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Research Interests

My research interests are in the areas of Riemannian geometry (non-negatively curved manifolds), group actions, Dehn functions and rigidity theorems in positive curvature. I also maintain an active interest in number theory and cryptography, and finite groups. In addition, I have an active program of undergraduate research projects. While most of these are expository, the occasionally rise to the level of original research (see below).

Here are some questions that I am thinking about off and on:

1. Let M be a 2-dimensional, complete Riemannian manifold without conjugate points. Suppose M satisfies Euclid's parallel postulate, i.e., given a line a point not on the line, there is a unique parallel line through that point. Is it true that M must be flat R^2?

2. Suppose X is an Alexandrov space with curvature bounded above and below (say +1 and -1) with the property that the injectivity radius equals the diameter? Is X in fact (locally) isometric to a compact, rank one symmetric space? This is a generalization of the classical Blaschke conjecture (which is still open for even dimensional manifolds).

3. Are there fundamental groups of arbitrarily large order among even dimensional manifolds with almost positive sectional curvature?

4. Suppose G is isomorphic to a finite cyclic group of order n. Given an element g in G, is there a fast way (polynomial time) to compute the order of g?

My work is currently supported by the National Science Foundation (DMS-1104352).

Publications

Caveat:You can click on "Article" next to each paper to view a copy. However, for copyright reasons, the version you download is not the published version. If you want the most up to date version, then please email me.

  1. On the fundamental groups of positively curved manifolds, Journal of Differential Geometry, 49 (1998) 179--182.  Article.
  2. Rank two fundamental groups of positively curved manifolds (with Karsten Grove), Journal of Geometric Analysis, 10 (2000) 679--682.  Article.
  3. Nilpotent numbers (with Jonathan Pakianathan), American Math. Monthly, 107 (2000), 631--634.  Article.
  4. Isometry groups of homogeneous, positively curved manifolds, Differential Geometry & Appl., 14(1) (2001), 57--78. Article.
  5. On complexes equivalent to S^3 bundles over S^4 (with Nitu Kitchloo), Int. Math. Research Notices, 8 (2001), 381--394.  Article.
  6. Strong inhomogeneity of Eschenburg spaces (with an Appendix written jointly with Mark Dickinson), Mich. Math. Journal,  50(1) (2002), 125--142.  Article.
  7. The diffeomorphism type of the Berger space SO(5)/SO(3) (with Sebastian Goette and Nitu Kitchloo), American Journal of Mathematics, 126(2) (2004), 395--416.  Article.
  8. Free isometric circle actions on compact, symmetric spaces (with Jost Eschenburg and Andreas Kollross), Geometriae Dedicata, 102 (2003), 35--44.  Article.
  9. Non-negatively and positively curved metrics on circle bundles (with Kris Tapp and Wilderich Tuschmann), Proceedings of the AMS, 133(8) (2005), 2449--2459.  Article.
  10. Spherical rank rigidity and Blaschke manifolds (with Ralf Spatzier and Burkhard Wilking), Duke Mathematical Journal, 1 (2005), 65--81.  Article.
  11. On the cohomogeneity and symmetries of Eschenburg and Bazaikin spaces (with Karsten Grove and Wolfgang Ziller), Asian J. of Mathematics (S. S. Chern memorial volume), (2006), 647--662.  Article.
  12. Snowflake groups, Perron-Frobenius eigenvalues and isoperimetric spectra (with Noel Brady, Martin Bridson and Max Forester), Geometry & Topology, 13 (2009), 141--187.  Article.
  13. Conjugate points in length spaces (with Christina Sormani), Advances in Mathematics, 220(3) (2009), 791--830.  Article.
  14. Riemannian submersions from simple, compact Lie groups (with Martin Kerin), Muenster Journal of Mathematics, 5 (2012), 25--40.  Article.

Preprints

  1. Square roots, continued fractions and the orbit of 1/0 in the hyperbolic plane (with Justin Roberts and Julian Rosen), 2006. Article.

Undergraduate Research Projects

Nearly every semester at Oklahoma I have been involved in undergrad research projects. Most are expository; typically students learn a new topic and write a report on what they have learned. Occasionally, they break new ground. The preprint above on square roots and continued fractions was written with two undergraduates (both of whom have completed their Ph.Ds in mathematics at this time). If you are an undergraduate and would like to learn some new mathematics or think about an unsolved problem, then feel free to contact me. Here are a couple of unsolved problems you can think about if you are so inclined.

  1. Euler Bricks: Given a rectangle in the plane with side lengths 'a' and 'b' its diagonal 'c' is the square root of a^2+b^2 (the Pythagorean theorem). One may ask: can one characterize rectangles where 'a', 'b' and 'c' are all integers? This was done a few hundred years ago and one can parametrize the infinitely many solutions in this case. The mystery starts when one jumps to 3 dimensions: is there a rectangular box of integer side lengths 'a', 'b', 'c', such that all face diagonals and all long diagonals are also integers? Such a box is called an Euler Brick. It is unknown whether such a brick exists, but recently a few undergraduates at Bucknell found infinitely many parallelepipeds with all integer lengths and tilting angle arbitrarily close to zero. Questions: are these the only possible examples? Is there an example with integer volume? Is there a way to take a limit of a subsequence of the above parallelepipeds?
  2. An interesting function: Consider the following function F: {positive integers} -> {positive integers}, defined as follows. Given a positive integer, let a(n) denote the largest factor of n less than n. So for example, a(7) = 1, a(24) = 12 etc. Then we define F(n) = n + a(n) - 1. It is easy to see that F(n) = n if and only if 'n' is a prime number. What is less clear is that starting with any positive integer 'n', repeatedly applying F leads to a prime number. For example, n=10: F(10) = 10+5-1 = 14; F(14) = 14+7-1 = 20; F(20) = 20+10-1 = 29, which is prime. The conjecture is that F stabilizes in finitely many iterations to yield a prime number. It has been checked for 'n' up to 100,000 and the longest iteration has length 78. Questions: are there arbitrarily long iterations? given a prime number 'p', is there a composite number 'n' which "hits" it after finitely many iterations? can one characterize numbers that stabilize in 2 steps or 3 steps?