Euclid tried (unsuccessfully!) to formulate a series of postulates for the geometry of a (flat, infinite) piece of paper. His abstract model was supposed to accurately reflect the world around us in that his postulates were to be "self-evident". However, his most famous postulate, the parallel postulate, turns out to be independent of the others. This leaves room for non-Euclidean geometries in which the other postulates are satisfied but which have too many or too few parallel lines.
This course investigates several types of geometry which differ fundamentally from the geometry of a piece of paper. We will discuss what it means to specify a geometry in terms of a set of axioms, and we will illustrate this concept with simple finite geometries. We will then study the classic alternatives to the parallel postulate, namely geometries with too many parallel lines (hyperbolic geometry) or too few parallel lines (elliptic geometry). Considerable time will be spent discussing specific models for these geometries in addition to their general properties.
We will also study the beautiful model of geometry called taxicab geometry, which is based on the notion that the "taxicab" distance between two points is not in a straight line "as a crow flies" but rather along streets "as a taxicab drives". This model has many surprising and "non-Euclidean" features, but actually does satisfy Euclid's postulates, including the parallel postulate. (It does not, however, satisfy the modern, corrected versions of Euclid's postulates and is thus classified as non-Euclidean.)
We will construct and investigate all of these models using computer graphics generated with Mathematica, and we will meet regularly in the Math Learning Center (MLC) Computer Lab. No prior experience with computers will be assumed. This course is also a Writing Intensive Course (WIC), and a 5-7 page paper on a topic in geometry will be required. The prerequisite for MTH 338 is MTH 252 or consent of the instructor. Every effort will be made to make the course self-contained, although prior background in Euclidean geometry and the ability to construct simple proofs would be helpful.
This course is a self-contained introduction to the many uses of differential forms. This approach emphasizes geometric content in a coordinate independent way. A good analogy is the use of vectors, rather than their components, to describe a given situation. While we will spend some time developing the necessary mathematical tools, the emphasis will be on applying these tools to concrete examples drawn from the physical sciences.
The formal prerequisites for MTH 434 are MTH 312 & MTH 341. However, the main prerequisite is a certain amount of scientific maturity, rather than background in a particular area. The only specific requirements are a working knowledge of multivariable calculus and linear algebra.
This course normally follows an introductory course in differential geometry, either MTH 435/436 or MTH 420. However, it is often possible, especially for students from other departments, to join the sequence at this point. The major focus so far has been on surfaces in ordinary Euclidean 3-space and curves in these surfaces. We will start with a brief review of the generalization of these concepts to abstract surfaces of arbitrary dimension, and the definition of various geometric objects, such as tensors and forms, on such surfaces.
As an application of this formalism, we will study Einstein's theory of relativity, one of the most mathematically elegant physical theories ever proposed. In essence, this theory says that gravity is curvature!
After briefly considering special relativity from a geometric point of view we will then study the Einstein field equations. We will examine several special solutions of these equations, including simple cosmological models, such as the Big Bang, and black holes.
The recommended prerequisite for MTH 437/537 is MTH 434. Students wishing to join the sequence at this point should contact me as far in advance as possible.
tevian@math.oregonstate.edu |