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
Office Hours
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.Link to Item
-
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.Full Text Open Access Copy
-
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.Link to Item
-
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.Full Text
-
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.Full Text
-
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.Full Text
-
Quigley, Callum, Savdeep Sethi, and Mark Stern. “Novel Branches of (0,2) Theories.” Jhep 1209, no. 064 (2012).Link to Item
-
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.Link to Item
-
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.Full Text
-
STERN, M. A. R. K. “B FIELDS FROM A LUDDITE PERSPECTIVE.” Quantum Theory and Symmetries, October 2004. https://doi.org/10.1142/9789812702340_0048.Full Text
-
Stern, Mark A. “Mechanical D Branes and B Fields,” October 2, 2003.Link to Item
-
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.Full Text
-
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.Link to Item
-
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.Full Text
-
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.Full Text
-
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.Full Text Open Access Copy
-
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.Full Text
-
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.Full Text
-
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.Full Text
-
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.Full Text Open Access Copy
-
Stern, M. “Lefschetz formulae for arithmetic varieties.” Inventiones Mathematicae 115, no. 1 (December 1, 1994): 241–96. https://doi.org/10.1007/BF01231760.Full Text
-
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.Full Text
-
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.Full Text
-
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.Full Text
-
Stern, M., and L. Saper. “L²-cohomology of arithmetic varieties.” Annals of Mathematics 132, no. 1 (1990): 1–69.Link to Item
-
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.Full Text
-
Saper, L., and M. Stern. “L²-cohomology of arithmetic varieties.” Proc Natl Acad Sci U.S.A. 84, no. 16 (August 1987): 5516–19.Link to Item
-
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.Link to Item
-
-
- Teaching & Mentoring
-
Recent Courses
- Scholarly, Clinical, & Service Activities
-
Presentations & Appearances
- Nahm transform for ALF spaces. U Florida Geometry Seminar. U Florida Mathematics Dept. . October 25, 2022 - October 25, 2022 2022
- Nahm transform and ALF spaces. Duke Geometry/Topolgy Seminar. Duke. September 19, 2022 - September 19, 2022 2022
- Nahm Transform Yoga. Geometry and physics of ALX metrics in gauge theory. AIM. July 29, 2022 - July 29, 2022 2022
- Bubbling threshholds for Yang Mills and harmonic maps in fat negatively curved spaces . Geometry, Analysis, and Quantum Physics of Monopoles. BIRS. February 5, 2021 - February 5, 2021 2021
- A coarse Cheeger inequality for 1-forms. UF Topology Seminar. University of Florida. January 14, 2020 - January 14, 2020 2020
- A Coarse Cheeger Inequality for one Forms. Geometry and Topology Seminar. Duke. March 18, 2019 - March 18, 2019 2019
- Monotonicity and Betti Number Bounds. Geometry Seminar. Harvard. October 23, 2018 - October 23, 2018 2018
- Nahm Transform for ALF spaces. Geometry Seminar. SUNY Stonybrook. March 6, 2018 2018
- Monotonicity and Betti Number Bounds. Geometry Seminar. CUNY Graduate Seminar. February 15, 2018 2018
- Monotonicity and Betti number bounds. The Analysis of Gauge-Theoretic Moduli Spaces. Banff International research Station. August 29, 2017 - August 29, 2017 2017
- Instantons on ALF Spaces. Gauge Theory and Special Geometry. Mathematical Congress of the Americas. July 25, 2017 - July 25, 2017 2017
- Spectral geometry of the 1 form Laplacian on hyperbolic manifolds.. Special Session on Geometry and Topology of Ball Quotients and Related Topics. AMS Eastern Sectional Meeting. May 6, 2017 - May 6, 2017 2017
- Instantons on ALF spaces. City University NY. CUNY . December 6, 2016 - December 6, 2016 2016
- Instantons on ALF spaces. Waterloo Geometry seminar. University of Waterloo. December 2, 2016 - December 2, 2016 2016
- In search of the second derivative. ICTP Elementary Notions Seminar. ICTP. October 28, 2016 - October 28, 2016 2016
- Instantons on ALF spaces. ICTP Geometry Seminar. ICTP. October 24, 2016 - October 24, 2016 2016
- Instantons on ALF spaces. Harvard Geometry Seminar. Harvard University. October 11, 2016 - October 11, 2016 2016
- Decay and Moduli of Yang Mills Instantons. Newton Institute. July 21, 2015 2015
- Introduction to nonlinear harmonic forms. U. Maryland. March 30, 2015 2015
- Introduction to nonlinear harmonic forms. Texas Geometry and Topology Conference. University of Houston. February 28, 2015 2015
- Introduction to nonlinear harmonic forms. Chinese University, Hong Kong. December 9, 2014 2014
- Introduction to nonlinear harmonic forms. Conference on Hodge Theory and L2-cohomology. Johns Hopkins. November 22, 2014 2014
- Instanton Decay. Conference on Heterotic Strings and (0,2) QFT. Texas A&M. April 2014 2014
- Symmetry, singularity, and decay of Yang Mills fields. Texas A&M. March 20, 2014 2014
- Stability, Hodge theory, and Grassmann embeddings. May 21, 2013 2013
- Hodge theory, Yang Mills, and Stability. January 28, 2013 2013
- Asymptotic Hodge Theory of Vector Bundles. December 18, 2011 2011
- Stability, dynamics, and the quantum Hodge theory of vector bundles. November 30, 2010 2010
- The Quantum Hodge Theory of Vector Bundles. March 12, 2010 2010
- Geometry of Stable Yang-Mills connections. October 20, 2009 2009
- Geometry of Minimal Yang-Mills connections. April 6, 2009 2009
- Geometry of Stable Yang-Mills connections. April 5, 2009 2009
- Minimal Energy Yang-Mills Connections and Interpolating Instantons. October 28, 2008 2008
-
Outreach & Engaged Scholarship
- Judge . 2018 Yau Science Award. Tsinghua University. December 12, 2018 - December 15, 2018 2018
- Judge. Dogrun-Yau Science Award. Tsinghua University. December 2016 - December 2016 2016
- Judge. 2015 Yau Math Award. Tsinghua University. December 2015 - December 2015 2015
- Judge. 2014 Yau Math Award. Tsinghua University. December 2014 - December 2014 2014
- Judge. Hang Lung Mathematics Award. City University Hong Kong. December 2014 - December 2014 2014
- Judge. 2011 Yau Math Award. Tsinghua University. December 2011 2011
Some information on this profile has been compiled automatically from Duke databases and external sources. (Our About page explains how this works.) If you see a problem with the information, please write to Scholars@Duke and let us know. We will reply promptly.