MARK STERN

B.S. Texas A & M University

Ph.D. Princeton University

Professor

Areas of Expertise: Geometry, Geometric Analysis, Lie Groups, Supersymmetry

Research Summary:

The focus of Professor Stern's research is the study of analytic realizations of topological and algebraic invariants of noncompact manifolds and conversely the topological or algebraic interpretation of naturally arising analytic invariants. A classic motivating example for compact manifolds is Hodge theory which relates harmonic forms on smooth compact manifolds to the singular cohomology, and when the manifold is Kähler, further relates harmonic forms to Dolbeault cohomology.

In collaboration with Professor Pardon, Professor Stern is investigating the L2 d- and delbar-cohomology of singular projective varieties. In their work they study the natural L2 extension of Hodge theory to projective varieties and seek to understand the deviation from compact Hodge theory in terms of the topological and algebraic properties of the singularities.

In recent work, Professor Stern has studied, in collaboration with S. Sethi and others, questions arising from string duality and the matrix model of M theory. The latter has led to a study of (i) non fredholm index theories, and (ii) constraints on supersymmetric Lagrangians arising in Yang Mills theory.

Recent Publications:

  1. L2-index Theorems on Locally Symmetric Spaces, Inventiones Math. 96 (2) (1989), 231-282.
  2. (with Leslie Saper), L2-cohomology of Arithmetic Varieties, Proc. Natl. Acad Sci. 84 (1987), 5516-5519.
  3. (with Leslie Saper), L2-Cohomology of Arithmetic Varieties, Ann. Math. 132 (1990), 1-69.
  4. Eta Invariants and Hermitian Locally Symmetric Spaces, J. Diff. Geom. 31 (1990), 771-789.
  5. Lefschetz formulae for arithmetic varieties, Invent. Math. 115 (1994), 241-296.
  6. (with William Pardon), L2-delbar-Cohomology of Complex Projective Varieties, J. of the A.M.S. 4 (1991), 603-621.
  7. Index Theory for Certain Complete Kähler Manifolds, J. Diff. Geom. 37 (1993), 467-503.
  8. A New Fixed Point Theorem for the Signature Complex, in preparation.
  9. Hodge Structures and L2-delbar-Cohomology of Arithmetic Varieties, preprint.
  10. L2-cohomology and index theory of noncompact manifolds, Proc. Symp. Pure Math. 54 (1993), Part 2, 559-575.
  11. (with William Pardon) Pure Hodge structures on the L2-cohomology of varieties with isolated singularities , alg-geom/9711003.

  12. (with S. Sethi and E. Zaslow), Monopole and dyon bound states in N=2 super symmetric Yang Mills, Nuclear Phys. B 457 (1995), 484-510.
  13. (with S. Sethi), A Comment on the Spectrum of H-Monopoles, Phys. Lett. B 398:47-51 (1997).
  14. (with S. Sethi), D-Brane Bound States Redux, Commun. Math. Phys. 194 :675-705 (1998).
  15. (with S. Sethi and S. Paban) Constraints from extended supersymmetry in quantum mechanics, Nucl. Phys. B. 534 :137-154 (1998).
  16. (with S. Sethi and S. Paban) Supersymmetry and higher derivative terms in the effective action of Yang-Mills theories , J. High Energy Physics. {06:012} (1998).
  17. (with S. Sethi and S. Paban) Summing up instantons in three dimensional Yang-Mills theories , hep-th 9808119.
  18. (with S. Sethi) Supersymmetry and the Yang-Mills effective action at finite N , manuscript.
Last modified February, 1999