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H. Blaine Lawson

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H. Blaine Lawson, Jr.
H. Blaine Lawson in Berkeley, 1972
Born (1942-01-04) January 4, 1942 (age 84)[1]
Norristown, Pennsylvania[2]
CitizenshipUnited States
Scientific career
Robert Osserman

Herbert Blaine Lawson Jr. is an American mathematician known for his work in minimal surfaces, calibrated geometry, algebraic cycles, foliations, several complex variables, Riemannian geometry, and partial differential equations. He is currently a Distinguished Professor of Mathematics at Stony Brook University.[3]

Lawson completed his undergraduate studies at Brown University in 1964, earning degrees in both applied mathematics and Russian literature.[4] He received his PhD from Stanford University in 1969, where he worked under the supervision of Robert Osserman.[5]

After completing his doctorate, Lawson joined the faculty at the University of California, Berkeley. He rose to the rank of full professor before moving to Stony Brook University in 1978, where he has remained.[3]

Lawson has held extended visiting positions at several international research institutes. These include the Institute for Advanced Study in Princeton,[6] the Institut des Hautes Études Scientifiques (IHÉS) near Paris,[7] the Instituto de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro,[8] the Research Institute for Mathematical Sciences (RIMS) at Kyoto University,[9] and the Tata Institute of Fundamental Research (TIFR) School of Mathematics in Mumbai.[10]

Research

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Minimal surfaces

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In 1970 Lawson constructed minimal embeddings of every compact surface into the Euclidean 3-sphere (with the exception of the real projective plane, which cannot be so embedded).[11] This gave, for example, embedded 3-dimensional cones in Euclidean 4-space of every possible topological type. This also led to interesting periodic surfaces of constant mean curvature in eEuclidean 3-space. His work in this area continued for years.[12][13][14] One nice result was with Jim Simons[15] where they showed how to use minimal integral currents for basic riemannnian geometry, and they proved that a stable minimal current (one whose second variation of mass is ≥ 0) in complex projective space, is a positive algebraic cycle.

Foliations

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Lawson found codimension-one foliations of higher dimensional spheres,[16] which answered a long-term question and engendered much subsequent work.

Compact Manifolds of Negative Curvature

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Together with S.-T. Yau[17] Lawson found basic theorems about these manifolds, such as the Splitting Theorem which says that if the fundamental group splits as a product of groups, then the manifold essentially splits as a direct metric product of manifolds. These results were independently found by Detlef Gromoll and Joseph A. Wolf[18]

Boundaries of Complex Analytic Varieties

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Together with F. Reese Harvey[19][20] Lawson characterized the compact oriented submanifolds of complex Euclidean space which bound complex analytic varieties. These submanifolds could have singularities, and the result has analogues in complex projective space minus a linear subspace of higher codimension. This was a vast geometric generalization of a classical result of S. Bochner.[21]

Calibrated Geometries

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In a 1982 Acta Mathematica paper of F. Reese Harvey and Blaine Lawson[22] found large classes of submanifolds (even with singularities) that are always homologically volume minimizing. This means that if one takes a compact piece M with boundary, then M has volume less than or equal to the volume of any M' such that M - M' bounds something of higher dimension. This paper was engendered by work of Herbert Federer.[23] It applied to submanifolds of certain Euclidean spaces, but also to more general manifolds with special geometries. It inspired Robert Bryant to discover G(2) and Spin(7) manifolds,[24] answering a long-standing question. It turned out the calibrated geometries discovered by Harvey-Lawson play a role in M-theory in modern particle physics. As a result there has been an enormous amount of work in this area.

