Mark A. Stern | Scholars@Duke

Mark A. Stern
Professor of Mathematics

The focus of Professor Stern's research is the study of analytic problems arising in geometry, topology, physics, and number theory.

In recent work, Professor Stern has studied analytical, geometric, and topological questions arising from Yang-Mills theory, Hodge theory, and number theory. These have led for example to a study of (i) stability questions arising in Yang Mills theory and harmonic maps, (ii) energy minimizing connections and instantons, (iii) new Hodge structures on vector bundles, (iv) new bounds for eigenvalues of Laplace Beltrami operators, and (v) new bounds for betti numbers.

Office Hours

Monday and Tuesday 2-3

Current Appointments & Affiliations

Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 1992

Contact Information

116 Physics Bldg, Durham, NC 27708
Box 90320, Durham, NC 27708-0320
stern@math.duke.edu (919) 660-2840
https://www.math.duke.edu/faculty/stern

Background
Education, Training, & Certifications
- Ph.D., Princeton University 1984
- B.S., Texas A&M University 1980
Duke Appointment History
- Chair, Department of Mathematics, Mathematics, Trinity College of Arts & Sciences 2006 - 2009
- Associate Chair, Department of Mathematics, Mathematics, Trinity College of Arts & Sciences 2003 - 2005
- Associate Professor with Tenure, Mathematics, Trinity College of Arts & Sciences 1989 - 1992
- Associate Professor, Mathematics, Trinity College of Arts & Sciences 1988 - 1989
- Assistant Professor, Mathematics, Trinity College of Arts & Sciences 1985 - 1988
Recognition
Awards & Honors
Research
Selected Grants
- Instanton Decay and Nonlinear Harmonic Forms awarded by Simons Foundation 2015 - 2020
- Chern awarded by National Science Foundation 2010 - 2014
- Positive Mass, Singularities, and Supersymmetry awarded by National Science Foundation 2005 - 2010
- Bound States, Singularities, and Supersymmetry awarded by National Science Foundation 2002 - 2006
- (98-0381) Non-Fredholm Index Theory, Matrix, Models, and Hodge Theory awarded by National Science Foundation 1998 - 2003
- (97-0445) Hodge Structures and L2 Cohomology awarded by National Science Foundation 1995 - 1999
- (96-0809) Hodge Structures and L2 Cohomology awarded by National Science Foundation 1995 - 1999
- (95-0287) Hodge Structures and L2 Cohomology awarded by National Science Foundation 1995 - 1997
- (94-0102) Presidential Young Investigator Award: Mathematical Sciences awarded by National Science Foundation 1989 - 1994
- (86-0081) Some New Spectral Invariants and Their Relationship to Automorphic Forms and Geodesics awarded by National Science Foundation 1986 - 1988
Publications & Artistic Works
Selected Publications
- Academic Articles
  - Lipnowski, M., and M. Stern. “Geometry of the Smallest 1-form Laplacian Eigenvalue on Hyperbolic Manifolds.” Geometric and Functional Analysis 28, no. 6 (December 1, 2018): 1717–55. https://doi.org/10.1007/s00039-018-0471-x.
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  - Cerbo, Luca F Di, and Mark Stern. “Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points,” April 18, 2017.
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  - Cherkis, Sergey A., Andres Larrain-Hubach, and Mark Stern. “Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem,” August 1, 2016.
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  - Stern, M. A. “Asymptotic Hodge Theory of Vector Bundles.” Communications in Analysis and Geometry 23, no. 3 (December 1, 2015): 559–609.
  - Stern, Mark. “$C^{\infty}$ Stability, Canonical Maps, and Discrete Dynamics,” October 31, 2014.
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  - Degeratu, A., and M. Stern. “Witten Spinors on Nonspin Manifolds.” Communications in Mathematical Physics 324, no. 2 (2013): 301–50. https://doi.org/10.1007/s00220-013-1804-0.
