Curriculum Vita

1  Personal Data

1. Name: David Lewis Johnson
2. Date of Birth: February 5, l951.
3. Citizenship: U. S.
4. Current Address:
   • Department of Mathematics
     Lehigh University 14 E. Packer Avenue
     Bethlehem, Pennsylvania 18015-3174
     Telephone: (610) 758-3759.
     E-Mail: dlj0 @lehigh.edu

5. Education:
   1. University of California, Berkeley, 1969-1973.
      A. B. Degree Received: June, 1973 (with honors)
      Major Subject: Mathematics
   2. Massachusetts Institute of Technology, 1973-1977
      Ph. D. Degree Received May, 1977
      Major Subject: Mathematics
      Thesis Advisor: Isadore M. Singer
      Thesis Title: A normal form for curvature

2  Professional Employment:

1. 9/84 - present Lehigh University, Associate Professor, (tenured 1986)

2. 1/92 - 6/92 University of Pennsylvania, Visitor
3. 9/83 - 8/84 Texas A & M University, Associate Professor
4. 9/77 - 8/83 Texas A & M University, Assistant Professor
5. 1/80 - 7/80 Rice University, Visiting Assistant Professor

3  Graduate student advising:

1. Chairman, Ph. D. Committee of Carla A. Schultes (nee Nelson) (completed, 5/91)

2. Chairman, Ph. D. Committee of Ismail Kocayusufoglu (completed, 10/93)
3. Chairman, Ph. D. Committee of Tracy Bowers, 1996 - present (expected completion, 6/01).
4. Chairman, MS committee of Vincent E. Coll, Jr., 1982-1984
5. Member, Ph. D. committee of Huajian Wang, 1993-1995
6. Member of several M.S. and Ph.D. committees for students in the College of Engineering and Applied Science, Lehigh.
7. Member of the Ph.D. committee of Jeff Spirko, Department of Physics, Lehigh, 1997 - present.

4  Colloquia and Lectures Presented:

1. Curvature and Euler characteristic of six-dimensional Kähler manifolds, AMS annual meeting, Biloxi, Mississippi, January, 1979, in the special session on global differential geometry
2. Feuilletagès totalment géodesiques (4 lectures), and Classification des éspaces fibrés vectoriels (3 lectures), Universidad de Valencia, June, 1979.
3. An obstruction to the geodesibility of a foliation of odd dimension, AMS annual meeting, San Antonio, Texas, January, 1980, in the special session on minimal submanifolds
4. Secondary characteristic classes of flat holomorphic vector bundles, AMS meeting in East Lansing, Michigan, November, 1982, in the special session on the geometry of foliations
5. Action of the automorphism group of a hermitian symmetric space on vector bundles, 1984 Joint Summer Research Conference on Integral Geometry, August, 1984.
6. Geometry of foliations, Joint U. S.-Spanish workshop Special Geometric Structures, Valencia, Spain, May 30 - June 1, 1985
7. Deformations of totally geodesic foliations, Georgia Topology Festival, August, 1985.
8. Volumes of flows, AMS Annual Meeting, San Antonio, January, 1987.
9. Volumes of foliations, Rutgers University, October, 1987.
10. Variational Problems in Differential Geometry, AMS Regional Meeting, Albuquerque, New Mexico, April, 1990
11. Topological Obstructions to the Existence of a Totally Geodesic Foliation, University of Pennsylvania, February, 1992
12. Regularity of volume-minimizing foliations, Seventh International Congress on Differential Geometry, Santiago de Compustela, Spain, July, 1994
13. Volume-minimizing graphs, University of Sao Paulo, Brazil, October, 1997 (series of 3 lectures).
14. Volume-minimizing Cartesian currents, IMPA, Rio de Janiero, Brazil, October, 1997.
15. Characteristic forms and boundary integrals, Lehigh, March 24, 1999.
16. Isoperimetric Inequalities, Universidad de Valencia, Spain, at an interenational conference in honor of Antonio M. naveira, July, 2001.