Manifolds of Positive Scalar Curvature

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In a series of three papers[25][26][27]Mikhael Gromov and Lawson used the Dirac operator and other techniques to prove global results about manifolds with positive scalar curvature ? > 0. The first work in this area was done by Rick Schoen and S.-T. Yau in this area.[28] Among many things Gromov and Lawson showed that for spin manifolds, the existence of a metric with ? > 0 depends only on the spin cobordism class of the manifold. They conjectured that a necessary and sufficient condition for ? > 0 was a KO-Theory analogue of the Aˆ-invariant. This conjecture was proved by Stephan Stoltz.[29]

Algebraic Cycles

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In his 1989 Annals of Mathematics paper "Algebraic Cycles and Homotopy Theory",[30] Lawson proved a theorem which showed that the limit of codimension-q algebraic cycles in complex projective n-space Pn is a finite product of spaces which classify integer cohomology in degrees 2, 4, ..., 2q. This basic theorem has analogs on any projective variety, and the homotopy groups of the resulting space gives a new homology theory in algebraic geometry. With Marie-Louise Michelsohn[31] they showed that the inclusion of the linear cycles in projective space leads to a map from K-theory to cohomology which is the total Chern class. Together with Eric Friedlander, a morphic cohomology was established for algebraic varieties, based on algebraic maps into cycles spaces on Pn, and this cohomology theory was shown to be dual the homology theory mentioned above.[32][33]

Lawson and Friedlander also proved a Moving Lemma for families of algebraic cycles.[34]

This cycle theory had many interesting applications in homotopy theory, which was worked out with Michelsohn, Paulo Lima-Filho, Charles Boyer, and Ben Mann.[35][36][37][38][39]

Spin Geometry

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Lawson and Michelsohn wrote a book,[40] published by Princeton Press, which presented the deep index theorems proved by Atiyah and Singer. The book gave the fundamentals of Spin manifolds, K-theory and KO-theory, Clifford algebras and their relation to Bott Periodicity, the construction of Atiyah-Singer-Dirac operators, detailed proofs of various index theorems, and many applications were given. This text has been used worldwide for many years.

Singular Connections and Characteristic Currents

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F. Reese Harvey and Lawson considered connections on vector bundles which were not smooth, and so the characteristic forms became singular currents.[41] This led to a long sequence of interesting results.[42] [43] [44] [45]

Differential Characters

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In[46] [47] Lawson, Harvey and Zweck established a Poincaré-Pontryagin Duality for the differential characters of Cheeger and Simons.[48] [49] They also gave a deRham-Federer theory of differential characters, that is, they gave various formulations of the theory using currents and forms. This was generalized to something called spark complexes[50] which gives rise to differential characters. It gave a theory which established the equivalence of many, quite different spark complexes. This axiomatic approach did not look at a product structure, and so it allowed the deep product of Jeff Cheeger[51] to be carried over to quite different contexts.

Projective Hulls and the Projective Gelfand Transformation

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These results[52] suggested a complex projective analogue of a theorem of John Wermer,[53] which was proved in a number of cases.

Potential Theory on Calibrated Manifolds

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Harvey and Lawson discovered that while calibrated manifolds do not have analogues of the holomorphic, or pluriharmonic, functions that exist in the Kaehler case, they always have plurisubharmonic functions and, in fact, each manifold has a potential theory (giving rise to these functions) which is, in a sense, dual to the calibrated structure.[54][55] The plurisubharmonics can be maximized on a domain, subject to boundary constraints, to give solutions to analogues of the Monge-Ampère Equation.

The Dirichlet problem on Riemannian manifolds

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Harvey and Lawson eventually realized that this situation on calibrated manifolds has a vast generalization leading to potential theories associated to quite general fully nonlinear differential equations.[56][57][58][59][60] This led to the establishment of viscosity solutions to quite general differential equations on manifolds, and to the discovery of some new geometric differential equations. For example, this gave a new Lagrangian Monge-Ampère operator on Symplectic manifolds with a Gromov metric.[61] The authors were able to solve the inhomogeneous Dirichlet Problem for inhomogeneous equations,[62][63] they were able to solve the complex Monge-Ampère equation on almost complex manifolds,[64] they developed a theory of tangents to subsolutions,[65][66] and much more.

