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. Lawson completed his undergraduate studies at Brown University in 1964, earning degrees in both applied mathematics and Russian literature. He received his PhD from Stanford University in 1969, where he worked under the supervision of Robert Osserman. 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. Lawson has held extended visiting positions at several international research institutes. These include the Institute for Advanced Study in Princeton, the Institut des Hautes Études Scientifiques (IHÉS) near Paris, the Instituto de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, and the Tata Institute of Fundamental Research (TIFR) School of Mathematics in Mumbai.
Research
Minimal surfaces 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). 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. One nice result was with Jim Simons 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 Lawson found codimension-one foliations of higher dimensional spheres, which answered a long-term question and engendered much subsequent work.
Compact Manifolds of Negative Curvature Together with S.-T. Yau 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
Boundaries of Complex Analytic Varieties Together with F. Reese Harvey 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.
Calibrated Geometries In a 1982 Acta Mathematica paper of F. Reese Harvey and Blaine Lawson 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. 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, 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 In a series of three papersMikhael 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. 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.
Algebraic Cycles In his 1989 Annals of Mathematics paper "Algebraic Cycles and Homotopy Theory", 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 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. Lawson and Friedlander also proved a Moving Lemma for families of algebraic cycles. This cycle theory had many interesting applications in homotopy theory, which was worked out with Michelsohn, Paulo Lima-Filho, Charles Boyer, and Ben Mann.
Spin Geometry Lawson and Michelsohn wrote a book, 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 F. Reese Harvey and Lawson considered connections on vector bundles which were not smooth, and so the characteristic forms became singular currents. This led to a long sequence of interesting results.
Differential Characters In
Lawson, Harvey and Zweck established a Poincaré-Pontryagin Duality for the differential characters of Cheeger and Simons.
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