggest readings, check progress, practice the talk, critique the paper, etc.
Bridget, Emily, Caroline
Because it is curved, the surface of the Earth cannot be represented on a flat 2D map perfectly. To work around this, mapmakers have developed various map projections with various advantages and disadvantages. Study the common projections (Mercator, Lambert, etc.), including their history/applications and formulas/theorems. Implement some in Mathematica. After that, find a digital elevation map and use it to create a true parametrization of the Earth's surface, including hills and valleys. Or study how remote sensing systems (GPS, GRACE, etc.) have revolutionized cartography and the geosciences.
In this class we focus on surfaces with a positive-definite inner product on each tangent plane. Altering the positive-definiteness assumption produces general relativity in two dimensions (one space and one time). Develop this theory, and then explain how it generalizes to the usual four-dimensional relativity. This is probably differential geometry's most famous application. The project is not for students who have studied general relativity before. Almost no physics is required, actually.
Until around 1948 the action principle, which we studied in our treatment of mechanics, was the fundamental axiom of physics and hence of science. But then Richard Feynman et al. developed an explanation for the action principle in terms of path integrals. Investigate the action principle in quantum mechanics and explain how it arises from path integrals. Perform Mathematica simulations to demonstrate the connection. Some experience with quantum mechanics, e.g. wavefunctions/amplitudes, will be helpful.
We're going to spend a little time in class talking about minimal surfaces, harmonic functions, etc. Using this paper and/or other references, study the subject in greater detail. Solve one or more problems posed at the end of the paper. This project requires comfort with complex analsyis and/or differential equations. It was proposed by Jesse.
In 20th century geometry the idea arose that spaces per se are not as important as the functions on those spaces. You will study this idea briefly in differential geometry (ring of smooth functions, Poisson bracket) and more deeply in algebraic geometry, where functions, tangent spaces, and even the spaces themselves are expressed in what appear to be completely different terms. This project uses ring theory heavily, and comfort with abstract mathematics is required.
Ciara, Joonhahn, Luyuan
As we discussed on the first day of class, the problem of unknotting DNA using enzymes such as topoisomerase II has been studied with White's Formula from differential geometry. Explain the formula and its application to DNA.
Kshipra, Arnav, Susanna, Mariya
In class we are going to discuss how differential geometry can be used to plan the motions of robots safely and efficiently. Investigate recent progress in this field and implement some algrorithms in Mathematica or some other language.
Euclidean geometry is essentially the study of R2 (and other Rn) with the standard inner product. Differential geometry helps us study other spaces with non-Euclidean geometries. Survey the standard examples (Möbius geometry, hyperbolic geometry, projective geometry, etc.) and thoroughly explain them in terms of differential geometry.
Accurate, efficient dynamical simulation is important to many fields, from serious physics research to video games. Usually numerical methods are used, since the differential equations are too difficult to solve exactly. But numerical methods lead to round-off errors which can have serious effects on sensitive systems. Techniques such as symplectic integration, which build on our treatment of mechanics, can be used to minimize such errors. Study one or more of these algorithms, implementing them in Mathematica or some other programming language.
Abstract manifolds, arbitrary dimensions, vector bundles, connections? Explain. Illustrate how these generalize the ideas we've studied in this course. This project requires comfort with abstract mathematics. I expect significant progress.
Okay, the consensus view nowadays seems to be that Gauss's geodetic survey of the Alps was probably not for detecting the curvature of space. But there is still a lot of beautiful math to study in this direction, namely using triangulations to detect curvature, building connections to topology, and higher-dimensional Gauss-Bonnet.
Morse theory, decompositions of surfaces, detecting geometry from data, smoothing data.
Pretty hard stuff, but some progress could be made.
Lagrangian reference frames, etc. tie in nicely with covariant derivative, etc.
How about winding numbers modulo 2, used to prove the Jordan curve theorem?