5  Grants and Contracts:

1. Short-term travel grant from the Universidad de Valencia, June, l979
2. Short-term travel grant from the Comite Conjunto Hispano-Norteamericano para la Cooperacion Cientifica y Tecnologica, for travel to Valencia, May, l980
3. College of Science Grant, Texas A & M Univ., June, l980
4. National Science Foundation Grant MCS-8002769, July, 1980 - June, 1982
5. Short-term travel grant from the Comite Conjunto Hispano-Norteamericano para la Cooperacion Cientifica y Tecnologica, for travel to Valencia, May -June, 1983
6. Joint U. S.-Spanish workshop grant to hold a workshop entitled Special Geometric Structures, May 30 -June 1, 1985, Valencia, Spain.
7. Hughes Foundation grant to improve the education of biological science students. Member of the group headed by Jeff Sands of the biology department. Mathematics Department portion of the grant ($40,000) to design a precalculus and calculus computer-based tutorial program. Grant period: 1989-1994.
8. Lehigh University computer rollover grant, with Lee Stanley and Clifford Queen, to purchase new computing equipment, 1990
9. College of Arts and Science award, disbursing a Commonwealth of Pennsylvania block grant, for computer equipment to improve instruction in calculus, 1990.

10. Grant from the Charles F. Dana Foundation for the development of calculus workshops, using a model developed by Uri Treisman at the University of California, Berkeley. Project Director is Susan Szczepanski, 1991-1995.
11. Mathematics faculty member, Integrated Mathematics and Science Teaching Institute (IMAST), a three-year institute for integrating the teaching of mathematics and science in grades K-12, funded through the Commonwealth Partnership. Project Director is Steven Krawiec, 1994-1997.
12. Travel grant from the University of Sao Paulo, Brazil, to deliver a series of lectures and work with mathematicians at USP, October, 1997.
13. Mellon Foundation grant (Clipper project) to evaluate the effectiveness of Web-based instruction. As a cooperative effort with faculty from Education, Chemistry, English, Engineering, and Economics (as well as Mathematics), we will be developing course materials for basic first-year courses, offering them to students who have been admitted to Lehigh but who have not yet enrolled (primarily those who have accepted early admission offers). They will be able to take legitimate lehigh courses in these subjects, for credit, thus freeing their schedules to allow them to take more advanced courses while in residence. We will also be evaluating their progress, in comparison with traditional Lehigh students and with current Lehigh students taking the courses on-line. A planning grant was awarded in June, 1999, and the first two years of the full 5-year grant were awarded in December, 1999. The PI of the grant is Steve Bronack of the College of Education. In addition to providing educational innovation in mathematics, this will also lead to publications in mathematics education.

6  Publications:

6.1  Published:

1. (with L. B. Whitt) Totally geodesic foliations on 3-manifolds, Proc. Amer. Math. Soc. 76 (1979), 355-357.
2. Sectional curvature and curvature normal forms, Michigan Math. J. 27 (1980), 275-294.
3. Kähler submersions and holomorphic connections, J. D. Geo. 15 (1980), 71-79 (with L. B. Whitt),
4. Totally geodesic foliations, J. Diff. Geo. 15 (1980), 225-235.

5. A curvature normal form for 4-dimensional Kähler manifolds, Proc. Amer. Math. Soc., 79 (1980), 462-464.
6. (with A. M. Naveira), A topological obstruction to the geodesibility of a foliation of odd dimension, Geometria Dedicata, 9 (1981), 347-352.

7. Curvature and Euler characteristic for six-dimensional Kähler manifolds, Illinois J. Math., 28 (1984), 654-675.
8. Secondary characteristic classes and intermediate Jacobians, Journal f{''ur die reine und angw. Math. 347 (1984), 134-145.
9. Deformations of totally geodesic foliations, Geometry and Topology: Manifolds, Varieties, (Proc., 1985 Georgia Topology Festival), C. McCrory and T. Shifrin, editors, (1987), 167-178.
10. Volumes of flows, Proc. AMS 104 (1988), 923-931.
11. Regularity of volume-minimizing graphs, (with P. Smith), Indiana University Mathematics Journal, 44 (1995), 45-85.
    • The main result of this paper is that there exist volume-minimizing graphs within each homology class of sections of a fiber bundle with compact fiber. These minimizing sections are not necessarily smooth, but are Cartesian currents, as defined by Giaquinta, Modica, and Soucek. It is shown that the set of points in the base space over which the minimizer is not a continuous graph is a set of Hausdorff co-dimension 3, except under certain topological conditions on the fiber, in which case the singular set on the base space can have co-dimension 2.

12. Regularity of mass-minimizing one-dimensional foliations (with P. Smith), conference proceedings of the Seventh International Congress on Differential Geometry, Santiago de Compustela, Spain, 1995.
    • This article treats a special case of the minimization problem in Regularity of volume-minimizing graphs, where the bundle in question is the unit tangent bundle, so that the sections are one-dimensional foliations of the base space. It is shown that, in all dimensions, the set of points in the base where a minimizing foliation has discontinuities is of Hausdorff co-dimension at least 3.

