Curriculum Vitæ: Lenhard Ng

As of November 2012. 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 (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: NSF CAREER grant DMS-0846346 “Symplectic Field Theory and Low-Dimensional Topology”, 2009-2014; Principal Investigator, research grant DMS-0706777 “Holomorphic Curves and Low-Dimensional Topology”, 2007-2010.
• American Institute of Mathematics: Five-Year Fellowship (one fellow named per year), 2001-2006.
• Department of Defense: National Defense Science and Engineering Graduate (NDSEG) fellow, 1996-1999.
• Fannie and John Hertz Foundation: selected as fellow, 1996.
• MIT Department of Mathematics: selected for Norman Levinson Fellowship, 1996.
• 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.
• Westinghouse Science Talent Search: third place, 1993.

Publications

• Topological strings, D-model, and knot contact homology (with M. Aganagic, T. Ekholm, and C. Vafa), preprint, 2013, arXiv:1304.5778.
• On transverse invariants from Khovanov homology (with R. Lipshitz and S. Sarkar), preprint, 2013, arXiv:1303.6371.
• A topological introduction to knot contact homology, preprint, 2012, arXiv:1210.4803.
• Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra (with D. Rutherford), Algebr. Geom. Topol., to appear.
• 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

Lecture series, minicourses, and colloquia

• 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

• 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.
• Special session on low-dimensional topology and geometry, AMS Southeastern Section Meeting, Wake Forest University, September 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.
• Special session on topology and symplectic geometry, AMS Western Section Meeting, UCLA, October 2010.
• 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.
• Special session on low dimensional topology and geometry, AMS Southeastern Sectional Meeting, North Carolina State University, April 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.
• Session on contact and symplectic geometry, AMS Eastern Sectional Meeting, Courant Institute, April 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.
• Session on symplectic and contact topology, AMS Southeastern Sectional Meeting, Georgia Institute of Technology, 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.

Seminar talks

• 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 workshops: “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.
• Organizer for the special session “Low dimensional topology and geometry”, AMS Southeastern Sectional Meeting, North Carolina State University, April 2009.
• Editor, Quantum Topology.
• Reviewed for: Mathematical Reviews; NSF grant panel; Algebraic & Geometric Topology, Communications in Contemporary Mathematics, Discrete Mathematics, Duke Mathematical Journal, Experimental Mathematics, Geometry & Topology, International Journal of Mathematics, International Mathematics Research Notices, Journal of Differential Geometry, Journal of the European Mathematical Society, Journal of Graph Theory, Journal of Knot Theory and its Ramifications, 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, 2009-2011.
• 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.
• Guest lecturer at Canada/USA MathCamp (for high school students), 1997.
• Conducted several undergraduate-level lectures as a staff member at the US Mathematical Olympiad Summer Program, 1994 and 1995; helped maintain an online archive of solutions for the Putnam Mathematical Competition (since 1997).