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  - Melnikov, I., C. Quigley, S. Sethi, and M. Stern. “Target spaces from chiral gauge theories.” Journal of High Energy Physics 2013, no. 2 (2013): 1–56. https://doi.org/10.1007/JHEP02(2013)111.
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  - Quigley, C., S. Sethi, and M. Stern. “Novel branches of (0, 2) theories.” Journal of High Energy Physics 2012, no. 9 (October 4, 2012). https://doi.org/10.1007/JHEP09(2012)0641.
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  - Quigley, Callum, Savdeep Sethi, and Mark Stern. “Novel branches of (0, 2) theories.” Journal of High Energy Physics 2012, no. 9 (September 2012). https://doi.org/10.1007/jhep09(2012)064.
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  - Stern, M. “Geometry of minimal energy yang-mills connections.” Journal of Differential Geometry 86, no. 1 (January 1, 2010): 163–88. https://doi.org/10.4310/jdg/1299766686.
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  - Stern, M. “Fixed point theorems from a de Rham perspective.” Asian Journal of Mathematics 13, no. 1 (October 26, 2009): 065–088.
  - STERN, M. A. R. K. “B FIELDS FROM A LUDDITE PERSPECTIVE.” Quantum Theory and Symmetries, October 2004. https://doi.org/10.1142/9789812702340_0048.
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  - Paban, S., S. Sethi, and M. Stern. “I. Non-commutativity and supersymmetry.” Journal of High Energy Physics 6, no. 3 (March 1, 2002): 183–200.
  - Stern, Mark A. “Quantum Mechanical Mirror Symmetry, D Branes, and B fields.” Eprint, 2002.
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  - Pardon, W., and M. Stern. “Pure hodge structure on the L2-cohomology of varieties with isolated singularities.” Journal Fur Die Reine Und Angewandte Mathematik 533 (December 1, 2001): 55–80.
  - Britto-Pacumio, R., A. Maloney, A. Strominger, and M. Stern. “Spinning bound states of two and three black holes.” Journal of High Energy Physics 5, no. 11 (January 1, 2001): 1–19. https://doi.org/10.1088/1126-6708/2001/11/054.
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  - Stern, M., and P. Yi. “Counting Yang-Mills dyons with index theorems.” Physical Review D Particles, Fields, Gravitation and Cosmology 62, no. 12 (December 15, 2000): 1–15.
    Open Access Copy
  - Sethi, S., and M. Stern. “The structure of the D0-D4 bound state.” Nuclear Physics B 578, no. 1–2 (July 3, 2000): 163–98. https://doi.org/10.1016/S0550-3213(00)00181-4.
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  - Sethi, S., and M. Stern. “Invariance theorems for supersymmetric Yang-Mills theories.” Advances in Theoretical and Mathematical Physics 4, no. 2 (March 1, 2000): 1–12.
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  - Sethi, S., and M. Stern. “Supersymmetry and the Yang-Mills effective action at finite N.” Journal of High Energy Physics 3, no. 6 (December 1, 1999).
  - Paban, S., S. Sethi, and M. Stern. “Summing up instantons in three-dimensional Yang-Mills theories.” Advances in Theoretical and Mathematical Physics 3, no. 2 (March 1, 1999): 1–18.
  - Paban, S., S. Sethi, and M. Stern. “Supersymmetry and higher derivative terms in the effective action of Yang-Mills theories.” Journal of High Energy Physics 2, no. 6 (December 1, 1998).
  - Paban, S., S. Sethi, and M. Stern. “Constraints from extended supersymmetry in quantum mechanics.” Nuclear Physics B 534, no. 1–2 (November 23, 1998): 137–54. https://doi.org/10.1016/S0550-3213(98)00518-5.
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  - Sethi, S., and M. Stern. “D-brane bound states redux.” Communications in Mathematical Physics 194, no. 3 (January 1, 1998): 675–705. https://doi.org/10.1007/s002200050374.
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  - Sethi, S., and M. Stern. “A comment on the spectrum of H-monopoles.” Physics Letters, Section B: Nuclear, Elementary Particle and High Energy Physics 398, no. 1–2 (April 10, 1997): 47–51. https://doi.org/10.1016/S0370-2693(97)00199-8.
