Lenhard Lee Ng
Eads Family Professor
My research mainly focuses on symplectic topology and lowdimensional topology. I am interested in studying structures in symplectic and contact geometry (Weinstein manifolds, contact manifolds, Legendrian and transverse knots), especially through holomorphiccurve techniques. One particular interest is extracting topological information about knots through cotangent bundles, and exploring relations to topological string theory. I have also worked in Heegaard Floer theory, quantum topology, and sheaf theory, especially as they relate to Legendrian and transverse knots.
Current Research Interests
Symplectic geometry, contact geometry, lowdimensional topology, knot theory, topological string theory.
Office Hours
Fall 2018:
 Mondays 2:003:00
 Fridays 3:004:00
Current Appointments & Affiliations
 Eads Family Professor, Mathematics, Trinity College of Arts & Sciences 2016  2021
 Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2016
Contact Information
 216 Physics Bldg, 120 Science Drive, Durham, NC 27708
 Box 90320, Durham, NC 277080320
 ng@math.duke.edu (919) 6606972
 http://www.math.duke.edu/~ng/
 Background

Education, Training, & Certifications
 Ph.D., Massachusetts Institute of Technology 2001

Duke Appointment History
 Associate Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2011  2016
 Assistant Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2006  2011
 Recognition

In the News

Awards & Honors
 Expertise

Subject Headings
 Research

Selected Grants
 Holomorphic Invariants in Symplectic Topology awarded by National Science Foundation 2017  2020
 Knots and contact topology through holomorphic curves awarded by National Science Foundation 2014  2018
 Simons Foundation Fellowship awarded by Simons Foundation 2015  2016
 CAREER: Symplectic Field Theory and LowDimensional Topology awarded by National Science Foundation 2009  2015
 Holomorphic Curves and LowDimensional Topology awarded by National Science Foundation 2007  2010
 Publications & Artistic Works

