Dr. Steven G. Krantz |
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Washington University in St. Louis College of Arts & Sciences Mathematics ProfessorAppointed: 1986 Washington University in St. Louis College of Arts & Sciences Mathematics ChairmanAppointed: 1999 |  |
Mailing AddressDepartment of Mathematics Washington University Campus Box 1146 St. Louis, Missouri 63130-4899United States | Contact Information |
Qualifications
Ph.D., Princeton University, Mathematics, 1974.
B.A., University of California, Santa Cruz, Mathematics, 1971.
Expertise and Research Interests
The research at this site will be on topics in analysis, probability, and finite mathematics. The program will run for ten weeks. Participants will be expected to provide regular reports on their work at informal meetings which will include all of the participants and appropriate faculty members.
The first concerns the study of Hodge theory for the de Rham complex using different inner products that the usual quadratic one. The work should provide new insights into the regularity theory for elliptic problems as well as insight into the structure of the harmonic space. The Hodge theory for the de Rham complex has important geometric interpretations. A second project involves the development of special function spaces adapted to specific boundary value problems on specific domains. Along with recent advances in operator theory, the idea of designing functions for particular applications should have far-reaching applications. In another direction, work will be don on basic elements of harmonic analysis and function theory on domains in complex Euclidean space. This includes the theory of hardy spaces, functions of bounded mean oscillation, the corona problem, and Lipschitz spaces. An important theme in the work focuses on the geometry of the boundary of the domain and how it determines the function theory on the interior. In particular questions of boundary uniqueness and rigidity for holomorphic functions and mappings will be taken up Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. When the equations are defined over domains in several complex variables, as they are in this project, the scope of the work increases considerably to take into account the large body of existing work available in the study of holomorphic functions of seve ral complex variables and the corresponding domains.
They include harmonic analysis and the phi-transform, complex analysis and differential equations, population genetics, image processing and combinatorial number theory. Seven faculty will be involved at various stages of the program. Regularly scheduled meetings together with informal discussions are used to develop interactions among the students and faculty. Although not required, students will be encouraged to prepare their results for presentation and publication. As Washington University is one of the preeminent research institutions in the country, the studentshave an exceptional opportunity to meet many of the visitors who pass through the campus during the summer.
In this project the principal investigator will continue working on three problems involving real and complex analysis on domains. The first problem concerns an examination of function spaces designed for boundary value problems for partial differential operators defined on domains in real Euclidean space. The second problem concerns the study of automorphism groups of domains in complex n-spaceand how their group- theoretic properties are determined by the geometry of the boundary of the domain. The third problem concerns various subproblems in the function theory of several complex variables; to wit, boundary uniqueness theorems for holomorphic mappings, biholomorphic exhaustions of domains by other domains, and the boundary behavior of holomorphic functions. Complex analysis is one of the oldest branches of modern mathematics. The principal objects of study in complex analysis are functions with special properties called holomorphic or analytic functions. Holomorphic functions are capable of expressing relationships and equivalences between complex quantities in two or more space dimensions. In this project the principal investigator will study the behavior of holomorphic functions at the boundaries of their domains of definition. The boundary behavior of holomorphic functions is very often the most interesting (and difficult|) aspect of the theory.
They will work under the direction of several faculty advisors all of whom have strong research programs under way. This program continues a successful program of undergraduate research at Washington University. It will be augmented during the next two years by a series of visits by leading mathematical researchers. The environment for this activity is particularly fecund at Washington University which has produced many of the top finalists and leading teams in the Putnam examinations over the past ten years. Included among the many topics which will be available for student projects is the study of the phi-transform. This concept is a very recent variation of the wavelet transform allowing one to decompose general functions into sums of simple terms. It has interesting computational characteristics which allow for simple but important questions to be undertaken early without need for extensive theoretical training. Work will also be done on the boundary behavior of conformal mappings and harmonic measure. This work will fit well into the development of students who have recently completed an undergraduate complex variables course. Another area of interest includes problems arising in the theory of several complex variables. Here, researchers can construct numerical models for automorphism groups and perform calculations of invariants associated with boundary differential invariants. The work only requires basic algebra and an understanding of symbol manipulation programs. In the process, students will develop concrete understanding of geometric structures and the analysis related to them. A more applied program developing mathematical models of molecular evolution will also be available. Students will consider questions concerning the distribution of nucleotides in the coding regions of individuals chosen from a random-mating population. Connections with algorithms for estimating phylogenies of creatures whose traits are known in the present will be investigated.
