Reading, reading, reading ..

Time grows limited tonight. Since there is too much to read carefully and understand in one sitting, I might as well tell you what I tried to read .. or will get to, if I ever plan well enough and have enough time to read it all.

But anyways, here is the list.

"Boundary Regularity and the Dirichlet Problem for Harmonic Maps" by R. Schoen and K. Uhlenbeck.

.. as suggested by my advisor. I'm up next for talking at AnSS (Analysis Study Seminar for those @ UM) and I need their results on smooth approximations of Sobolev mappings between smooth manifolds.

Right now I'm encountering trouble understanding how degree, initially a topological notion, applies to Sobolev functions and their Jacobians, which are analytic notions.

"Degree and Sobolev Spaces" by H. Brezis, Y. Li, P. Mironescu, and L. Nirenberg.

.. as a possible means to understand the notion of degree and how it applies to the Schoen-Uhlenbeck counterexample.

Edit (as of October 6th, 1:30 pm): it doesn't help with Schoen and Uhlenbeck. This BLMN paper is quite interesting, but just not what I'm looking for. Maybe I'll have time to read it later in life

"Approximation of Metric Space Valued Sobolev Mappings: Four Counterexamples and a Theorem" by P. Hajlasz.

.. as added material for the aforementioned talk @ ANSS .. next week? The clock is indeed running, but happily Hajlasz writes very well and clearly.

Correction (as of October 6th, 8 pm): I'll be talking @ AnSS in two (2) weeks's time, so now I can read this Hajlasz paper in a more leisurely manner, which is nice. Somehow one never enjoys reading if there is an enforced deadline .. \:

Some typed lecture notes by R. Canary, and to come, "Fundamental Polyhedrons and limit sets of Kleinian groups" by L. Ahlfors.

.. which I've chosen as my student talk for Prof. Canary's hyperbolic 3-manifolds course. Getting away with an analysis talk in a topology seminar is a chance not to be missed! For shame: I even get to talk about harmonic functions on hyperbolic manifolds .. q:

Now if only I can get to the details at some point ..

"Collapsed Riemannian Manifolds with Bounded Sectional Curvature" by X. Rong, and as a reference, parts from A Course on Metric Geometry by D. Burago, Y. Burago, and S. Ivanov

.. as a means to an end: I promised a few fellow students I'd give a joint talk at SGTS (Student Geometry & Topology Seminar) with my cheerful, tall Scottish friend, John. Happily he'll know more about it and I can have him field all the hard questions. q:

Measure Theory and Fine Properties of Functions by L.C. Evans and R. Gariepy, as well as the Mattila book: the exact name escapes me at the moment.

.. as the agreed reading for my Oral Prelim Exam this mid-December. Yikes!

Even when I break up my time for these reading tasks, somehow the whole of my time seems far less than the sum of its parts, and that sum isn't that much to speak of, anyways.

How did I ever get anything done at all, before this point?

Maybe I never did. I don't feel like I know anything anyways, so perhaps I am Achilles in Xeno's paradox, racing just to reach a finite point but never getting there .. \: