Ralph Elwood Howard

Department of Mathematics University of South Carolina Columbia, South Carolina 29208 howard@math.sc.edu (803)-777-2913 (office) (803)-799-8401 (home)

Date/Place of Birth: 25 December 1950 Glendale, California

Citizenship:

United States of America

Education

Ph.D California Institute of Technology  1977–1982
M.S. California State University, Northridge 1973–1974
B.A. California State University, Northridge 1970–1973

Professional Experience

Professor, University of South Carolina  1999–Present
Associate Professor, University of South Carolina  1988–1999
Assistant Professor, University of South Carolina  1984–1988
Research Associate, Michigan State University  1982–1984
Visiting Assistant Professor, Duke University  Fall 1987
Visiting Associate Professor, Royal Institute of Technology, Stockholm Fall 1993& Spring 1994

Grant Support

1986 NSF Summer Support (though Department of Engineering) Michael A. Sutton,

principal investor

1986 University of South Carolina Research and Productive Scholarship

Grant

1988–1990 NSF Summer Support, Joint grant with Marek Kossowski
1992  Summer Support on EPSCoR grant
1994–1998 Summer Support on DEPSCoR grants, Björn Jawerth, principal investor
1999–2000 Summer Support on DEPSCoR grants, Ron DeVore, principal investor

Colloquium and Seminar Talks

University of South Carolina  1982
Eastern Michigan State University  1984
Michigan State University  1984
University of Rochester (2 Talks)  1985
University of Oklahoma (2 Talks)  1985
University of Missouri  1986
University of North Carolina  1988
Emory University (2 Talks)  1988
Benedict College  1988
Saint Olaf College  1989
University of Minnesota  1989
University of Massachusetts, Amherst  1990
University of South Carolina, Aiken  1991
University of Utah  1992
Utah State University  1992
Dynamics Seminar Royal Institute of Technology, Stockholm 1993
Analysis Seminar Royal Institute of Technology, Stockholm 1993
Analysis Seminar University of Uppsala, Sweden  1994
Physics Seminar University of Stockholm  1994
University of Tennessee, Knoxville  1994
University of Miami  1995

Physics Department University of South Carolina 1996
University of Miami (2 Talks)  1997
South Carolina State University  1999
University of Georgia  1999
University of Georgia  2001
University of Georgia  2002

Talks at Conferences

Brunswick, Maine  1984  Los Angles, California  1985
San Antonio, Texas  1987  Salt Lake City, Utah  1987
Waterloo, Ontario Canada 1987  Chapel Hill, North Carolina 1989
Corvallis, Oregon  1989  San Francisco, California  1991
Bethlehem, Pennsylvania 1992  Athens, Georgia  1992
Athens, Georgia  1994  Orlando, Florida  1996
Columbia, Missouri  1996  Memphis Tennessee  1997
Athens, Georgia  1998  Athens, Georgia  1999
Lowell, Massachusetts  2000  Wichita, Kansas  2001^{∗}
Pisa, Italy  2002  Boston, Massachusetts  2002
Baltimore, Maryland  2003  Charleston, South Carolina 2003
Columbia, Missouri  2003  Lawrenceville, New Jersey 2004
Snowbird, Utah  2004  Wuhan, China  2004^{∗}

(∗ Plenary talk.) Graduate Students

Master’s Students:

1990 Somasundaram Velummylum The Calculus of Variations and Zoll Surfaces
1991 Louis Kenneth Reintz  The Periodic Symmetry Groups of the Plane
1991 Gary Allen Cohen  Intersection Theory
1999 Aseem Raval  Mathematics Related to Spinning Tops
2001 Michael S. Venn  Analysis on Finite Gel’fand Spaces
2004 Annamaria Rusu  Reflection Groups

Ph.D. Students:

1991 Shiying Zhao^{∗} On the Boundary Behavior of Subharmonic Functions

in Nontangential Acessible Domains

(∗ Supervised jointly with Manfred Stoll.) Reviewing for Journals

Journal  Number of Manuscripts
American Journal of Mathematics.  1
American Mathematical Monthly  2
Annals of Mathematics  1
Australian Mathematical Society. Journal.  1
Contemporary Mathematics.  1
The Bulletin of the London Mathematical Society  1
Discrete and Computational Geometry  2
Illinois Journal of Mathematics  1
Manuscripta Mathematica  1
Journal of Mathematical Physics  1
Proceedings of the American Mathematical Society.  2
The Rocky Mountain Journal of Mathematics  1
Transactions of the American Mathematical Society  1
Tsukuba Journal of Mathematics  2

Reviewing of Grants

Agency  Number of Proposals
National Science Foundation  13
USC Research and Productive Scholarship Awards Program  4
Idaho Board of Education  1

Conferences Organized

Organized and ran Seventh Annual Southeast Geometry Conference which took place at the University of South Carolina May 17 and May 18 of 1996. There were about 25 participants with 13 speakers. Some of the schools represented by speakers where The Royal Institute of Technology (Stockholm), The University of Pennsylvania, The University of Rochester, Duke University, The University of Miami, The University of Georgia.

Co-organizer, with Mohammad Ghomi, of a special session, Geometry of Curves and Surfaces, at the 2001 Spring Southeastern Section Meeting of the American Mathematical Society. This special session doubled as the Twelfth Annual Southeast Geometry Conference co-organized along with Joe Fu.

On the organizing committee for Fifteenth Spring Southeast Geometry Conference held March 19–21 at the University of Georgia in Athens, GA.

Books or Monographs

1. The kinematic formula in Riemannian homogeneous spaces, Memoirs of the Amer. Math. Soc.

Volume 109 Number 509 (1993) Publications

1. Dimension inequalities on the range of a multilinear function with vector space range, Linear and Multilinear Algebra 8 (1978–1980) pp. 247–252

2. Linear maps that preserve matrices annihilated by a polynomial, Linear Algebra Appl. 30 (1980) pp. 167–176

3. The nonexistence of stable submanifolds, verifolds, and harmonic maps in sufficiently pinched simply connected Riemannian manifolds, Michigan Math. J. 32 (1985) pp. 321–334

4. (with S. W. Wei) Inequalities relating sectional curvatures of a submanifold to the size of its second fundamental form and applications to pinching theorems for submanifolds, Proc. Amer. Math. Soc. 94 (1985) pp. 699–702

5. (with S. W. Wei) Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space, Trans. Amer. Math. Soc. 294 (1986) pp. 319– 331

6. On the Gehrig link problem and the isoperimetric inequality of Bombieri and Simon, J. Analyse Math. 47 (1986) pp. 243–253

7. (with J. C. Bezdek, R.J. Hathaway, C. A. Wilson)Coordinate descent and clustering, Control and Cybernet 15 (1986) pp. 195–204

8. Classical integral geometry in Riemannian homogeneous spaces, Contemporary Mathematics 63 (1987) pp. 179–203

9. (with J. C. Bezdek, R.J. Hathaway, C. A. Wilson, M. P. Windham) Local convergence analysis of a grouped variable version of coordinate decent, J. Optim. Theory Appl. 54 (1987) pp. 471– 477

10. A note on Roe’s characterization of the sine function, Proc. Amer. Math. Soc. 105 (1989) pp. 658–663

11. (with R. A. DeVore, C. Micchelli) Optimal nonlinear approximation, Manuscripta Math. 63 (1989) pp. 469–478

12. (with P. R. Goodey) Processes of flats induced by higher dimensional processes, Advances in Math. 80 (1990) pp. 92–109

13. (with A. R. Schep) Norms of positive operators on L^{p}-spaces, Proc. Amer. Math. Soc. 109 (1990) pp. 135–146 14. (with P. R. Goodey)Processes of flats induced by higher-dimensional flats II, Contemporary

Mathematics 113 (1990) pp. 111–119

15. (with M. A. Sutton, J. R. Dickerson, S. R. McNeill) A constrained, least squares approach for hybrid stress analysis of elastic bodies, Engineering Analysis with Boundary Elements 8 (1991)

16. (with M. L. Reese) Characterization of eigenfunctions by boundedness conditions, Canadian Mathematical Bulletin 35 (1992) pp. 204–213

17. (with A. Treibergs) A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mountain Jour. Math., 25 (1995), 635–684.