13. Locally volume-minimizing codimension-one foliations of S³ , (with Ismail Kocayusufoglu), Algebras, Groups and Geometry, 15 (1998).
14. Locally volume-minimizing codimension-one foliations of the solid torus, (with Ismail Kocayusufoglu), Turkish Journal of Mathematics, 22 (1998), 207-222.
15. Some Sharp Isoperimetric Theorems for Riemannian Manifolds (with Frank Morgan). Indiana University Mathematics Journal, 49 (2000), 1017-1042.

6.2  Submitted and in Preparation:

1. Errata and addenda to ``Regularity of volume-minimizing graphs'' (with P. Smith), in preparation.
   • This article corrects the proofs of two results from Regularity of volume-minimizing graphs, based on the extension of the more significant of those results in the next article.

2. Partial regularity of mass-minimizing Cartesian currents, (with Penny Smith), in preparation.
   • Extends and completes the analysis of a basic regularity results for mass-minimizing Cartesian sections, filling the more serious gap in ``Regularity of volume-minimizing graphs'' and placing the analysis there and in papers that article was based on in a correct theoretical context.

3. Regularity of volume-minimizing foliations of 3-manifolds, (with P. Smith), in preparation.
   • This article improves the regularity estimates obtained in Regularity of volume-minimizing graphs, and Regularity of mass-minimizing one-dimensional foliations in the case of a 3-manifold, showing that such volume-minimizing foliations have no singular points. The paper is completed, awaiting only the acceptance of the previous two papers before submission.

4. Regularity of mass-minimizing framings of 3-manifolds, (with P. Smith), in preparation.
   • This article, using methods similar to those in Regularity of volume-minimizing foliations of 3-manifolds, finds mass (energy)-minimizing framings of the tangent bundle of a 3-manifold, and shows that such minimal framings are continuous.

5. Obstructions to homogeneity of a vector bundle, in preparation.
   • This article, which is actually a rather old result which was not published, shows that the natural action of the group of holomorphic automorphisms of a Hermitian symmetric space X on the moduli space of stable vector bundles over X has orbits whose isotropy subgroup is the complexification of a compact group. This theorem allows the immediate determination of the modulus of such bundles in a number of cases, and gives sharper bounds than otherwise possible on the dimension of the moduli.

6. Volume-minimizing foliations on spheres, (with F. Brito and A. M. Naveira), in preparation.
   • This article is still incomplete, but has as one interesting result the fact that the standard Hopf fibrations with S³ -fibers are volume-minimizing among all three-dimensional foliations on S⁴ⁿ⁺³ . It actually shows much more, in that the condition of integrability of the distribution is not used. In addition, several geometric bounds are established for volumes of foliations on spheres.
   • This article establishes several improved isoperimetric results. My major contribution to the work is a theorem (Theorem 4.4) showing that an upper bound on the sectional curvature provides an asymptotic lower bound for the isoperimetric ratios given as the isoperimetric ratio of the spaceform with that curvature.

7. Deformations of Riemannian Foliations (with Mark Kellum). In preparation.
   • In this article we establish that the representation-theory construction of deformations of Riemannian foliations (work of Kellum to appear in Michigan Math J) can be compared with cohomological methods (my earlier work, such as in (9) above) to show that, on the one hand, the representations do generate all deformations of Riemannian foliations, and that the cohomological approach does give, for each infinitesimal deformation, a real family of foliations. We have established this symbiosis in certain classes of Riemannian foliations and are continuing to work on expanding the class of foliations to which this comparison applies successfully.

     Progress on this manuscript has been hindered by Kellum's career shift, into an industry position.

8. Boundary Characteristic Classes, In preparation.
   • In the work involved in (6) and (7) above, I have noticed a relationship, which I had not previously known, between Chern-Simons secondary characteristic classes and boundary terms of integrals of primary characteristic classes on manifolds with boundary. A precursor of this idea appears in Chern's 1946 paper on integral curvatures, but it seems to apply, with some caveats, to an arbitrary characteristic class. I intend to explore this relationship to see whether formulas like Morgan and I developed for (7) and as I constructed in (6), both using the Euler form, can be applied in some way to such things as the signature form. Heuristically, it should be possible. This might lead to a more computational formula for the eta invariant in some cases.