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  - Sethi, S., M. Stern, and E. Zaslow. “Monopole and dyon bound states in N = 2 supersymmetric Yang-Mills theories.” Nuclear Physics, Section B 457, no. 3 (December 25, 1995): 484–510. https://doi.org/10.1016/0550-3213(95)00517-X.
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  - Stern, M. “Lefschetz formulae for arithmetic varieties.” Inventiones Mathematicae 115, no. 1 (December 1, 1994): 241–96. https://doi.org/10.1007/BF01231760.
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  - Stern, M. “Index theory for certain complete kähler manifolds.” Journal of Differential Geometry 37, no. 3 (January 1, 1993): 467–503. https://doi.org/10.4310/jdg/1214453896.
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  - Pardon, W. L., and M. A. Stern. “L² -∂-cohomology of complex projective varieties.” Journal of the American Mathematical Society 4, no. 3 (January 1, 1991): 603–21. https://doi.org/10.1090/S0894-0347-1991-1102582-6.
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  - SAPER, L., and M. STERN. “L2-COHOMOLOGY OF ARITHMETIC VARIETIES.” Annals of Mathematics 132, no. 1 (July 1990): 1–69. https://doi.org/10.2307/1971500.
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  - Stern, M. “Eta invariants and Hermitian locally symmetric spaces.” Journal of Differential Geometry 31, no. 3 (January 1, 1990): 771–89. https://doi.org/10.4310/jdg/1214444634.
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  - Stern, M. “L²-index theorems on locally symmetric spaces.” Inventiones Mathematicae 96, no. 2 (June 1, 1989): 231–82. https://doi.org/10.1007/BF01393964.
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  - Saper, L., and M. Stern. “L²-cohomology of arithmetic varieties.” Proc Natl Acad Sci U.S.A. 84, no. 16 (August 1987): 5516–19.
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  - Cerbo, Luca F Di, and Mark Stern. “Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence,” n.d.
    Open Access Copy Link to Item
  - Paban, Sonia, Savdeep Sethi, and Mark Stern. “Non-commutativity and Supersymmetry.” Journal of High Energy Physics 2002, no. 03 (n.d.): 012–012. https://doi.org/10.1088/1126-6708/2002/03/012.
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  - Stern, Mark A. “Mechanical D Branes and B Fields,” n.d.
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  - Stern, Mark. “Spectral asymmetry for manifolds of special holonomy,” n.d.
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- Book Sections
  - Stern, Mark. “"Nonlinear Harmonic Forms and Indefinite Bochner Formulas " in Hodge Theory and L^2-Analysis,” Vol. 39. Beijing: Higher Education Press, 2017.
  - Stern, M. A. “Geometry of Stable Yang Mills Connections.” In Advances in Geometric Analysis. International Pressof Boston Incorporated, 2012.
  - Saper, L., and M. Stern. “Appendix to: On the shape of the contribution of a fixed point on the boundary. The case of Q-rank one, by M. Rapoport.” In The Zeta Functions of Picard Modular Surfaces Based on Lectures Delivered at a CRM Workshop in the Spring of 1988, edited by R. Langlands and D. Ramakrishnan, 489–91. Montréal: Centre De Recherches Mathématiques, 1992.
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Teaching & Mentoring
Recent Courses
- MATH 493: Research Independent Study 2019
- MATH 555: Ordinary Differential Equations 2019
- MATH 633: Complex Analysis 2019
- MATH 790-90: Minicourse in Advanced Topics 2019
- MATH 635: Functional Analysis 2018
- MATH 653: Elliptic Partial Differential Equations 2018
- MATH 356: Elementary Differential Equations 2017
Scholarly, Clinical, & Service Activities
Presentations & Appearances
Outreach & Engaged Scholarship

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Education, Training, & Certifications

Duke Appointment History

Awards & Honors

Selected Grants

Selected Publications

Academic Articles

Book Sections

Recent Courses

Presentations & Appearances

Outreach & Engaged Scholarship