## Jeffrey Lee Ph.D

Associate Professor
Department of Mathematics and Statistics
Texas Tech University

Office:  MATH 239     Office hours: 10-11 M-F

Email: jeffrey.lee@ttu.edu

## Mini-Biography

• Ph.D. from University of California (Los Angeles) 1987
• Research Assistance Professor at Purdue University 1988 and 1989
• Came to Texas Tech as assistant professor 1990, granted tenure 1996
• Avid guitarist

## Research Interests

• Differential Geometry, Geometric Analysis, Geometric Control, Mathematical Physics, Spectral Geometry. Learning and Teaching of Mathematics, Mathematics Outreach

Research Interests: Differential Geometry, Geometric Analysis, Geometric Control, Mathematical Physics, Spectral Geometry.
Learning and Teaching of Mathematics, Mathematics Outreach.

Old News: My book Manifolds and Differential Geometry  can be found at the AMS website or at outlets such as Amazon

Online Supplement to "Manifolds and Differential Geometry" (Work in progress but stalled)

The errata by itself is here.

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Some rough notes on the first steps in classical mechanics. (For what it's worth).
There is also a short (3 or 4 pages) primer on manifolds here

Change of Basis notes 2

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Selected Publications (not in chronological order) ( Lee, Jeffrey )

1. (with Lance Drager, Efton Park and Ken Richardson),  Smooth distributions are finitely generated, Annals of Global Analysis and Geometry, vol. 41, no. 3, (2012), 357-369
A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle. Click here to get a version from the arXiv.

2. (Book 671 pages) Manifolds and differential geometry, Graduate Studies in Mathematics, 107, American Mathematical Society, Providence, RI, 2009. xiv+671 pp. ISBN: 978-0-8218-4815-9 (Reviewer: Richard H. Escobales Jr.), 53-01 (58-01)

3. (with Drager, Lance D. Martin, Clyde F.) On the geometry of the smallest circle enclosing a finite set of points. J. Franklin Inst. 344 (2007), no. 7, 929--940. 51M04 (68U05)

4. (with Drager, Lance D and R. Byerly, Robert) Observability of Finite Dynamical Systems, IEEE Transactions on Information Theory,  (2003)

5.  (with Richardson, Ken Lichnerowicz and Obata Theorems for foliations, Pacific Journal of Mathematics, vol. 206, no. 2, 339-357 (2002).

6. (with Richardson, Ken) Riemannian foliations and eigenvalue comparison. Ann. Global Anal. Geom. 16 (1998), no. 6, 497--525. (Reviewer: James F. Glazebrook) 53C12 (58G25)

7. (with Ruymgaart, Frits H.) Nonparametric curve estimation on Stiefel manifolds. J. Nonparametr. Statist. 6 (1996), no. 1, 57--68. (Reviewer: P. E. Jupp) 62G05 (62H99)

8. Geometry detected by a finite part of the spectrum. Progress in inverse spectral geometry, 15--22, Trends Math., Birkhäuser, Basel, 1997. (Reviewer: Ruth Gornet) 58J50 (58J53)

9. Dimension, volume, and spectrum of a Riemannian manifold. Illinois J. Math. 37 (1993), no. 1, 14--32.

10. Finite inverse spectral geometry. Geometry and nonlinear partial differential equations (Fayetteville, AR, 1990), 85--100, Contemp. Math., 127, Amer. Math. Soc., Providence, RI, 1992.

11. (with Donnelly, Harold) Domains in Riemannian Manifolds and Inverse Spectral Geometry, Pacific Journal of Mathematics, Vol. 150, No. 1 (1991).

12. (with Donnelly, Harold) Heat kernel remainders and inverse spectral theory. Illinois J. Math. 35 (1991), no. 2, 316--330

13. Eigenvalue comparison for tubular domains. Proc. Amer. Math. Soc. 109 (1990), no. 3, 843--848.

14. The gaps in the spectrum of the Laplace-Beltrami operator. Houston J. Math. 17 (1991), no. 1, 1--24.

15. Hearing the volume of a drum in hyperbolic space. Indiana Univ. Math. J. 39 (1990), no. 3, 585--615

16. (with Mao, Yiping) Two-point boundary value problems for nonlinear differential equations. Rocky Mountain J. Math. 26 (1996), no. 4, 1499--1515

17. (with  Weinberg, David A.) A note on canonical forms for matrix congruence. Linear Algebra Appl. 249 (1996), 207--215.

18. (with Page, Robert; Pantrangenaru, Vic, Ruymgaart, F). Nonparametric density estimation on homogeneous spaces in high level image analysis analysis, Bioinformatics, Images and Wavlets; Program and Abstracts. Aykroyd, Barber and Mardia Eds. pp 37-40 http://www.amsta.leeds.ac.uk/Statistics/workshop/lasr2004/Proceedings/paige.pdf

19. Online Supplement for Manifolds and Differential Geometry.

A couple more articles:

1. Ship

2. Airplane

 “In many cases, proofs based on coordinate free local representations in charts are clearer than proofs which are repleate with the claws of a rather unpleasant prying insect such as $\Gamma^i_{jkl}$.” - Serge Lang.