Blaine Lawson

H. Blaine Lawson, Jr.

Distinguished Professor,
Department of Mathematics

Office: Math 5-109, (631) 632-8285; Fax 632-7631
Email: blaine@math.stonybrook.edu

Secretary : Lynne Barnett, Math 5-116, 632-8290

Atiyah-Singer Index Theorem Course : Syllabus,

Topology/Geometry II Course : Syllabus,

Complex Geometry Course : Syllabus,

Differential Geometry Course : Syllabus,

Differential Topology Course : Syllabus,

Algebraic Topology Course : Syllabus,

Complex Analysis Course : Syllabus, Homework-1, Homework-2, Homework-3, Homework-4, Homework-5 Homework-6 Homework-7 Midterm Homework-9 Homework-10 Homework-11 Homework-12 Final

Differential Geometry Course : Syllabus, Notes on geodesic convexity
Notes on Minimal Surfaces

Geometric Measure Theory Course : Syllabus,
Notes on sheaves of currents, Poincare Duality via forms and currents
The Compactness Theorem for Integral Currents

Course on the Viscosity Approach to Nonlinear PDE's : Syllabus,
The User's Guide,
Survey Paper
Dirichlet Duality
Notes on the Differentiation of Quasi-Convex Functions
Viscosity Theory for Universal Equations in Geometry

Geometry-Topology II : Syllabus

Riemannian Geometry II (Spring 2015) : Syllabus

Index Theorems for Elliptic Operators (Fall 2015) : Syllabus

Slides from some Recent Lectures :
Jim Simons' Work on the Theory of Minimal Varieties,
Analytic and Homological Aspects of Differential Cohomology
Reflections on the Early Mathematical Life of Bob Osserman
Restriction and Removable Singularities for Fully Nonlinear Partial Differential Equations
Potential Theory for Nonlinear Partial Differential Equations
Luminy Lecture on Tangents, Densities, and Riesz Characteristics
A Monge-Ampere Operator in Symplectic Geometry
The Special Lagrangian Potential Equation

Giulio Minervini's Thesis : Thesis

A Theory of Characteristic Currents associated with a Singular Connection (with Reese Harvey) : Book

Recent Preprints and Reprints

1. (with F. Reese Harvey) "Finite Volume Flows and Morse Theory", Annals of Math. vol. 153 no. 1 (2001), 1-25, available in PostScript or as dvi file or as a pdf file

2. (with Paulo C. Lima-Filho and Marie-Louise Michelsohn) "On Equivariant Algebraic Suspension", J. of Algebraic Geometry vol. 7 (1998), 627-650, available in PostScript or as dvi file or as pdf file

3. (with Eric M. Friedlander) "Moving Algebraic Cycles of Bounded Degree", Inventiones Math. vol. 132 (1998), 91-119, available in PostScript or as dvi file or as pdf file

4. (with Eric M. Friedlander) "Duality Relating Spaces of Algebraic Cocycles and Cycles", Topology, vol. 36, no. 2 (1997), 533-565, available in PostScript or as dvi file or as pdf file

5. (with Paulo C. Lima-Filho and Marie-Louise Michelsohn) "Algebraic Cycles and Equivariant Cohomology Theories", Proc. London Math. Soc., vol. 73, no. 3 (1996), 679-720, available in PostScript or as dvi file or as pdf file

6. (with F. Reese Harvey) "Geometric Residue Theorems", Amer. J. Math., vol. 117, no. 4 (1995), 829-873, available in PostScript or as dvi file or as pdf file

7. (with Paulo C. Lima-Filho and Marie-Louise Michelsohn) "Algebraic Cycles and the Classical Groups, Part I : Real Cycles", Topology, vol. 42 (2003), 467-506, available in PostScript or as pdf file

8. (with Paulo C. Lima-Filho and Marie-Louise Michelsohn) "Algebraic Cycles and the Classical Groups, Part II : Quarternionic Cycles", Geometry and Topology, vol. 9 (2005), 1187-1220, available in PostScript or as pdf file

9. (with F. Reese Harvey) ``Morse Theory and Stokes' Theorem'', available in PostScript or as pdf file

10. (with F. Reese Harvey) "Singularities and Chern-Weil Theory I -- The Local MacPherson Formula", Asian J. Math. vol. 4 no. 1 (2000), 71-96, available in PostScript or as dvi file or as pdf file

11. (with F. Reese Harvey) "Singularities and Chern-Weil Theory II -- Geometric Atomicity", Duke Math. J., vol. 119 (2003), 119-158, available in PostScript or as pdf file

12. (with F. Reese Harvey) "Lefschetz-Pontrjagin Duality for Differential Characters", Anais da Academia Brasileira de Ciencias, vol. 73 (2001), 145-160, available in pdf file