Selected Publications

Journal Articles
 Ekholm, T, Ng, L, and Shende, V. "A complete knot invariant from contact homology." Inventiones Mathematicae 211, no. 3 (March 1, 2018): 11491200. Full Text Open Access Copy
 Cieliebak, K, Ekholm, T, Latschev, J, and Ng, L. "Knot contact homology, string topology, and the cord algebra." Journal de l'Ecole Polytechnique  Mathematiques 4 (January 1, 2017): 661780. Full Text
 Ng, L, Rutherford, D, Shende, V, and Sivek, S. "The cardinality of the augmentation category of a Legendrian link." Mathematical Research Letters 24, no. 6 (2017): 18451874. Link to Item
 Cornwell, C, Ng, L, and Sivek, S. "Obstructions to Lagrangian concordance." Algebraic & Geometric Topology 16, no. 2 (April 26, 2016): 797824. Full Text
 Ekholm, T, and Ng, L. "Legendrian contact homology in the boundary of a subcritical Weinstein 4manifold." Journal of Differential Geometry 101, no. 1 (September 2015): 67157. Full Text
 Lipshitz, R, Ng, L, and Sarkar, S. "On transverse invariants from Khovanov homology." Quantum Topology 6, no. 3 (January 1, 2015): 475513. Full Text
 Aganagic, M, Ekholm, T, Ng, L, and Vafa, C. "Topological strings, Dmodel, and knot contact homology." Advances in Theoretical and Mathematical Physics 18, no. 4 (January 1, 2014): 827956. Full Text
 Ng, L. "A topological introduction to knot contact homology." Bolyai Society Mathematical Studies 26 (January 1, 2014): 485530. Full Text
 Ekholm, T, Etnyre, J, Ng, L, and Sullivan, M. "Filtrations on the knot contact homology of transverse knots." Mathematische Annalen 355, no. 4 (2013): 15611591. Full Text
 Ekholm, T, Etnyre, JB, Ng, L, and Sullivan, MG. "Knot contact homology." Geometry and Topology 17, no. 2 (2013): 9751112. Full Text
 Ng, L, and Rutherford, D. "Satellites of Legendrian knots and representations of the ChekanovEliashberg Algebra." Algebraic and Geometric Topology 13, no. 5 (2013): 30473097. Full Text
 Ekholm, T, Etnyre, J, Ng, L, and Sullivan, M. "Filtrations on the knot contact homology of transverse knots." Mathematische Annalen (2012): 131. Full Text
 Ng, L. "On arc index and maximal ThurstonBennequin number." Journal of Knot Theory and Its Ramifications 21, no. 4 (2012).
 Chongchitmate, W, and Ng, L. "An atlas of Legendrian knots (Accepted)." Experimental Mathematics (2011). (Academic Article)
 Etnyre, J, Ng, L, and Vertesi, V. "Legendrian and transverse twist knots (Accepted)." Journal of the European Mathematical Society (2011). (Academic Article)
 Ng, L. "Combinatorial knot contact homology and transverse knots." Advances in Mathematics 227, no. 6 (2011): 21892219. Full Text
 Khandhawit, T, and Ng, L. "A family of transversely nonsimple knots." Algebraic and Geometric Topology 10, no. 1 (2010): 293314. Full Text
 Ng, L. "Rational symplectic field theory for Legendrian knots." Inventiones Mathematicae 182, no. 3 (2010): 451512. Full Text
 Lenhard, NG. "A skein approach to bennequintype inequalities." International Mathematics Research Notices 2008, no. 1 (2008). Full Text
 Ng, L, Ozsvath, P, and Thurston, D. "Transverse knots distinguished by knot Floer homology." Journal of Symplectic Geometry 6, no. 4 (2008): 461490. (Academic Article)
 Ng, L. "Framed knot contact homology." Duke Mathematical Journal 141, no. 2 (2008): 365406. Full Text
 Muratori, MC, Stanley, JC, Ng, L, Ng, J, Gross, MUM, Tao, T, and Tao, B. "Insights from SMPY's greatest former child prodigies: Drs. Terence ("Terry") Tao and Lenhard ("Lenny") Ng reflect on their talent development." Gifted Child Quarterly 50, no. 4 (2006): 307324.
 NG, LL, and Sabloff, JM. "The correspondence between augmentations and rulings for legendrian knots." Pacific Journal of Mathematics 224, no. 1 (2006): 141150. Full Text
 Lenhard, NG, and Gadgil, S. "Knot and braid invariants from contact homology II." Geometry and Topology 9 (2005).
 Ng, L, and Traynor, L. "Legendrian solidtorus links." Journal of Symplectic Geometry 2, no. 3 (2005): 411443. (Academic Article)
 Ng, L. "Knot and braid invariants from contact homology I." Geometry and Topology 9 (2005): 247297. Full Text
 Ng, L. "A Legendrian Thurston–Bennequin bound from Khovanov homology." Algebraic & Geometric Topology 5 (2005): 16371653. (Academic Article)
 Ng, LL. "Computable Legendrian invariants." Topology 42, no. 1 (2003): 5582. Full Text
 Ng, L, Etnyre, J, and Sabloff, J. "Invariants of Legendrian links and coherent orientations." J. Symplectic Geom. 1, no. 2 (2002): 321367. (Academic Article)
 Ng, L. "Maximal Thurston–Bennequin number of twobridge links." Algebr. Geom. Topol. 1 (2001): 427434. (Academic Article)
 Kedlaya, KS, and Ng, LL. "The rook on the halfchessboard, or how not to diagonalize a matrix." American Mathematical Monthly 105, no. 9 (1998): 819824.
 Ng, LL. "Hamiltonian Decomposition of Lexicographic Products of Digraphs." Journal of Combinatorial Theory. Series B 73, no. 2 (1998): 119129. Full Text
 Ng, L, and Schultz, M. "kOrdered Hamiltonian Graphs." Journal of Graph Theory 24, no. 1 (1997): 4557.
 Ng, LL. "Hamiltonian decomposition of complete regular multipartite digraphs." Discrete Mathematics 177, no. 13 (1997): 279285.
 Chantraine, B, Ng, L, and Sivek, S. "Representations, sheaves, and Legendrian $(2,m)$ torus links." . Link to Item
 Ekholm, T, and Ng, L. "Higher genus knot contact homology and recursion for colored HOMFLYPT polynomials." Link to Item
 Etnyre, JB, and Ng, LL. "Legendrian contact homology in $\mathbb{R}^3$." . Link to Item
 Ng, L, Rutherford, D, Shende, V, Sivek, S, and Zaslow, E. "Augmentations are Sheaves (Submitted)." Open Access Copy Link to Item
 Ng, L, and Thurston, D. "Grid Diagrams, Braids, and Contact Geometry." Proceedings of Gokova GeometryTopology Conference 2008, 120136 . Link to Item
 Ng, L. "Plane curves and contact geometry." Proceedings of G\~A{\P}kova GeometryTopology Conference 2005 (2006), 165174 . Link to Item
 Ng, L. "Conormal bundles, contact homology and knot invariants." Geom. Topol. Monogr. 8: 129144. Full Text Link to Item
 Ng, LL. "The Legendrian satellite construction." Appears as part of"Legendrian solidtorus links", J. Symplectic Geom. 2 (2005), No. 3, 411443 . Link to Item
 Ng, LL. "Legendrian mirrors and Legendrian isotopy." Link to Item