Awards and honors

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He was a 1975 recipient of the American Mathematical Society's Leroy P. Steele Prize for Exposition, and was elected to the National Academy of Sciences in 1995. He is a former recipient of both the Sloan Fellowship and the Guggenheim Fellowship, and has delivered two invited addresses at International Congresses of Mathematicians, one on geometry, and one on topology. He has served as Vice President of the American Mathematical Society, and is a foreign member of the Brazilian Academy of Sciences. For 2026 he was awarded the Leroy P. Steele Prize for Lifetime Achievement.[67]

In 2012 he became a fellow of the American Mathematical Society.[68] He was elected to the American Academy of Arts and Sciences in 2013.[69]

Major publications

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Books

  • Lawson, H. Blaine Jr. (1980). Lectures on minimal submanifolds. Vol. I. Mathematics Lecture Series. Vol. 9 (Second edition of 1977 original ed.). Wilmington, DE: Publish or Perish, Inc. ISBN 0-914098-18-7. MR 0576752. Zbl 0434.53006.
  • Lawson, H. Blaine Jr. (1974). Minimal varieties in real and complex geometry. Séminaire de mathématiques supérieures. Vol. 57. Montréal: Les Presses de l'Université de Montréal. ISBN 0840502486. Zbl 0328.53001.

See also

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References

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  1. ↑ Date information sourced from Library of Congress Authorities data, via corresponding Library of Congress Linked Data Service: linked authority record n80139222.
  2. ↑ "Lawson, Herbert Blaine". American Men and Women of Science. Vol. 4 (21st ed.). Bowker. 2009. ISBN 978-0-7876-6527-2.
  3. 1 2 "H. Blaine Lawson, Jr. - Stony Brook Mathematics". Stony Brook University. Retrieved 2026-06-08.
  4. ↑ Wen, Melissa (2026-02-27). "SBU's H. Blaine Lawson wins lifetime achievement award from math society". TBR News Media. Retrieved 2026-06-07.
  5. ↑ "H. Blaine Lawson Jr. - Mathematics Genealogy Project". North Dakota State University. Retrieved 2026-06-08.
  6. ↑ "Herbert Blaine Lawson - Institute for Advanced Study". Institute for Advanced Study. Retrieved 2026-06-14.
  7. ↑ "Institut des Hautes Études Scientifiques". Institut des Hautes Études Scientifiques. Retrieved 2026-06-14.
  8. ↑ "Visiting researchers - IMPA" (PDF). Instituto de Matemática Pura e Aplicada. Retrieved 2026-06-07.
  9. ↑ "RIMS". Research Institute for Mathematical Sciences. Retrieved 2026-06-07.
  10. ↑ "TIFR Mumbai". Tata Institute of Fundamental Research. Retrieved 2026-06-07.
  11. ↑ Lawson, H. Blaine (1970). "Complete minimal surfaces in S3". Annals of Mathematics. 92 (2): 335–374. JSTOR 1970625. MR 0270280.
  12. ↑ Lawson, H. Blaine (1970). "The unknottedness of minimal embeddings". Inventiones Mathematicae. 11: 183–187. doi:10.1007/BF01404649. MR 0287447.
  13. ↑ Hsiang, Wu-Yi; Lawson, H. Blaine (1971). "Minimal submanifolds of low cohomogeneity". Journal of Differential Geometry. 5: 1–38. MR 0298593.
  14. ↑ Lawson, H. B.; Osserman, R. (1977). "Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system". Acta Mathematica. 139 (1–2): 1–17. MR 0452745.
  15. ↑ Lawson, H. Blaine; Simons, James (1973). "On stable currents and their application to global problems in real and complex geometry". Annals of Mathematics. 98 (2): 427–450. doi:10.2307/1970913. MR 0324529.
  16. ↑ Lawson, H. Blaine (1971). "Codimension-one foliations of spheres". Annals of Mathematics. 94 (2): 494–503. doi:10.2307/1970767. JSTOR 1970767. MR 0287570.