18. (with P. R. Goodey and M. Reeder) Processes of Flats Induced by Higher Dimensional Process III, Geom. Dedicata, 61 pp. 257–269, (1996)

19. (with L. Andersson and M. Dahl) Boundary and Lens Rigidity of Lorentzian Surfaces, Trans. Amer. Math. Soc., 348 pp. 2307–2329 (1996)

20. (with Paul Sisson) Capturing the Origin with Random Points: Generalizations of a Putnam Problem, College Math. Jour., 27 (1996) pp. 186–192

21. The Sharp Sobolev Inequality and the Banchoff-Pohl Inequality on Surfaces, Proc. of the Amer. Math. Soc. 126 (1998) pp. 2779–2787.

22. (with Lars Andersson and Greg Galloway) A Strong Maximum Principle for Weak Solutions of Quasi-Linear Elliptic Equations with Applications to Lorentzian and Riemannian Geometry, Comm. Pure Appl. Math., 51 (1998) pp. 581–624.

23. (with Lars Andersson and Greg Galloway) The Cosmological Time Function, Classical and Quantum Gravity 15(1998) pp. 309–322

24. (With Lars Andersson) Comparison and rigidity theorems in semi-Riemannian geometry, Comm. in Analysis and Geometry 6 (1998) pp. 819–877.

25. (With S. J. Dilworth and J. W. Roberts) Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls, Advances in Mathematics 148 (1999) pp. 1–43.

26. Blaschke’s Rolling Theorem for Manifolds with Boundary, Manuscripta Mathematica, 99 (1999) pp. 471–483.

27. (with Piotr T. Chrusciel, Erwann Delay, and Greg Galloway) Regularity of Horizons and the Area Theorem, Annales Henri Poincaré 2 (2001) pp. 109–178.

28. (with Emil Cornea and Per-Gunnar Martinsson) Solutions Near Singular Points to the Eikonal and Related First Order Non-linear Partial Differential Equations in Two Independent Variables, Differential and Integral Equations, 14 (2001) pp. 1441–1468.

29. (With S. J. Dilworth and James W. Roberts) On the Size of Approximately Convex Sets in Normed Spaces, Studia Mathematica 140 (2000) pp. 213–241.

30. (with S. J. Dilworth and James W. Roberts) Extremal Approximately Convex Functions and the Best Constants in a Theorem of Hyers and Ulam, Advances in Mathematics 172 (2002) pp. 1–14.

31. (with P. T. Chrusciel, J. Fu, and G. Galloway) On Fine Differentiability Properties of Horizons and Applications to Riemannian Geometry, Journal of Geometry and Physics 41 (2002) pp. 1– 12.

32. (with S. J. Dilworth and J. W. Roberts) A General Theory of Almost Convex Functions, To appear in in Trans. Amer. Math. Soc. (40 pages)

33. (With G. Károlyi and L. A. Szákely) Towards a Katona Type Proof for the 2-Intersecting Erdös-Ko-Rado Theorem, Electronic Journal of Combinatorics 8 (2001) R31 (8 pages)

34. (with A. Abrams, J. Cantarella, J. Fu, and M. Ghomi) Circles Minimize most Knot Energies, Topology 42 (2003) pp. 381–394.

35. Constructing Complete Projectively Flat Connections, To Appear in Rocky Mountain Journal of Mathematics. (11 pages)

36. (with Wenxiong Chen, Erwin Lutwak, Deane Yang, and Gaoyong Zhang) A Generalized Affine Isoperimetric Inequality, To appear in Journal of Geometric Analysis. (17 pages)

37. Convex Bodies of Constant Width and Constant Brightness, Submitted.

38. (with Daniel Hug) Smooth Convex Bodies with Proportional Projection Functions, Submitted.