Curriculum Vitae: Lenhard Ng

As of March 2015. For contact information, please click here.

Research interests

Low dimensional topology, symplectic and contact geometry, differential geometry, mathematical physics. Particular interests: holomorphic curves, symplectic field theory, knot theory.

Current position

Associate Professor, Mathematics Department, Duke University, 2011-present (Assistant Professor, 2006-2011).

Education

• Massachusetts Institute of Technology (1996-2001)
  Ph.D. in mathematics, June 2001, Tomasz Mrowka, advisor.
• Harvard University (1993-1996)
  A.B. summa cum laude in mathematics and physics, June 1996.

Past employment and positions

• Research Member, Mathematical Sciences Research Institute, spring 2010.
• Visiting Assistant Professor, Princeton University, spring 2007.
• American Institute of Mathematics Five-Year Fellow (postdoctoral), 2001-2006.
• Lecturer in Mathematics and Visiting Scholar, Stanford University, 2002-2006.
• Member, Institute for Advanced Study, 2001-2002, for program “Symplectic Geometry and Holomorphic Curves”.

Selected awards, grants, and honors

• National Science Foundation: Principal Investigator on the following grants:
  • DMS-1406371 “Knots and Contact Topology Through Holomorphic Curves”, 2014-2017;
  • DMS-0846346 (CAREER) “Symplectic Field Theory and Low-Dimensional Topology”, 2009-2014;
  • DMS-0706777 “Holomorphic Curves and Low-Dimensional Topology”, 2007-2010.
• Simons Foundation: Simons Fellow in Mathematics, 2015-2016.
• American Institute of Mathematics: Five-Year Fellow, 2001-2006.
• Department of Defense: National Defense Science and Engineering Graduate (NDSEG) fellow, 1996-1999.
• AMS/MAA/SIAM Frank and Brennie Morgan Prize for undergraduate research in mathematics: honorable mention (second place), 1996.
• William Lowell Putnam mathematical competition: Putnam Fellow, 1993, 1994, 1995.
• International Mathematical Olympiad: gold medal, 1992 (Moscow), 1993 (Istanbul); silver medal, 1991 (Sigtuna, Sweden).
• USA Mathematical Olympiad: first place, 1992; winner, 1991, 1993.