  17. ↑ Lawson, H. B.; Yau, Shing-Tung (1972). "Compact manifolds of nonpositive curvature". J. Differential Geometry. 7: 211–228. doi:10.4310/jdg/1214430828. MR 0334083.
  18. ↑ Gromoll, Detlef; Wolf, Joseph A. (1971). "Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature". Bull. Amer. Math. Soc. 77: 545–552. doi:10.1090/S0002-9904-1971-12747-7. MR 0281122.
  19. ↑ Harvey, F. Reese; Lawson, H. Blaine (1975). "On boundaries of complex analytic varieties. I". Annals of Mathematics. 102 (2): 223–290. doi:10.2307/1971032. JSTOR 1971032. MR 0425173.
  20. ↑ Harvey, F. Reese; Lawson, H. Blaine (1977). "On boundaries of complex analytic varieties. II". Annals of Mathematics. 106 (2): 213–238. doi:10.2307/1971093. JSTOR 1971093. MR 0499285.
  21. ↑ Bochner, S. (1943). "Analytic and meromorphic continuation by means of Green's formula". Annals of Mathematics. 44 (4): 652–673. doi:10.2307/1969103. JSTOR 1969103. MR 0009206.
  22. ↑ Harvey, F. Reese; Lawson, H. Blaine (1982). "Calibrated geometries". Acta Mathematica. 148: 47–157. doi:10.1007/BF02392726. MR 0666108.
  23. ↑ Federer, H. (1969). Geometric Measure Theory. New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. MR 0257325.
  24. ↑ Bryant, Robert L. (1987). "Metrics with exceptional holonomy". Annals of Mathematics. 126 (3): 525–576. doi:10.2307/1971360. JSTOR 1971360. MR 0916718.
  25. ↑ Gromov, Mikhael; Lawson, H. Blaine (1980). "The classification of simply connected manifolds of positive scalar curvature". Annals of Mathematics. 111 (3): 423–434. doi:10.2307/1971103. JSTOR 1971103. MR 0577131.
  26. ↑ Gromov, Mikhael; Lawson, H. Blaine (1980). "Spin and scalar curvature in the presence of a fundamental group. I". Annals of Mathematics. 111 (2): 209–230. doi:10.2307/1971198. JSTOR 1971198. MR 0569070.
  27. ↑ Gromov, Mikhael; Lawson, H. Blaine (1983). "Positive scalar curvature and the Dirac operator on complete Riemannian manifolds". Publications Mathématiques de l'Institut des Hautes Études Scientifiques (58): 83–196. doi:10.1007/BF02953774. MR 0720933.
  28. ↑ Schoen, Richard; Yau, Shing-Tung (1979). "On the structure of manifolds with positive scalar curvature". Manuscripta Mathematica. 28 (1–3): 159–183. doi:10.1007/BF01647970. MR 0535700.
  29. ↑ Stolz, Stephan (1992). "Simply connected manifolds of positive scalar curvature". Annals of Mathematics. 136 (3): 511–540. doi:10.2307/2946598. JSTOR 2946598. MR 1189863.
  30. ↑ Lawson, H. Blaine (1989). "Algebraic cycles and homotopy theory". Annals of Mathematics. 129 (2): 253–291. doi:10.2307/1971448. JSTOR 1971448. MR 0986794.
  31. ↑ Lawson, H. Blaine; Michelsohn, Marie-Louise (1988). "Algebraic cycles, Bott periodicity, and the Chern characteristic map". The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987). Proceedings of Symposia in Pure Mathematics. Vol. 48. Providence, RI: American Mathematical Society. pp. 241–263. MR 0974339.
  32. ↑ Friedlander, Eric M.; Lawson, H. Blaine (1992). "A theory of algebraic cocycles". Annals of Mathematics. 136 (2): 361–428. arXiv:math/9204230. doi:10.2307/2946609. JSTOR 2946609. MR 1185123.
  33. ↑ Friedlander, Eric M.; Lawson, H. Blaine (1997). "Duality relating spaces of algebraic cocycles and cycles". Topology. 36 (2): 533–565. doi:10.1016/0040-9383(96)00011-0. MR 1415605.
  34. ↑ Friedlander, Eric M.; Lawson, H. Blaine (1998). "Moving algebraic cycles of bounded degree". Inventiones Mathematicae. 132 (1): 91–119. doi:10.1007/s002220050219. MR 1618633.