The first concerns the study of Hodge theory for the de Rham complex using different inner products that the usual quadratic one. The work should provide new insights into the regularity theory for elliptic problems as well as insight into the structure of the harmonic space. The Hodge theory for the de Rham complex has important geometric interpretations. A second project involves the development of special function spaces adapted to specific boundary value problems on specific domains. Along with recent advances in operator theory, the idea of designing functions for particular applications should have far-reaching applications. In another direction, work will be don on basic elements of harmonic analysis and function theory on domains in complex Euclidean space. This includes the theory of hardy spaces, functions of bounded mean oscillation, the corona problem, and Lipschitz spaces. An important theme in the work focuses on the geometry of the boundary of the domain and how it determines the function theory on the interior. In particular questions of boundary uniqueness and rigidity for holomorphic functions and mappings will be taken up Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. When the equations are defined over domains in several complex variables, as they are in this project, the scope of the work increases considerably to take into account the large body of existing work available in the study of holomorphic functions of seve ral complex variables and the corresponding domains.
They include harmonic analysis and the phi-transform, complex analysis and differential equations, population genetics, image processing and combinatorial number theory. Seven faculty will be involved at various stages of the program. Regularly scheduled meetings together with informal discussions are used to develop interactions among the students and faculty. Although not required, students will be encouraged to prepare their results for presentation and publication. As Washington University is one of the preeminent research institutions in the country, the studentshave an exceptional opportunity to meet many of the visitors who pass through the campus during the summer.
In this project the principal investigator will continue working on three problems involving real and complex analysis on domains. The first problem concerns an examination of function spaces designed for boundary value problems for partial differential operators defined on domains in real Euclidean space. The second problem concerns the study of automorphism groups of domains in complex n-spaceand how their group- theoretic properties are determined by the geometry of the boundary of the domain. The third problem concerns various subproblems in the function theory of several complex variables; to wit, boundary uniqueness theorems for holomorphic mappings, biholomorphic exhaustions of domains by other domains, and the boundary behavior of holomorphic functions. Complex analysis is one of the oldest branches of modern mathematics. The principal objects of study in complex analysis are functions with special properties called holomorphic or analytic functions. Holomorphic functions are capable of expressing relationships and equivalences between complex quantities in two or more space dimensions. In this project the principal investigator will study the behavior of holomorphic functions at the boundaries of their domains of definition. The boundary behavior of holomorphic functions is very often the most interesting (and difficult|) aspect of the theory.
They will work under the direction of several faculty advisors all of whom have strong research programs under way. This program continues a successful program of undergraduate research at Washington University. It will be augmented during the next two years by a series of visits by leading mathematical researchers. The environment for this activity is particularly fecund at Washington University which has produced many of the top finalists and leading teams in the Putnam examinations over the past ten years. Included among the many topics which will be available for student projects is the study of the phi-transform. This concept is a very recent variation of the wavelet transform allowing one to decompose general functions into sums of simple terms. It has interesting computational characteristics which allow for simple but important questions to be undertaken early without need for extensive theoretical training. Work will also be done on the boundary behavior of conformal mappings and harmonic measure. This work will fit well into the development of students who have recently completed an undergraduate complex variables course. Another area of interest includes problems arising in the theory of several complex variables. Here, researchers can construct numerical models for automorphism groups and perform calculations of invariants associated with boundary differential invariants. The work only requires basic algebra and an understanding of symbol manipulation programs. In the process, students will develop concrete understanding of geometric structures and the analysis related to them. A more applied program developing mathematical models of molecular evolution will also be available. Students will consider questions concerning the distribution of nucleotides in the coding regions of individuals chosen from a random-mating population. Connections with algorithms for estimating phylogenies of creatures whose traits are known in the present will be investigated.
Other Expertise
Krantz is presently engaged in research on applications of techniques of differential geometry and mathematical analysis to aesthetic rhinoplasty, liposuction, and other plastic surgery issues. This work is funded by the National Science Foundation, andthere are other grants pending. Krantz's collaborators are plastic surgeon Michael Cedars and electrical engineer Thomas Lu.