Publications

• Augmentations are sheaves (with D. Rutherford, V. Shende, S. Sivek, and E. Zaslow), preprint, arXiv:1502.04939.
• Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold (with T. Ekholm), J. Differential Geom., to appear, arXiv:1307.8436.
• Topological strings, D-model, and knot contact homology (with M. Aganagic, T. Ekholm, and C. Vafa), Adv. Theor. Math. Phys. 18 (2014), no. 4, 827-956.
• On transverse invariants from Khovanov homology (with R. Lipshitz and S. Sarkar), Quantum Topol., to appear, arXiv:1303.6371.
• A topological introduction to knot contact homology, in Contact and Symplectic Topology, Bolyai Soc. Math. Stud. 26 (Springer, Berlin, 2014).
• Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra (with D. Rutherford), Algebr. Geom. Topol. 13 (2013), no. 5, 3047-3097.
• Knot contact homology (with T. Ekholm, J. Etnyre, and M. Sullivan), Geom. Topol. 17 (2013), 975-1112.
• An atlas of Legendrian knots (with W. Chongchitmate), Exp. Math. 22 (2013), no. 1, 26-37.
• Combinatorial knot contact homology and transverse knots, Adv. Math. 227 (2011), no. 6, 2189--2219.
• Filtrations on the knot contact homology of transverse knots (with T. Ekholm, J. Etnyre, and M. Sullivan), Math. Annalen 355 (2013), no. 4, 1561-1591.
• Legendrian and transverse twist knots (with J. Etnyre and V. Vértesi), J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 451-512.
• Grid diagrams, braids, and contact geometry (with D. Thurston), Proceedings of Gökova Geometry-Topology Conference 2008, 120--136 (Gökova Geometry-Topology Conference (GGT), Gökova, 2009).
• Rational Symplectic Field Theory for Legendrian knots, Invent. Math. 182 (2010), no. 3, 451--512.
• A family of transversely nonsimple knots (with T. Khandhawit), Algebr. Geom. Topol. 10 (2010), no. 1, 293--314.
• A skein approach to Bennequin type inequalities, Int. Math. Res. Not. 2008, Art. ID rnn116, 18 pp.
• Transverse knots distinguished by knot Floer homology (with P. Ozsváth and D. Thurston), J. Symplectic Geom. 6 (2008), no. 4, 461--490.
• On arc index and maximal Thurston-Bennequin number, J. Knot Theory Ramifications 21 (2012), no. 4, 1250031, 11 pp.
• A Legendrian Thurston-Bennequin bound from Khovanov homology, Algebr. Geom. Topol. 5 (2005), 1637--1653.
• The correspondence between augmentations and rulings for Legendrian knots (with J. Sabloff), Pacific J. Math. 224 (2006), no. 1, 141--150.
• Plane curves and contact geometry, Proceedings of Gökova Geometry-Topology Conference 2005, 162--171 (Gökova Geometry-Topology Conference (GGT), Gökova, 2006).
• Conormal bundles, contact homology, and knot invariants, in The interaction of finite type and Gromov-Witten invariants at the Banff International Research Station (2003), Geom. Topol. Monogr. 8 (2006), 33--48.
• Framed knot contact homology, Duke Math. J. 141 (2008), no. 2, 365--406.
• Legendrian solid-torus links (with L. Traynor), J. Symplectic Geom. 2 (2005), no. 3, 411--443.
• Knot and braid invariants from contact homology II, Geom. Topol. 9 (2005), 1603--1637.
• Knot and braid invariants from contact homology I, Geom. Topol. 9 (2005), 247--297.
• Problems in low dimensional contact geometry (with J. Etnyre), in Topology and Geometry of Manifolds, Proc. Sympos. Pure Math. 71 (2003), 337--357.
• Invariants of Legendrian links and coherent orientations (with J. Etnyre and J. Sabloff), J. Symplectic Geom. 1 (2002), no. 2, 321--367.
• Computable Legendrian invariants, Topology 42 (2003), no. 1, 55--82.
• Maximal Thurston-Bennequin number of two-bridge links, Algebr. Geom. Topol. 1 (2001), 427--434.
• The rook on the half-chessboard, or how not to diagonalize a matrix (with K. Kedlaya), Amer. Math. Monthly 105 (1998), no. 9, 819--824.
• Hamiltonian decomposition of lexicographic products of digraphs, J. Combin. Theory Ser. B 73 (1998), no. 2, 119--129.
• Hamiltonian decomposition of complete regular multipartite digraphs, Discrete Math. 177 (1997), no. 1-3, 279--285.
• k-ordered hamiltonian graphs (with M. Schultz), J. Graph Theory 24 (1997), no. 1, 45--57.

Selected invited talks

Invited addresses, lecture series, minicourses, and colloquia

• Minicourse “Conormal bundles, knot invariants, and topological strings”, SwissMAP Geometry & Topology Conference, Engelberg, Switzerland, January 2015.
• Colloquium, Princeton University, December 2014.
• Minicourse “Grid diagrams and contact geometry”, Combinatorial Link Homology Theories, Braids, and Contact Geometry, ICERM, August 2014.
  
• Colloquium, Northwestern University, May 2014.
  
• Colloquium, University of California at Santa Barbara, November 2013.
• Colloquium, Australian National University, June 2013.
• Colloquium, University of Southern California, October 2012.
• Invited Address “From holomorphic curves to knot invariants via the cotangent bundle”, American Mathematical Society Southeastern Section Meeting, Tulane University, October 2012.
• Minicourse “Knot contact homology and applications”, Contact and Symplectic Topology Summer School and Conference, Alfréd Rényi Institute of Mathematics, Budapest, July 2012.
• Minicourse “Knots and the topology of three-manifolds”, conference “O Gosto pela Matematica”, Fundaçao Calouste Gulbenkian, Lisbon, July 2010.
• Minicourse “Legendrian knots”, Introductory Workshop: Homology Theories of Knots and Links, Mathematical Sciences Research Institute, January 2010.
• Minicourse “Contact geometry and three-dimensional topology”, Summer Graduate Workshop: Symplectic and Contact Geometry and Topology, Mathematical Sciences Research Institute, August 2009.
• Minicourse “Legendrian Symplectic Field Theory”, Holomorphic Curves: Algebraic Structures and Geometric Application, Stanford University, August 2008.
• Colloquium, Dartmouth College, May 2008.
• Colloquium, University of North Carolina at Chapel Hill, November 2006.
• Salomon Bochner Lectures in Mathematics, Rice University, October 2006.
• Minicourse “Invariants of knots and links via SFT”, Workshop on Symplectic Field Theory, Universität Leipzig, August 2006.
• Colloquium, University of California at Berkeley, February 2006.
• Colloquium, University of Illinois at Urbana-Champaign, January 2006.
• Colloquium, University of Illinois at Chicago, January 2006.
• Colloquium, Rutgers University, January 2006.
• Colloquium, University of Toronto, January 2006.
• Colloquium, University of Wisconsin, November 2005.
• Colloquium, Columbia University, September 2005.
• Minicourse, Courbes Holomorphes et Topologie de Contact, summer school sponsored by CNRS, Berder, France, June 2003.

Conference talks

• Quantum curves and quantum knot invariants, Banff International Research Station, June 2014.
• Low-dimensional topology after Floer, Université de Montréal, July 2013.
• Low dimensional topology, Simons Center for Geometry and Physics, Stony Brook, May 2013.
• Southern California Topology Colloquium, Caltech, March 2013.
• Interactions of gauge theory with contact and symplectic geometry in dimensions 3 and 4, Banff International Research Station, March 2013.
• Graduate Workshop on Symplectic and Contact Topology (2 talks), Simons Center for Geometry and Physics, Stony Brook, October 2012.
• Conference on Holomorphic Curves and Low Dimensional Topology, Stanford University, August 2012.
• Tech Topology Conference, Georgia Institute of Technology, December 2011.
• Conference on contact and symplectic topology, Université de Nantes, June 2011.
• Interactions between contact & symplectic topology and gauge theory in dimensions 3 and 4, Banff International Research Station, March 2011.
• Categorification and Low Dimensional Topology, Stony Brook University, June 2010.
• Knots, Contact Geometry and Floer Homology, University of Tokyo, May 2010.
• Introductory Workshop: Symplectic and Contact Geometry and Topology, Mathematical Sciences Research Institute, August 2009.
• Workshop on Mirror Symmetry, Symplectic Geometry, and Related Topics, MIT, June 2009.
• Georgia International Topology Conference, University of Georgia, May 2009.
• Interactions of Geometry and Topology in Dimensions 3 and 4, Banff International Research Station, March 2009.
• Illinois/Indiana Symplectic Geometry Conference, University of Illinois at Urbana-Champaign, March 2009.
• AMS special session on categorification and link homology, Joint Mathematics Meetings, Washington, DC, January 2009.
• Math Institutes Modern Mathematics Workshop, Society for Advancement of Chicanos and Native Americans in Science annual conference, Salt Lake City, October 2008.
• Legendrian and Transverse Knots, American Institute of Mathematics, September 2008.
• Gökova Geometry-Topology Conference, Gökova, Turkey, May 2008.
• Knots in Washington XXVI, George Washington University, April 2008.
• Towards Relative Symplectic Field Theory, CUNY, September 2007.
• Communicating Mathematics, University of Minnesota at Duluth, July 2007.
• Interactions of Geometry and Topology in Low Dimensions, Banff International Research Station, March 2007.
• Low Dimensional Topology, Research Program, Park City Mathematics Institute, July 2006.
• Conference on 3-manifold Topology in honor of Peter Shalen, Université de Montréal, June 2006.
• Workshop “Around Khovanov homology”, Université du Québec à Montréal, October 2005.
• Holomorphic Curves workshop, Institute for Advanced Study, June 2005.
• Gökova Geometry/Topology Conference, Gökova, Turkey, June 2005.
• The Interaction of Finite Type and Gromov-Witten Invariants, Banff International Research Station, November 2003.
• Holomorphic Curves in Contact Geometry, American Institute of Mathematics, August 2003.
• Symplectic Geometry and Physics Workshop, Institute for Pure and Applied Mathematics, UCLA, March 2003.
• Holomorphic Curves and Low Dimensional Topology, Institute for Advanced Study, March 2002.
• Georgia International Topology Conference, University of Georgia, May 2001.
• Conference, Stanford University/American Institute of Mathematics program in contact geometry, Stanford University, December 2000.
• AMS sectional meeting special sessions: Georgetown University, March 2015; Temple University, October 2013; Wake Forest University, September 2011; UCLA, October 2010; North Carolina State University, April 2009; Courant Institute, April 2003; Georgia Institute of Technology, March 2002.

Seminar talks

• 2014: University at Buffalo, Columbia University, Uppsala University, Princeton University.
  
• 2013: Australian National University, UC Santa Barbara, MIT.
• 2011: MIT, Columbia University, Haverford College, Simons Center for Geometry and Physics (Stony Brook).
• 2010: Princeton University.
• 2009: ETH Zürich (2).
• 2008: Stanford University, MIT.
• 2007: Columbia University (2), Princeton University, SUNY Stony Brook, University of Virginia, MIT, Harvard University.
• 2006: University of Texas, Duke University.
• 2005: Bryn Mawr College/Haverford College, University of California-Berkeley (2), Columbia University (2), Georgia Institute of Technology, University of Wisconsin, University of Southern California, University of Illinois at Urbana-Champaign.
• 2004: University of Southern California/California Institute of Technology, Stanford University.
• 2003: University of California-Berkeley, Stanford University, University of Pennsylvania, Harvard University, Haverford College.
• 2002: Columbia University, Stanford University.
• 2001: University of North Carolina, Princeton University.
• 2000: Stanford University, Harvard University.

Service and miscellaneous

• Organizer for the program “Symplectic Geometry and Topology”, Institut Mittag-Leffler, Fall 2015.
  
• Organizer for the conferences: “Cyclic Homology and Symplectic Topology”, American Institute of Mathematics, November 2009; “Algebraic Structures in the Theory of Holomorphic Curves”, Mathematical Sciences Research Institute, November 2009; “27th Annual Geometry Festival”, Duke University, April 2012.
• Organizer for the special sessions: “Low dimensional topology and geometry”, AMS Southeastern Sectional Meeting, North Carolina State University, April 2009; “Interactions of geometry and topology in low dimensions”, AMS Southeastern Sectional Meeting, Tulane Unversity, October 2012.
• Editor, Quantum Topology.
• Reviewed for: NSF grant panel (2); Mathematical Reviews; Advances in Mathematics, Algebraic & Geometric Topology, Communications in Contemporary Mathematics, Compositio Mathematica, Discrete Mathematics, Duke Mathematical Journal, Experimental Mathematics, Geometry & Topology, International Journal of Mathematics, International Mathematics Research Notices, Involve, Journal of Differential Geometry, Journal of Graph Theory, Journal of Knot Theory and Its Ramifications, Journal of Symplectic Geometry, Journal of the European Mathematical Society, Journal of Topology, Proceedings of the National Academy of Sciences, Rose-Hulman Undergraduate Mathematics Journal, Topology and Its Applications, Transactions of the American Mathematical Society.
• Member, Subcommittee for the United States Mathematical Olympiad, Mathematical Association of America, 2009-2014.
• Member, AMS Southeastern Section Program Committee, American Mathematical Society, 2014-present.
  
• Instructor for Berkeley Math Circle (for high school students), 2010; San José Math Circle (for middle school students), 2006; Stanford Math Circle (for high school students), 2005.
• Contributor to the online archive of solutions for the Putnam Mathematical Competition, 1997-present.