13. (with F. Reese Harvey and J. Zweck) "The de Rham-Federer Theory of Differential Characters and Character Duality", Amer. J. Math. vol. 125 (2003), 791-847, available in PostScript or as a pdf file

14. "Cycles and Spectra", Bull. Braz. Math.Soc. New Series vol. 34 (2003), 77-105, available in PostScript or as pdf file

15. (with F. Reese Harvey) "From Sparks to Grundles - Differential Characters", Comm. in Anal. and Geom. vol. 14 (2006), 1-34, available as pdf file

16. (with F. Reese Harvey) "Boundaries of varieties in projective manifolds", J.Geom. Anal. vol. 14 (2004), 673-695, available in PostScript or as pdf file

17. (with F. Reese Harvey) "D-bar sparks, I", (to appear in J. London Math. Soc), available in PostScript or as pdf file

18. (with F. Reese Harvey) "Projective Hulls and the Projective Gelfand Transformation", Asian J. Math. vol. 10 (2006), 279-318, available in PostScript or as pdf file

19. (with E. Friedlander) "Graph Mappings and Poincare Duality", Math. Ann. vol. 343 (2009), 431-461, available in PostScript or as pdf file

20. (with F. Reese Harvey) "Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part I", in ``The Many Facets of Geometry: a Tribute to Nigel Hitchin'' J.-P. Bourguignon, O. Garcia-Prada, Simon Salamon, Eds., Oxford Univ. Press, Oxford, 2008, available in Postscript or as pdf file

21. (with F. Reese Harvey) "Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part II", in ``Inspired by S. S. Chern'', Phillip Griffiths, Ed., Nankai Tracts in Mathematics, vol 11, World Scientfic Publishing Co., Singapore, 2006, available in Postscript or as pdf file

22. (with F. Reese Harvey) "Plurisubharmonic Functions in Calibrated Geometries", available in Postscript or as pdf file

23. (with F. Reese Harvey) "Remarks on the Alexander-Wermer Theorem for Curves", Mathematica Contemporanea vol. 31 (2006), 53-64, available in Postscript or as pdf file

24. (with F. Reese Harvey) "Boundaries of Positive Holomorphic Chains and the Relative Hodge Question", Asterisque, vol. 328 (2009), 207-222,available in Postscript or as pdf file

25. (with F. Reese Harvey and John Wermer) "The Projective Hull of Certain Curves in C^n", Asterisque 322 (2008), 241-254, available in Postscript or as pdf file

26. (with F. Reese Harvey) "A introduction to potential theory in calibrated geometry", Amer. J. Math. 131 no. 4 (2009), 893-944, available in Postscript or as pdf file

27. (with F. Reese Harvey) "Duality of positive currents and plurisubharmonic functions in calibrated geometry", Amer. J. Math. 131 no. 5 (2009), 1211-1240, available in Postscript or as pdf file

28. (with John Wermer) "On the Complement of the Projective Hull in C^n", available in Postscript or as pdf file

29. (with F. Reese Harvey) "Dirichlet duality and the nonlinear Dirichlet problem", Comm. on Pure and Applied Math. 62 (2009), 396-443, available in Postscript or as pdf file

30. (with F. Reese Harvey) "Plurisubharmonicity in a General Geometric Context", Geometry and Analysis, vol. 1 (2010), 363-401, available in Postscript or as pdf file

31. (with John Wermer) "Analytic disks and the projective hull", available in Postscript or as pdf file

32. (with F. Reese Harvey) "Dirichlet duality and the nonlinear Dirichlet problem on riemannian manifolds", J. Diff. Geom., vol. 88 (2011), 395-482, available in Postscript or as pdf file

33. (with F. Reese Harvey) "Hyperbolic polynomials and the Dirichlet problem", available in Postscript or as pdf file

34. (with F. Reese Harvey) "Split Special-Lagrangian geometry", pp. 42-90 in ``Metric and Differential Geometry'', eds. X. Dai and X. Rong, Progress in Mathematics vol. 297, Birkh\"auser Press, Basel, 2012, available in Postscript or as pdf file

35. (with F. Reese Harvey) "The Restriction Theorem for Fully Nonlinear Subequations", Ann. Institut Fourier, Vol. 64 no. 1 (2014), p. 217-265, available in Postscript or as pdf file

36. (with F. Reese Harvey) "Potential Theory on Almost Complex Manifolds", Ann. Institut Fourier, Vol. 65 no. 1 (2015), p. 171-210, available in Postscript or as pdf file

37. (with F. Reese Harvey) "p-convexity, p-plurisubharmonicity and the Levi problem", Indiana Univ. Math. J. {\bf 62} No. 1 (2014), 149-170, available in Postscript or as pdf file

