Notes from a class titled Harmonic Analysis on Homogeneous Spaces given at the Royal Institute of Technology, Stockholm in the spring of 1994. They are elementary, incomplete, and disorganized, but redeem themselves by having lots of miss-prints and an index.
A straight tunnel is cut through the Earth, but not necessary though its center. Arthur Dent starts at rest at one end of the tunnel and slides, without friction or air resistance and only under the force of gravity, until he reaches the other end of the tunnel. Question: How long does the trip take Arthur?
Given a planet of uniform density and a point on its surface, how can we reshape the planet to maximize the gravitational force at the given point.
An exposition of the analytic proof of Milnor, as simplified by Rogers, of the Brouwer Fixed Point Theorem. All that is required is the inverse function theorem and the change of variable formula for multiple integrals.
For a smooth compact manifold M let S(M) be the space of infinitely differentiable real valued functions on M. Given two manifolds M and N with dim(M)> dim(N) it is shown that there are injective linear linear maps of finite order from S(M) to S(N). I had conjecture that all such maps have infinite dimensional kernel, which, if it had been true, would have explained many non-injectiveity results in integral geometry.
If the set C of compact subsets of Rn is given made into a metric space using the Hausdorff distance, then set of K in C with dense cut locus dense in Rn is a dense G-delta set in C.
If f(x)=f1(x)+f2(x)+f3(x)+... is a pointwise convergent series of function each of which is monotone increasing, then we give a proof of Fubini's result that the derivative is given by f '(x)=f1'(x)+f2'(x)+f3'(x)+... almost everywhere. This is used to an easy example of function that is continuous, strictly increasing, but has derivative zero almost everywhere.
An exposition of a recent result of Mohammad Ghomi who has shown that if M is smooth compact oriented surface in R3 so that all the shadow sets of M are simply connected, then M is the boundary of a convex set. Also given is his example showing that "simply connected" can not be weakened to "connected".
Let c(t) be a regular curve in the Euclidean space R3. Then the tantrix is the curve on the sphere S2 given by t(t)=c'(t)/|c'(t)|. A proof of a folk theorem characterizing the tantrix curves of closed curves is given and extended to higher dimensions and the case of curves symmetric with respect to a group action.
If a light is shined on a surface, the boundary between the light and the shadow is the shadow curve. These are notes, at an elementary level, on the geometry of shadow curves.
Notes giving a detailed proof of Alexandrov's Theorem that a convex suction has second derivatives almost everywhere. This done by use of Rademacher's theorem (a Lipschitz function between Euclidean spaces has first derivatives almost everywhere) whose proof is also included.
A proof is given of the result of Fritz John that if K is a convex body in Rn and E is the ellipsoid of maximum volume in K then K is contained in c+n(K-c). When K is symmetric about the origin this can be improved so that K is contained in n1/2E.
An expanded version of a preprint above (written with Steve Dilworth and Jim Roberts). The differences are extra figures, some exposition of results in the literature, and an alternate proof of one theorem.
Some notes based on lectures of Kostya Oskolkov giving a simplified proof of a theorem of Halász bounding the number of points a "random" sum of a set of n vectors, not clustered about any hyperplane, that can lie in a sufficiently small set.
A version of the Gronwall inequality that estimates the difference between solutions to two differential equations in terms of the difference between their initial conditions and how how much the two equations differ.
The Loewner-Pu inequality is generalized to Riemannian metrics g on an n dimensional torus Tn that are C-quasi-conformal to a flat metric where the resulting inequality relates C and some isosytolic constants. For n > 2 this can be combined with recent results of Babenko and Katz to give examples of smooth metrics that have arbitrarily large "quasi-conformal distance" from the set of flat metrics on Tn.
This is a preliminary and expanded version of one of the preprints above. It also contains an extra section giving a strengthened version of McKean's lower bound on the first eigenvalue of the Laplacian on a negatively curved surface.
This is an edited version of a proof, in the from of exercises with detailed hints, of the classical inverse function and the inverse function theorem for Lipschitz maps between Banach spaces that was given to a graduate class in differential equations as homework.
A self contained proof of the theorem of E. Hopf that a Riemannian metric on the two dimensional torus without conjugate points is flat. An extension to higher dimensions, due to L. Green, is also given. The proof follows that of Green.
A very elementary account of the local version of the Nash isometric embedding theorem using the method of Gunther.
A proof of the result of Kuiper that a compact simply connected locally conformal flat Riemannian manifold is globally conformally equivalent to the standard sphere. The proof here, while it follows the basic outline of Kuiper's proof, requires less smoothness of the metric.
If all principle curvatures of a complete hypersurface M of Euclidean space Rn have absolute value at most one then at any point of M the graphing radius is at least one. This is used to prove results on the inradius of domains in terms of the topology and the size of the principle curvatures.
If M is a two dimensional sphere with a Riemannian metric that has curvature in the interval [0,1] then every simple closed geodesic of M has length at least 2
with equality iff M is either
a standard sphere or a capped cylinder.