My office hours are Mon 1-2pm in room 7210, AP&M building. My email is justin@math.ucsd.edu and my phone is 534-2649.
Discussion section is at 6pm on Wednesdays, in the same room Centre Hall 220. The TA is Ben Wilson (bwilson@math.ucsd.edu). His office hours are TBA.
Homework will be set most weeks, due in the section.
Midterm will be on a date to be decided! Let's just see how it goes for now!
Final exam is in room TBA on Monday June 9th, 8am-11am.
Grade Does 30% HW, 20% MT, 50% final seem reasonable?
Books I will start by following Milnor's classic "Topology from the differentiable viewpoint" and then proceed to Barden and Thomas's "An introduction to differential manifolds". I'll try to make copies of these available, since the first is very short and the second apparently on back-order.
Prerequisites Not much more than linear algebra. It's possible to teach this course without assuming any knowledge of point-set topology (though open sets and basic metric spaces will be used) so that's what I intend to do. I'll try to explain any other analytical or algebraic stuff that we need along the way. The course is also more or less independent of 150A, though that course helps very much in terms of context.
1. We'll develop the basic theory of smooth manifolds in arbitrary dimensions, thinking of them first as subsets of R^n and ultimately as abstract objects in their own right. To do this we start with a bit of basic calculus of smooth functions R^n -> R^m, the inverse and implicit function theorems, and so on before getting to the general definitions. A key goal is trying to understand the simple problem: when is the set of solutions of an equation f(x_1, x_2, ..., x_n)=0 actually a smooth manifold of dimension n-1?
2. We'll see that topological issues enter the theory right from the start. The idea of degree of a map between two smooth manifolds encodes many ideas like winding number, linking number, Euler characteristic and so forth. Simple applications of the idea of "taking preimage of a regular value" will give proofs of a bunch of simple theorems of topology, such as the fundamental theorem of algebra, the Brouwer fixed point theorem, and the Hairy Ball theorem.
3. We investigate integration on manifolds. If we try to mimic the Riemann integral of a function on a general n-manifold, we run into the problem that the small pieces we divide the manifold into do not have a well-defined volume, and therefore we just can't integrate functions! The solution is to replace functions by n-forms which incorporate the "dx"s necessary to make sense of the volume of a small parallelepiped. Such things can be integrated nicely. Algebra leads us automatically to the idea of p-forms for arbitrary 0<= p <= n, and this gives the correct generalisation of vector calculus: we get operators generalising grad, div and curl in higher dimensions. The analogues of "the curl of a grad is zero" are always true, but the analogues of "if curl of something is zero, then it is a gradient" are not always true, and the extent to which they fail defines a topological invariant called de Rham cohomology with beautiful properties. We can also go back and use integration to give a different definition of degree, with many useful consequences that say "some integral is an integer".
4. Finally, if there's time, we will do a bit of geometry of higher-dimensional manifolds, making sense of curvature and using it to formulate Einstein's equations.