 Teaching & Mentoring

Recent Courses

Advising & Mentoring

Current Ph.D. advisees:
Chester Lian
Orsola CapovillaSearle

Current Ph.D. advisees:
 Scholarly, Clinical, & Service Activities

Presentations & Appearances
 Recent developments in knot contact homology. Princeton/IAS Symplectic Geometry Seminar. Institute for Advanced Study. December 11, 2017 2017
 The Legendrian conormal as a complete knot invariant. Symplectic geometry, gauge theory, and categorification seminar. Department of Mathematics. October 27, 2017 2017
 Legendrian contact homology and a variety of augmentations. Uppsala Summer Geometry School. Uppsala University. August 14, 2017  August 18, 2017 2017
 Recent developments in knot contact homology. Conference "Holomorphic curves and symplectic topology". Institut MittagLeffler. August 10, 2017 2017
 How knots are captured by the symplectic geometry of the cotangent bundle. Colloquium. Department of Mathematics. April 27, 2017 2017
 Studying knots through symplectic geometry and cotangent bundles. Colloquium. Department of Mathematics. February 17, 2017 2017
 Workshop "Homological mirror symmetry: methods and structures". Institute for Advanced Study. November 7, 2016 2016
 Minicourse "Knot contact homology and augmentation varieties", William Rowan Hamilton Geometry and Topology Workshop. Trinity College Dublin. August 22, 2016  August 23, 2016 2016
 Conference "Knots in the Triangle". NC State University. May 1, 2016 2016
 Topology/Geometry Seminar. University of Oregon. April 19, 2016 2016
 Colloquium. University of Oregon. April 18, 2016 2016
 Conference "Interactions of Gauge Theory with Contact and Symplectic Topology in Dimensions 3 and 4". Banff International Research Station. March 23, 2016 2016
 Topology Seminar. Princeton University. February 25, 2016 2016
 Symplectic geometry, gauge theory, and categorification seminar. Columbia University. February 19, 2016 2016
 Workshop "Augmentations and Legendrians at the IAS". Institute for Advanced Study. February 11, 2016 2016
 Colloquium. University of California, San Diego. January 21, 2016 2016
 Colloquium. University of Toronto. January 21, 2016 2016
 Colloquium. Rice University. December 3, 2015 2015
 Colloquium. University of Virginia. November 30, 2015 2015
 Symplectic Geometry Seminar. Institut MittagLeffler. October 29, 2015 2015
 Conference "Recent challenges in contact geometry". June 9, 2015  June 10, 2015 2015
 Conference "Physics and mathematics of knot homologies". June 1, 2015 2015
 Colloquium, Princeton University. December 10, 2014 2014
 Conference "Quantum Curves and Quantum Knot Invariants". June 19, 2014 2014
 Geometry and Physics Seminar, Uppsala University. May 22, 2014 2014
 Colloquium, Northwestern University. May 7, 2014 2014
 Symplectic Geometry, Gauge Theory, and Categorification Seminar, Columbia University. May 2, 2014 2014
 Geometry and Topology Seminar, University at Buffalo. April 11, 2014 2014
 Geometry and Topology Seminar, MIT. November 18, 2013 2013
 Geometry, Topology, and Physics Seminar, UCSB. November 8, 2013 2013
 Colloquium, UCSB. November 7, 2013 2013
 AMS Special Session on Contact and Symplectic Topology. October 12, 2013 2013
 Conference "Lowdimensional topology after Floer". July 10, 2013 2013
 Algebra and Topology Seminar, Australian National University. June 11, 2013 2013
 Colloquium, Australian National University. June 6, 2013 2013
 Conference "Low dimensional topology". May 21, 2013 2013
 Southern California Topology Colloquium. March 30, 2013 2013
