Adam S. Levine, Associate Professor

My research is in low-dimensional topology, the study of the shapes of 3- and 4-dimensional spaces (manifolds) and of curves and surfaces contained therein. Classifying smooth 4-dimensional manifolds, in particular, has been a deep challenge for topologists for many decades; unlike in higher dimensions, there is not enough "wiggle room" to turn topological problems into purely algebraic ones. Many of my projects reveal new complications in the topology of 4-manifolds, particularly related to embedded surfaces. My main tools come from Heegaard Floer homology, a powerful package of invariants derived from symplectic geometry. I am also interested in the interrelations between different invariants of knots in 3-space, particularly the connections between knot invariants arising from gauge theory and symplectic geometry and those coming from representation theory.

Contact Info:

Office Location:	120 Science Drive, Durham, NC 27708
Email Address:
Web Page:	http://www.math.duke.edu/~alevine

Teaching (Fall 2025):

• MATH 221D.001, LINEAR ALGEBRA Synopsis

  Physics 119, MW 01:25 PM-02:40 PM

• MATH 221D.01D, LINEAR ALGEBRA Synopsis

  Allen 326, F 01:25 PM-02:40 PM

• MATH 221D.02D, LINEAR ALGEBRA Synopsis

  Social Sciences 119, F 03:05 PM-04:20 PM

• MATH 721D.001, LINEAR ALGEBRA & APPLICA Synopsis

  Physics 119, MW 01:25 PM-02:40 PM

• MATH 721D.01D, LINEAR ALGEBRA & APPLICA Synopsis

  Allen 326, F 01:25 PM-02:40 PM

• MATH 721D.02D, LINEAR ALGEBRA & APPLICA Synopsis

  Social Sciences 119, F 03:05 PM-04:20 PM

Office Hours:

Please email me for office hours.

Education:

Ph.D.	Columbia University	2010
A.B.	Harvard University	2005

Keywords:

Floer homology • Knot theory • Low-dimensional topology • Topology

Recent Publications   (More Publications)

1. Hedden, M; Levine, AS, A surgery formula for knot Floer homology, Quantum Topology, vol. 15 no. 2 (January, 2024), pp. 229-336 [doi]  [abs]
2. Levine, AS, A note on rationally slice knots, New York Journal of Mathematics, vol. 29 (January, 2023), pp. 1363-1372  [abs]
3. Hom, J; Levine, AS; Lidman, T, KNOT CONCORDANCE IN HOMOLOGY COBORDISMS, Duke Mathematical Journal, vol. 171 no. 15 (October, 2022), pp. 3089-3131 [doi]  [abs]
4. Gujral, OS; Levine, AS, Khovanov homology and cobordisms between split links, Journal of Topology, vol. 15 no. 3 (September, 2022), pp. 973-1016 [doi]  [abs]
5. Baldwin, JA; Dowlin, N; Levine, AS; Lidman, T; Sazdanovic, R, Khovanov homology detects the figure-eight knot, Bulletin of the London Mathematical Society, vol. 53 no. 3 (June, 2021), pp. 871-876 [doi]  [abs]

Recent Grant Support

• Four-Manifolds and Categorification, National Science Foundation, 2022/08-2026/07.