# Mark A. Stern

- Professor of Mathematics

**External address:**116 Physics Bldg, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2840

**Office Hours:**

Monday and Tuesday 2-3

### Research Areas and Keywords

##### Analysis

geometric analysis, elliptic partial differential equations

##### Geometry: Differential & Algebraic

geometric analysis, Yang Mills theory, Hodge theory, Index theory

##### Mathematical Physics

Yang Mills theory, String theory

##### PDE & Dynamical Systems

geometric analysis, elliptic partial differential equations

##### Topology

Index theory

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.

### Selected Grants

Instanton Decay and Nonlinear Harmonic Forms awarded by Simons Foundation (Principal Investigator). 2015 to 2020

Chern-Simons Invariance Instantons and Mass awarded by National Science Foundation (Principal Investigator). 2010 to 2014

Positive Mass, Singularities, and Supersymmetry awarded by National Science Foundation (Principal Investigator). 2005 to 2010

Bound States, Singularities, and Supersymmetry awarded by National Science Foundation (Principal Investigator). 2002 to 2006

Non-Fredholm Index Theory, Matrix, Models, and Hodge Theory awarded by National Science Foundation (Principal Investigator). 1998 to 2003

Hodge Structures and L2 Cohomology awarded by National Science Foundation (Principal Investigator). 1995 to 1999

Hodge Structures and L2 Cohomology awarded by National Science Foundation (Principal Investigator). 1995 to 1999

(94-0102) Presidential Young Investigator Award: Mathematical Sciences awarded by National Science Foundation (Principal Investigator). 1989 to 1994

(86-0081) Some New Spectral Invariants and Their Relationship to Automorphic Forms and Geodesics awarded by National Science Foundation (Principal Investigator). 1986 to 1988

Stern, Mark. *"Nonlinear Harmonic Forms and Indefinite Bochner Formulas " in
Hodge Theory and L^2-Analysis*. Vol. 39, Higher Education Press, 2017.

Stern, M. A. “Geometry of Stable Yang Mills Connections.” *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.” *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, Centre De Recherches Mathématiques, 1992, pp. 489–91.

Cherkis, Sergey A., et al. “Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem.” *Journal of Differential Geometry*, International Press, Dec. 2019.

Cerbo, Luca F. Di, and Mark Stern. “Price Inequalities and Betti Number Growth on Manifolds without
Conjugate Points.” *Communications in Analysis and Geometry*, International Press, Aug. 2019.
Open Access Copy

Lipnowski, M., and M. Stern. “Geometry of the Smallest 1-form Laplacian Eigenvalue on Hyperbolic Manifolds.” *Geometric and Functional Analysis*, vol. 28, no. 6, Dec. 2018, pp. 1717–55. *Scopus*, doi:10.1007/s00039-018-0471-x.
Full Text

Stern, M. A. “Asymptotic Hodge Theory of Vector Bundles.” *Communications in Analysis and Geometry*, vol. 23, no. 3, International Press, Dec. 2015, pp. 559–609.

Stern, Mark. *$C^{\infty}$ Stability, Canonical Maps, and Discrete Dynamics*. Oct. 2014.

Degeratu, A., and M. Stern. “Witten Spinors on Nonspin Manifolds.” *Communications in Mathematical Physics*, vol. 324, no. 2, 2013, pp. 301–50. *Scival*, doi:10.1007/s00220-013-1804-0.
Full Text

Melnikov, I., et al. “Target spaces from chiral gauge theories.” *Journal of High Energy Physics*, vol. 2013, no. 2, 2013, pp. 1–56. *Scival*, doi:10.1007/JHEP02(2013)111.
Full Text

Quigley, C., et al. “Novel branches of (0, 2) theories.” *Journal of High Energy Physics*, vol. 2012, no. 9, Oct. 2012. *Scopus*, doi:10.1007/JHEP09(2012)0641.
Full Text

Quigley, Callum, et al. “Novel branches of (0, 2) theories.” *Journal of High Energy Physics*, vol. 2012, no. 9, Springer Science and Business Media LLC, Sept. 2012. *Crossref*, doi:10.1007/jhep09(2012)064.
Full Text

Stern, M. “Geometry of minimal energy Yang-Mills connections.” *Journal of Differential Geometry*, vol. 86, no. 1, 2010, pp. 163–88.