38. (with F. Reese Harvey) "Geometric plurisubharmonicity and convexity - an introduction", Advances in Math. vol. 230 (2012), 2426-2458, available in Postscript or as pdf file

39. (with F. Reese Harvey) "The equivalence of viscosity and distributional subsolutions for convex subequations - a strong Bellman principle", Bulletin Brazilian Math. Soc. {\bf 44} No. 4 (2013), 621-652, available in Postscript or as pdf file

40. (with F. Reese Harvey) "Existence, uniqueness, and removable singularities for nonlinear partial differential equations in geometry", pp. 102-156 in ``Surveys in Differential Geometry 2013'', vol. 18, H.-D. Cao and S.-T. Yau eds., International Press, Somerville, MA, 2013, available in Postscript or as pdf file

41. (with F. Reese Harvey) "Removable singularities for nonlinear subequations", Indiana Univ. Math. J., {\bf 63}, No. 5 (2014), 1525-1552, available in Postscript or as pdf file

42. (with F. Reese Harvey) "Characterizing the strong maximum principle for constant coefficient subequations", Rendiconti di Matematica, available in Postscript or as pdf file

43. (with F. Reese Harvey) "The AE Theorem and Addition Theorems for quasi-convex functions", available in Postscript or as pdf file

44. (with F. Reese Harvey) "Notes on the differentiation of quasi-convex functions", available in Postscript or as pdf file

45. (with F. Reese Harvey) "Tangents to subsolutions - existence and uniqueness, Part I", Ann.Fac. des Sciences de Toulouse, Mathematiques, 27 (2018), 777-848, available in Postscript or as pdf file

46. (with F. Reese Harvey) "Tangents to subsolutions - existence and uniqueness, Part II", J. Geom. Analysis, 27 (2017), 2190-2223, available in Postscript or as pdf file

47. (with F. Reese Harvey and Szymon Plis) "Smooth approximation of plurisubharmonic functions on almost complex manifolds", Math. Ann. 366, issue 3 (2016), 929-940, available in Postscript or as pdf file

48. (with F. Reese Harvey) "The Dirichlet problem with prescribed interior singularities", Advances in Math., 303 (2016), 1319-1357, available in pdf file or as published manuscript

49. (with F. Reese Harvey) Preface to the thesis of Giulio Minervini: "A Current Approach to Morse and Novikov Theories", Rendiconti di Matematica, Serie VII, 36 (2015), 89-94, available in Postscript or as pdf file

50. (with F. Reese Harvey) "Lagrangian potential theory and a Lagrangian equation of Monge-Ampere type",pp. 217- 257 in Surveys in Differential Geometry, edited by H.-D. Cao, J. Li, R. Schoen and S.-T. Yau, vol. 22, International Press, Somerville, MA, 2018. available in pdf file

51. (with F. Reese Harvey) "Pluriharmonics in general potential theories", Contemporary Mathematics, vol. 735 (2019), 145-168, available in Postscript file or as pdf file

52. (with F. Reese Harvey and Szymon Plis) "The Richberg technique for subequations", Comm. in Anal. and Geom., vol. 28 no. 8 (2020), 1787-1806, available in Postscript file or as pdf file

53. (with F. Reese Harvey) "The inhomogeneous Dirichlet problem for natural operators on manifolds", Ann. Inst. Fourier, Grenoble, Tome 69, no. 7 (2019), 3017-3064, available in Postscript file or as pdf file

54. (with F. Reese Harvey) "A generalization of pde's from a Krylov point of view", Advances in Math. vol. 372 (2020) 107298, available in pdf file

55. (with F. Reese Harvey) "Pseudoconvexity for the special Lagrangian potential equation", Cal. of Var. and PDE, vol. 60 no. 6 (2021), available in Postscript file or as pdf file

56. "Reflections on the Early Work of Eugenio Calabi", pp. 3-21 in "Eugenio Calabi - Collected Works'', editors: Jean-Pierre Bourguignon, Xiuxiong Chen and Simon Donaldson, Springer-Verlag, 2020. available in Postscript file or as pdf file

57. (with Reese Harvey) "Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive", ArXiv:2207.01729 available in Postscript file or as pdf file

58. "Spin^h-Manfolds", available in pdf file

59. (with Misha Gromov) "Perspectives in Scalar Curvature, Vol.'s I and II", World Scientific Press, 2023.

60. (with Marco Cirant, Reese Harvey and Kevin Payne) "Comparison Principles for General Potential Theories and PDEs", Annals of Math. Studies No. 218, Princeton Univ. Press. 2023.

61. (with Misha Gromov) "The Atiyah-Singer Operator and Scalar Curvature" (in honor of Is Singer), available in pdf file