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 bounds for eigenvalues of Laplace Beltrami operators, and (iv) new bounds for betti numbers.

Current Research Interests

Currently I am interested in various aspects of the structure of yang mills moduli spaces. I am also studying various aspects of the interaction of geometry and topology and the spectrum of the Laplace beltrami operator on forms. This includes spectral estimates motivated by questions in number theory and betti number estimates motivated by questions in geometric topology.

Office Hours

Wednesday: 10-11, Thursday : 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
Previous Appointments & Affiliations
- 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 - 2022
- Chern-Simons Invariance Instantons and Mass 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
- Non-Fredholm Index Theory, Matrix, Models, and Hodge Theory awarded by National Science Foundation 1998 - 2003
- Hodge Structures and L2 Cohomology awarded by National Science Foundation 1995 - 1999
- Hodge Structures and L2 Cohomology awarded by National Science Foundation 1995 - 1999
- Presidential Young Investigator Award: Mathematical Sciences awarded by National Science Foundation 1989 - 1994
- 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
  - Cerbo, Luca F Di, and Mark Stern. “Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points.” Communications in Analysis and Geometry 30, no. 2 (November 29, 2022): 297–334.
    Open Access Copy Link to Item
  - Cherkis, Sergey A., Andres Larrain-Hubach, and Mark Stern. “Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem.” Journal of Differential Geometry 119, no. 1 (December 9, 2021): 1–72.
    Open Access Copy Link to Item
  - Cherkis, Sergey, Andrés Larraín-Hubach, and Mark Stern. “Instantons on multi-Taub-NUT Spaces II: Bow Construction,” March 23, 2021.
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  - Cerbo, Luca F Di, and Mark Stern. “On the Betti Numbers of Finite Volume Hyperbolic Manifolds,” September 23, 2020.
    Open Access Copy Link to Item
  - Cerbo, Luca F Di, and Mark Stern. “Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence,” September 12, 2019.
    Open Access Copy Link to Item
  - 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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  - Stern, M. A., and Benoit Charbonneau. “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 (January 1, 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.” Jhep 1209, no. 064 (2012).
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  - Stern, M. “Geometry of minimal energy Yang-Mills connections.” Journal of Differential Geometry 86, no. 1 (2010): 163–88.
  - Stern, Mark. “Spectral asymmetry for manifolds of special holonomy,” February 12, 2009.
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  - Stern, M. “Fixed point theorems from a de Rham perspective.” Asian Journal of Mathematics 13, no. 1 (January 1, 2009): 065–088. https://doi.org/10.4310/ajm.2009.v13.n1.a4.
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  - 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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  - Stern, Mark A. “Mechanical D Branes and B Fields,” October 2, 2003.
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  - Paban, Sonia, Savdeep Sethi, and Mark Stern. “Non-commutativity and Supersymmetry.” Journal of High Energy Physics 2002, no. 03 (March 6, 2002): 012–012. https://doi.org/10.1088/1126-6708/2002/03/012.
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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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  - 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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  - 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 (2001): 55–80.
  - 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 (January 1, 2000): 487–501. https://doi.org/10.4310/atmp.2000.v4.n2.a8.
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  - 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 (January 1, 1999): 343–61. https://doi.org/10.4310/atmp.1999.v3.n2.a6.
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  - 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 (1998): 675–705.
  - 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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  - 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., and L. Saper. “L²-cohomology of arithmetic varieties.” Annals of Mathematics 132, no. 1 (1990): 1–69.
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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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  - Cherkis, Sergey, Andrés Larraín-Hubach, and Mark Stern. “Instantons on multi-Taub-NUT Spaces II: Bow Construction (Accepted).” Journal of Differential Geometry, n.d.
    Open Access Copy
- 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 621: Differential Geometry 2023
- MATH 633: Complex Analysis 2022
- MATH 635: Functional Analysis 2022
- MATH 790-90: Minicourse in Advanced Topics 2022
- MATH 799: Special Readings 2022
- MATH 532: Basic Analysis II 2021
- MATH 555: Ordinary Differential Equations 2021
Scholarly, Clinical, & Service Activities
Presentations & Appearances
Outreach & Engaged Scholarship

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Education, Training, & Certifications

Previous Appointments & Affiliations

Awards & Honors

Selected Grants

Selected Publications

Academic Articles

Book Sections

Recent Courses

Presentations & Appearances

Outreach & Engaged Scholarship