  35. ↑ Boyer, Charles P.; Lawson, H. Blaine; Lima-Filho, Paulo; Mann, Benjamin M.; Michelsohn, Marie-Louise (1993). "Algebraic cycles and infinite loop spaces". Inventiones Mathematicae. 113 (2): 373–388. MR 1228130.
  36. ↑ Lawson, H. Blaine; Lima-Filho, Paulo; Michelsohn, Marie-Louise (1996). "Algebraic cycles and equivariant cohomology theories". Proceedings of the London Mathematical Society. 73 (3): 679–720. doi:10.1112/plms/s3-73.3.679. MR 1407465.
  37. ↑ Lawson, H. Blaine; Lima-Filho, P.; Michelsohn, M.-L. (1998). "On equivariant algebraic suspension". Journal of Algebraic Geometry. 7 (4): 627–650. MR 1642736.
  38. ↑ Lawson, H. Blaine; Lima-Filho, Paulo; Michelsohn, Marie-Louise (2003). "Algebraic cycles and the classical groups. I. Real cycles". Topology. 42 (2): 467–506. doi:10.1016/S0040-9383(02)00018-6. MR 1941445.
  39. ↑ Lawson, H. Blaine; Lima-Filho, Paulo; Michelsohn, Marie-Louise (2005). "Algebraic cycles and the classical groups. II. Quaternionic cycles". Geometry & Topology. 9: 1187–1220. arXiv:math/0507451. doi:10.2140/gt.2005.9.1187. MR 2174264.
  40. ↑ Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton Mathematical Series. Vol. 38. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08542-5. MR 1031992.
  41. ↑ Harvey, F. Reese; Lawson, H. Blaine (1993). "A theory of characteristic currents associated with a singular connection". Astérisque (213): 268. MR 1230025.
  42. ↑ Harvey, Reese; Lawson, H. Blaine (1995). "Geometric residue theorems". American Journal of Mathematics. 117 (4): 829–873. doi:10.2307/2374951. JSTOR 2374951. MR 1342833.
  43. ↑ Harvey, F. Reese; Lawson, H. Blaine (2000). "Singularities and Chern-Weil theory. I. The local MacPherson formula". Asian Journal of Mathematics. 4: 71–95. doi:10.4310/AJM.2000.v4.n1.a6. MR 1802913.
  44. ↑ Harvey, F. Reese; Lawson, H. Blaine (2001). "Finite volume flows and Morse theory". Annals of Mathematics. 153 (1): 1–25. doi:10.2307/2661371. JSTOR 2661371. MR 1826410.
  45. ↑ Harvey, F. Reese; Lawson, H. Blaine (2008). "D-bar sparks". Proceedings of the London Mathematical Society. 97 (1): 1–30. doi:10.1112/plms/pdm039. MR 2434089.
  46. ↑ Harvey, Reese; Lawson, H. Blaine; Zweck, John (2003). "The de Rham-Federer theory of differential characters and character duality". American Journal of Mathematics. 125 (4): 791–847. doi:10.1353/ajm.2003.0025. MR 1993742.
  47. ↑ Harvey, Reese; Lawson, H. Blaine (2001). "Lefschetz-Pontrjagin duality for differential characters". Anais da Academia Brasileira de Ciências. 73 (2): 145–160. doi:10.1590/S0001-37652001000200001.
  48. ↑ Cheeger, Jeff; Simons, James (1973). "Differential characters and geometric invariants". Lecture Notes Summer Institute on Differential Geometry. Stanford: American Mathematical Society. doi:10.1007/BFb0075216. MR 0827262.
  49. ↑ Cheeger, Jeff; Simons, James (1985). "Differential characters and geometric invariants". Geometry and Topology (College Park, Md., 1983/84). Lecture Notes in Mathematics. Vol. 1167. Berlin: Springer. pp. 50–80. MR 0827262.
  50. ↑ Harvey, Reese; Lawson, H. Blaine (2006). "From sparks to trundles, differential characters". Communications in Analysis and Geometry. 14 (1): 1–34. arXiv:math/0306193.
  51. ↑ Cheeger, Jeff (1973). "Multiplication of differential characters". Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971 & Convegno di Geometria, INDAM, Rome, 1972). London: Academic Press. pp. 441–445. MR 0488077.