Keywords
COS Keywords:
Analysis and Functional Analysis, Differential Geometry, Finite Mathematics, Harmonic Analysis, Mathematics, Partial Differential Equations, Probability.Additional Terms:
Classical Analysis, Differential Geometry, Function Theory of One Complex Variable, Function Theory of Several Complex Variables, Harmonic Analysis, Lie Groups, Mathematics, Partial Differential Equations.Languages
(Reading, Writing, Speaking)
English: (Fluent, Fluent, Fluent)
German: (Functional, Basic, Functional)
French: (Basic, Basic, Basic)
Memberships
American Mathematical Society
Mathematical Association of America
Textbook and Academic Authors Association
Honors and Awards
Chauvenet Prize,
Mathematical Association of America,
mathematical writing
Beckenbach Book Award,
Mathematical Association of America,
mathematical writing
Outstanding Academic Book Award,
Current Review for Academic Libraries,
problem solving
Distinguished Teaching Award,
UCLA Alumni Foundation,
teaching
Previous Positions
1984-1986, Professor,
Pennsylvania State University,
College of Science,
Mathematics
1980-1984, Professor, Associate,
Pennsylvania State University,
College of Science,
Mathematics
1974-1980, Professor, Assistant,
University of California, Los Angeles,
School of Arts and Sciences,
Mathematics
Funding Received
- National Science Foundation (NSF): GIG: Research and Training in Computational Harmonic Analysis, $150,000, Sep 31, 1996 to Aug 2000.
- National Science Foundation (NSF): Research Group in Analysis, $284,499, Aug 1, 1996 to Jul 31, 2000.
- National Science Foundation (NSF), 9424206, Mathematical Sciencs REU Site: Problems in Analysis, Probability, and Finite Mathematics, $60,000 (Estimated), April 1, 1995-March 31, 1997 (Estimated)
- 9400772, Mathematical Sciences: Harmonic Analysis on Domains, $28,500 (Estimated), August 1, 1994-June 30, 1995 (Estimated)
- 9300553, Mathematical Sciences: REU: Problems in Analysis, Probability, and Finite Mathematics, $60,000 (Estimated), April 1, 1993-September 30, 1995 (Estimated)
- 9101104, Mathematical Sciences: Real and Complex Analysis on Domains, $156,000 (Estimated), July 15, 1991-December 31, 1994 (Estimated)
- 9106223, Mathematical Sciences: Problems in Analysis, Probability, and Finite Mathematics, $60,000 (Estimated), April 1, 1991-September 30, 1993 (Estimated)
Publications
- Steven G. Krantz and Harold R. Parks, The Implicit Function Theorem, 2002
- Robert E. Greene and Steven G. Krantz, Function Theory of One Complex Variable, 2002
- Steven G. Krantz, Handbook of Logic and Proof Techniques for Computer Science, 2002
- Steven G. Krantz and Harold R. Parks, A Primer of Real Analytic Functions, 2002
- Krantz, SG, Fatou theorems old and new: an overview of the boundary behavior of holomorphic functions, Proceedings of a Conference on Several Complex Variables, 1999
- Krantz SG, Kim KT, A crash course in the function theory of several complex variables, Contemporary Math, 1998
- Krantz, S.G., Fontana, L. & Peloso, M., 'Hodge theory in the Sobolev topology for the de Rham complex', Memoirs of the AMS, 131, viii + 100, 1998
- Krantz SG, Isaev A, Finitely smooth Reinhardt domains with non-compact automorphism group, Illinois Math. J., 41, 412-420, 1997
- Krantz SG, Li SY, Rochberg R, Analysis of some function spaces associationd to Hankel operators, Ill. J. Math, 41, 398-411, 1997
- Krantz, S.G., Li, Song-Y., Rochberg, R., 'The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces', Integral Equations and Operator Theory, 28, 196-213, 1997
- Krantz SG, Isaev A, On boundary orbit accumulation set for a domain with non-compact automorphism group, Michigan Math. Journal, 43, 611-617, 1996
- Krantz SG, FuS, Isaev A, Examples of domains with non-compact automorphism groups, Math. Research Letters, 3, 609-617, 1996
- Krantz SG, Fu S, Isaev A, Finite type conditions on Reinhardt domains, Complex Variables, 31, 609-617, 1996
- Krantz SG, Li, SY, Factorization of functions in subspaces of L1 and applications to the corona problem, Indiana Journal, 45, 83-102, 1996
