Mark A. Stern, Professor

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

In recent and ongoing work, Professor Stern has studied analytical, geometric, and topological questions arising in Yang-Mills theory. These include analyzing the moduli space of Yang Mills instantons on gravitational instantons, analyzing the asymptotic structure of instantons (proving a nonlinear analog of the inverse square law of electromagnetism), and analyzing the structure of singularities of instantons and of  harmonic maps.  

 In addition, Professor Stern has recently studied questions arising in the interplay between geometric group theory and Lp and L2 cohomology. This work includes finding new bounds on L2 betti numbers of negatively curved manifolds, and new growth,
stability, and vanishing results for Lp and L2 cohomology of symmetric and locally symmetric spaces.

Contact Info:

Office Location:	116 Physics Bldg, Durham, NC 27708
Email Address:
Web Page:	https://www.math.duke.edu/faculty/stern

Teaching (Spring 2025):

• MATH 690-20.01, DIFFERENTIAL GEOMETRY (TOP) Synopsis

  Physics 227, WF 08:30 AM-09:45 AM

Office Hours:

Monday: 2-3,  Tuesday : 10-11

Education:

Ph.D.	Princeton University	1984
B.S.	Texas A&M University	1980

Specialties:

Geometry
Mathematical Physics

Research Interests: Geometric Analysis, Yang-Mills theory, Hodge theory, string theory

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

In recent work, Professor Stern has studied analytical, geometric, and topological questions arising from Yang-Mills theory, string theory, and Hodge 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) the analysis of harmonic spinors on singular spin structures, and (v) non fredholm index theories and exotic fixed point theorems.

Curriculum Vitae

Current Ph.D. Students   (Former Students)

Postdocs Mentored

• Akos Nagy (2017 - 2020)  
• Goncalo Oliveira (2014 - 2017)  
• Luca Di Cerbo (2011 - 2014)  
• Benoit Charbonneau (2007 - 2010)  
• Bianca Santoro (2009)  
• Anda Degeratu (December 17, 2002 - 2005)  

Recent Publications   (More Publications)

1. Di Cerbo, LF; Stern, M, Harmonic Forms, Price Inequalities, and Benjamini–Schramm Convergence, Journal of Geometric Analysis, vol. 35 no. 1 (January, 2025) [doi]  [abs]
2. Cherkis, S; Larraín-Hubach, A; Stern, M, Instantons on multi-Taub-NUT Spaces II: Bow Construction, Journal of Differential Geometry, vol. 127 no. 2 (June, 2024), pp. 433-503, International Press [doi]  [abs]
3. Cherkis, SA; Larraín-Hubach, A; Stern, M, Instantons on multi-Taub-NUT Spaces III: Down Transform, Completeness, and Isometry (August, Preprint, 2023)
4. Di Cerbo, LF; Stern, M, Price inequalities and Betti number growth on manifolds without conjugate points, Communications in Analysis and Geometry, vol. 30 no. 2 (January, 2022), pp. 297-334, International Press [doi]  [abs]
5. Cherkis, SA; Larrain-Hubach, A; Stern, M, Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem, Journal of Differential Geometry, vol. 119 no. 1 (December, 2021), pp. 1-72, International Press  [abs]

Recent Grant Support

• The Geometry and Analysis of Yang-Mills Instantons., Simons Foundation, 2023/09-2028/08.      
• Instanton Decay and Nonlinear Harmonic Forms, Simons Foundation, 3553857, 2015/09-2022/08.      
• Bound states, singularities, and supersymmetry, NSF, 2002/07.