Adam S. Levine

Adam S. Levine
  • Assistant Professor of Mathematics
External address: 120 Science Drive, 211 Physics Building, Durham, NC 27708
Phone: (919) 660-2802

Research Areas and Keywords

3-manifolds, 4-manifolds, knot theory, Heegaard Floer homology, categorification

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.

Education & Training
  • Ph.D., Columbia University 2010

  • A.B., Harvard University 2005

Selected Grants

Low-Dimensional topology, Floer Homology, and Categorification awarded by National Science Foundation (Principal Investigator). 2017 to 2020

Levine, A. S., and T. Lidman. “SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS.” Forum of Mathematics, Sigma, Jan. 2019. Scopus, doi:10.1017/fms.2019.11. Full Text Open Access Copy

Baldwin, John A., et al. “Khovanov homology and knot Floer homology for pointed links.” Journal of Knot Theory and Its Ramifications, vol. 26, no. 02, World Scientific Pub Co Pte Lt, Feb. 2017, pp. 1740004–1740004. Crossref, doi:10.1142/s0218216517400041. Full Text Open Access Copy

Greene, Joshua, and Adam Levine. “Strong Heegaard diagrams and strong L–spaces.” Algebraic & Geometric Topology, vol. 16, no. 6, Mathematical Sciences Publishers, Dec. 2016, pp. 3167–208. Crossref, doi:10.2140/agt.2016.16.3167. Full Text Open Access Copy

Hedden, Matthew, and Adam Simon Levine. “Splicing knot complements and bordered Floer homology.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal), vol. 2016, no. 720, Walter de Gruyter GmbH, Jan. 2016. Crossref, doi:10.1515/crelle-2014-0064. Full Text Open Access Copy

LEVINE, A. D. A. M. S. I. M. O. N. “NONSURJECTIVE SATELLITE OPERATORS AND PIECEWISE-LINEAR CONCORDANCE.” Forum of Mathematics, Sigma, vol. 4, Cambridge University Press (CUP), 2016. Crossref, doi:10.1017/fms.2016.31. Full Text Open Access Copy

Levine, Adam, et al. “Nonorientable surfaces in homology cobordisms.” Geometry & Topology, vol. 19, no. 1, Mathematical Sciences Publishers, Feb. 2015, pp. 439–94. Crossref, doi:10.2140/gt.2015.19.439. Full Text Open Access Copy

Levine, A. S., and Daniel Ruberman. “Generalized Heegaard Floer correction terms.” Proceedings of the Gökova Geometry Topology Conference 2013, International Press, May 2014, pp. 76–96. Open Access Copy

Baldwin, John A., and Adam Simon Levine. “A combinatorial spanning tree model for knot Floer homology.” Advances in Mathematics, vol. 231, no. 3–4, Elsevier BV, Oct. 2012, pp. 1886–939. Crossref, doi:10.1016/j.aim.2012.06.006. Full Text Open Access Copy

Levine, Adam Simon. “Slicing mixed Bing-Whitehead doubles.” Journal of Topology, vol. 5, no. 3, Wiley, Sept. 2012, pp. 713–26. Crossref, doi:10.1112/jtopol/jts019. Full Text

Levine, Adam Simon. “Knot doubling operators and bordered Heegaard Floer homology.” Journal of Topology, vol. 5, no. 3, Wiley, Sept. 2012, pp. 651–712. Crossref, doi:10.1112/jtopol/jts021. Full Text