- Krantz SG, Li SY, Rochberg R, The effect of boundary regularity on the singular numbers of Friedrichs operators on Bergman spaces, Michigan Math. Journal, 43, 337-348, 1996
- Krantz SG, Fu S, Isaev A, Reinhardt domains with non-compact automorphism groups, Math. Research Letters, 3, 109-122, 1996
- Krantz, S.G. & Li, Song-Ying, 'Explicit Solutions for the corona problem with Lopschitz data in the polydisc', Pacific Journal of Mathematics, 174, 443-458, 1996
- Krantz, S.G. & Yu, J., 'On the Bergman invariant and curvatures of the Bergman metric', Illinois Journal of Math, 40, 226-244, 1996
- Krantz SG, Fontana L, Peloso M, Hodge theory in the Sobolev topology, Electronic Research Announcements of the American Mathematical Society, 1, 103-107, 1995
- Krantz, S.G. & Huang, X., 'On a problem of Moser', Duke J. Math., Duke Jour. Math, 78, 213-228, 1995
- Krantz, S.G., Huang, X., Ma. D. & Pan, Y., 'A Hopf lemma for holomorphic functions and applications', Complex Variables, 26, 273-276, 1995
- Krantz, S.G. & Greene, R.E., 'Stability of the Caratheodory and Kobayashi metrics and applications to biholomorphic mappings', Proceedings of Symposia in Pure Mathematics', 41, 77-93, 1994
- Krantz, S.G. & Burns, D., 'Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary', Journal of the A.M.S., 7, 661-676, 1994
- Krantz, S.G., 'A unique continuation problem for holomorphic mappings', Comm. P.D.E., 18, 241-263, 1993
- Krantz, S.G. & Greene, R.E., 'Techniques for Studying the Automorphism Groups of Weakly Pseudoconvex Domains, Proceedings of the Special Year at the Mittag-Leffler Institute', Annals of Mathematics Studies, 1992
- Krantz, S.G., 'On the area inside a circle', Missouri Journal of Mathmatics, 1992
- Krantz, S.G., 'Proper holomorphic mappings and the Cowen-Douglas class', Proc. Amer. Math. Society, 1992
- Krantz, S.G., 'Invariant metrics and the boundary behavior of the holomorphic functions on domains in Cn', Journal of Geometric Analysis, 1, 71-98, 1991
- Krantz, S.G. & Paulsen, W., 'Eigenvalue asymptotics for the N-beam Euler-Bernoulli coupled beam equation with dissipative joints', Journal of Symbolic Computation, 11, 369-418, 1991
- Krantz, S.G. & Parks, H.R., 'On the vector sum of two convex sets', Canadian Journal of Mathematics, 1991
- Krantz SG, Gray RH, Damewood MD, Wallach EE, Time trends in risk factors and clinical outcome of ectopic pregnancy, Fertility and Sterility, 54(1), 42-6, July 1990
- Krantz, S.G., 'On a theorem of Stein I', Trans. AMS, 320, 625-642, 1990
- Krantz, S.G., 'Mathematical Anacdotes', Mathematical Intelligencer, 12, 32-38, 1990
- Krantz, S.G., Chen, G., Wayne, C.E. & West, H.H., 'Analysis, designs and behaviors of dissipative joints for coupled beams', SIAM Journal on Applied Mathematics, 49, 1665-1693, 1989
- Krantz, S.G., Chen, G., Ma, D.,Wayne, C.E. & West, H.H., 'The Euler-Bernoulli beam equation with boundary energy dissipation', Operator Methods for Optimal Control Problems', 67-96, 1988
- Krantz, S.G., Chen, G., Russell, D., Wayne, C.E., West, H. & Zhou, J., 'Modelling, analysis and testing of dissipative beam joints-experiments and data smoothing', Math. Comp. Modelling, 11, 1011-1016, 1988
- Krantz, S.G. & Greene, R.E., 'Characterizations of certain weakly pseudo-convex domains with non-compact automorphism groups', Complex Analysis Seminar, 1268, 121-157, 1987
- Krantz, S.G. & Bell, S.R., 'Smoothness to the boundary of conformal mappings', Rocky Mountain J. Math., 17, 23-40, 1987
- Krantz, S.G., 'Fatou theorems on domains in Cn', Bulletin of the American Mathematical Society, 16, 93-96, 1987
- Krantz, S.G., 'Recent progress and future directions in several complex variables', Complex Analysis Seminar, 1268, 1-23, 1987
- Krantz, S.G. & Greene, R.E., Biholomorphic self-maps of domains', Complex Analysis II, 1276, 136-207, 1987
- Krantz, S.G. & Parsons, T.D., 'Antisocial subcoverings of self-centered covers', American Mathematical Monthly, 93, 45-48, 1986