  52. ↑ Harvey, F. Reese; Lawson, H. Blaine (2006). "Projective hulls and the projective Gelfand transform". Asian Journal of Mathematics. 10 (3): 607–646. doi:10.4310/AJM.2006.v10.n3.a5. MR 2253160.
  53. ↑ Wermer, John (1958). "The hull of a curve in Cn". Annals of Mathematics. 2. 68: 550–561. doi:10.2307/1970155. MR 0100102.
  54. ↑ Harvey, F. Reese; Lawson, H. Blaine (2009). "Duality of positive currents and plurisubharmonic functions in calibrated geometry". American Journal of Mathematics. 131 (5): 1211–1239. doi:10.1353/ajm.0.0074. MR 2555839.
  55. ↑ Harvey, F. Reese; Lawson, H. Blaine (2009). "An introduction to potential theory in calibrated geometry". American Journal of Mathematics. 131 (4): 893–944. doi:10.1353/ajm.0.0067. MR 2543918.
  56. ↑ Cirant, M.; Harvey, F. R.; Lawson, H. B.; Payne, K. (2023). Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and PDEs. Annals of Mathematics Studies. Vol. 218. Princeton, NJ: Princeton University Press. doi:10.2307/jj.1895841.
  57. ↑ Harvey, F. Reese; Lawson, H. Blaine (2009). "Dirichlet duality and the nonlinear Dirichlet problem". Communications on Pure and Applied Mathematics. 62 (3): 396–443. doi:10.1002/cpa.20265. MR 2487853.
  58. ↑ Harvey, F. Reese; Lawson, H. Blaine (2011). "Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds". Journal of Differential Geometry. 88 (3): 395–482. doi:10.4310/jdg/1321366356. MR 2844439.
  59. ↑ Harvey, F. Reese; Lawson, H. Blaine (2013). "Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry". Surveys in Differential Geometry. Geometry and Topology. Surveys in Differential Geometry. Vol. 18. Somerville, MA: International Press. pp. 103–156. doi:10.4310/SDG.2013.v18.n1.a3. MR 3087918.
  60. ↑ Harvey, F. Reese; Lawson, H. Blaine (2016). "The Dirichlet problem with prescribed interior singularities". Advances in Mathematics. 303: 1319–1357. doi:10.1016/j.aim.2016.08.024. MR 3552552.
  61. ↑ Harvey, F. Reese; Lawson, H. Blaine (2018). "Lagrangian potential theory and a Lagrangian equation of Monge-Ampère type". Surveys in Differential Geometry 2017. Celebrating the 50th Anniversary of the Journal of Differential Geometry. Surveys in Differential Geometry. Vol. 22. Somerville, MA: International Press. pp. 217–257. MR 3838119.
  62. ↑ Harvey, F. Reese; Lawson, H. Blaine (2019). "The inhomogeneous Dirichlet problem for natural operators on manifolds". Annales de l'Institut Fourier. 69 (7): 3017–3064. arXiv:1805.11121. doi:10.5802/aif.3344. MR 4286829.
  63. ↑ Harvey, F. Reese; Lawson, H. Blaine; Pliś, Szymon (2016). "Smooth approximation of plurisubharmonic functions on almost complex manifolds". Mathematische Annalen. 366 (3–4): 929–940. MR 3563228.
  64. ↑ Harvey, F. Reese; Lawson, H. Blaine (2015). "Potential theory on almost complex manifolds". Annales de l'Institut Fourier. 65 (1): 171–210. MR 3449151.
  65. ↑ Harvey, F. Reese; Lawson, H. Blaine (2017). "Tangents to subsolutions existence and uniqueness, II". Journal of Geometric Analysis. 27 (3): 2190–2223. MR 3667427.
  66. ↑ Harvey, F. Reese; Lawson, H. Blaine (2018). "Tangents to subsolutions: existence and uniqueness, Part I". Annales de la Faculté des Sciences de Toulouse Mathématiques. 6. 27 (4): 777–848. arXiv:1408.5797. doi:10.5802/afst.1583. MR 3884610.
  67. ↑ Leroy P. Steele Prize for Lifetime Achievement 2026
  68. ↑ List of Fellows of the American Mathematical Society, retrieved 2013-01-27.
  69. ↑ Newly elected members Archived 2013-05-01 at the Wayback Machine, American Academy of Arts and Sciences, April 2013, retrieved 2013-04-24.
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