- Krantz, S.G., 'Integral formulas in complex analysis', Chapter in The Beijing Lectures in Harmonic Analysis, 112, 185-240, 1986
- Krantz, S.G., 'Functions of One Complex Variables and Analytic Spaces', The Encyclopedia of Physical Science and Technology, 5, 698-722, 1986
- Krantz, S.G. & Greene, R.E., 'Normal families and the semicontinuity of isometry and automorphism groups', Math. Zeitschrift, 190, 455-467, 1985
- Krantz, S.G., Duncan, J. & Parks, H.R., 'Non-linear conditions for differentiability of functions', Journal d'Analyse Mathematique, 45, 46-68, 1985
- Krantz, S.G., Cima, J.A. & Suffridge, T., 'A reflection principle for proper holomorphic mappings of strongly pseudoconvex domains and applications', Math. Zeitschrift, 186, 1-8, 1984
- Krantz, S.G. & Cima, J. A., 'The Lindelof principle and normal functions of several complex variables', Duke Mathematical Journal, 50, 303-328, 1983
- Krantz, S.G., 'Lipschitz spaces, smoothness of functions, and approximation theory', Expositiones Math, 3, 193-260, 1983
- Krantz SG, Lipschitz spaces on stratified groups, Transactions of the American Mathematical Society, 269, 39-66, 1982
- Krantz, S.G., 'Fractional integration on Hardy spaces', Studia Math., 73, 87-94, 1982
- Krantz, S.G. & Greene, R.E., 'Deformations of complex structure, estimates for the []-equation, and stability of the Bergman kernel', Advances in Math., 43, 1-86, 1982
- Krantz, S.G. & Greene, R.E., 'The automorphism groups of strongly pseudoconvex domains', Math. Annalen, 261, 425-446, 1982
- Krantz, S.G. & Greene, R.E., 'The stability of the Bergman kernel and the geometry of the Bergman metric', Bulletin of the American Mathematical Society, 4, 111-115, 1981
- Krantz SG, Geometric Lipschitz spaces and applications to complex function theory and nilpotent groups, Journal of Functional Analysis, 34, 456-471, 1980
- Krantz, S.G., 'Boundary Values and estiamtes for holomorphic functions of several complex variables', Duke Mathematical Journal, 47, 81-98, 1980
- Krantz, S.G., 'Holomorphic functions of bounded mean oscillation and mapping properties of the Szego projection', Duke Mathematical Journal, 47, 743-761, 1980
- Krantz SG, Jewell NP, Toepllitz operators and related function algegras on certain pseudo-convex domains, Trans. Mer. Math. Soc., 252, 297-312, 1979
- Krantz SG, Smoothness of harmonic and holomorphic functions, Proc. Symp. Pure Math, 35, 63-67, 1979
- Krantz, S.G., 'Finite type conditions and elliptic boundary value problems', Journal of Differential Equations, 34, 239-260, 1979
- Krantz, S.G. & Fornaess J.E., 'Continuously varying peaking functions', Pacific Journal of Mathematics, 83, 341-347, 1979
- Krantz, S.G., 'Analysis on the Heisenberg group and estimates for functions in Hardy classes of several complex variables', Math. Annalen, 244, 243-262, 1979
- Krantz, S.G., 'Estimates for integral kernels of mixed type, fractional integration operators and optimal estimates for the []-operator', Manuscripta Mathematica, 30, 21-52, 1979
- Krantz, S.G. & Greene, R.E., 'Stability properties of the Bergman kernel and curvature properties of bounded domains', Recent Developments in Several Complex Variables, 179-198, 1979
- Krantz, S.G., 'Intrinsic Lipschitz classes on manifolds with applications to complex functions theory and estimates for the [] and [] equations', Manuscripta Mathematica, 24, 351-378, 1978
- Krantz, S.G., 'Optimal Lipschitz and Lp regularity for the equation [] = f on strongly pseudo-convex domains', Math. Annalen, 219, 233-260, 1976
- Krantz, S.G., 'Structure and interpolation theorems for certain Lipschitz spaces and estimates for the []-equation', Duke Mathematical Journal, 43, 417-439, 1976
- Krantz, S.G. Pending, 'Survey of some recent ideas concerning automorphism groups of domains', Proceedings of a Conference in Honor of Pierre Dolbeault
- Krantz SG, Isaev AV, Characterizations of Reinhardt domains by their automorphism groups, Journal of the Korean Math Society
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