 Conference "Interactions of gauge theory with contact and symplectic geometry in dimensions 3 and 4". March 25, 2013 2013
 Colloquium, University of Southern California. October 31, 2012 2012
 Graduate Workshop on Symplectic and Contact Topology, Simons Center. October 18, 2012 2012
 Invited Address, AMS Southeastern Section Meeting. October 14, 2012 2012
 Conference on Holomorphic Curves and Low Dimensional Topology. August 8, 2012 2012
 Minicourse, Contact and Symplectic Topology Summer School and Conference. July 1, 2012 2012
 Tech Topology Conference. December 10, 2011 2011
 AMS Special Session on LowDimensional Topology and Geometry. September 24, 2011 2011
 Conference on Contact and Symplectic Topology. June 17, 2011 2011
 Seminar, Simons Center. May 9, 2011 2011
 Philadelphia Area Topology (Contact/Hyperbolic) Seminar. April 7, 2011 2011
 Conference "Interactions between contact & symplectic topology and gauge theory in dimensions 3 and 4". March 23, 2011 2011
 Geometric Topology Seminar, Columbia University. March 4, 2011 2011
 Geometry and Topology Seminar, MIT. February 7, 2011 2011
 Topology Seminar, Princeton University. November 11, 2010 2010
 AMS Special Session on Topology and Symplectic Geometry. October 10, 2010 2010
 Conference "O Gosto pela Matematica". July 1, 2010 2010
 Conference "Categorification and Low Dimensional Topology". June 24, 2010 2010
 Conference "Knots, Contact Geometry and Floer Homology". May 27, 2010 2010
 Academic Sponsors Day, Mathematical Sciences Research Institute. March 5, 2010 2010
 Minicourse, Introductory Workshop: Homology Theories of Knots and Links. January 1, 2010 2010
 Introductory Workshop: Symplectic and Contact Geometry and Topology. August 18, 2009 2009
 Minicourse, Summer Graduate Workshop: Symplectic and Contact Geometry and Topology. August 1, 2009 2009
 Workshop on Mirror Symmetry, Symplectic Geometry, and Related Topics. June 26, 2009 2009
 Georgia International Topology Conference. May 29, 2009 2009
 Topology Seminar, ETH Zurich. May 13, 2009 2009
 Symplectic Geometry Seminar, ETH Zurich. May 11, 2009 2009
 AMS Session on Low Dimensional Geometry and Topology. April 4, 2009 2009
 Conference "Interactions of Geometry and Topology in Dimensions 3 and 4". March 25, 2009 2009
 Illinois/Indiana Symplectic Geometry Conference. March 14, 2009 2009
 AMS Special Session on Categorification and Link Homology. January 6, 2009 2009
 Symplectic Geometry Seminar, MIT. December 1, 2008 2008
 Math Institutes Modern Mathematics Workshop. October 1, 2008 2008
 Symplectic Geometry Seminar, Stanford University. October 1, 2008 2008
 Conference "Legendrian and Transverse Knots". September 1, 2008 2008
 Minicourse, Conference "Holomorphic curves: Algebraic structures and geometric application". August 1, 2008 2008
 Colloquium, Dartmouth College. May 1, 2008 2008
 Gokova Geometry/Topology Conference. May 1, 2008 2008
 Conference "Knots in Washington XXVI". April 1, 2008 2008

Service to the Profession
 Coorganizer. Fourth Duke Mathematical Journal conference. Mathematics Department. April 2018 2018
 Collaborating editor. American Mathematical Monthly problems section. Mathematical Association of America. 2017 2017
 Special session organizer, "Low dimensional topology". American Mathematical Society Southeastern Section meeting. November 12, 2016  November 13, 2016 2016
 Organizer. Triangle Topology Seminar. 2016 2016
 Conference organizer. Symplectic Geometry and Topology. Uppsala University. September 21, 2015  September 25, 2015 2015
 Program organizer. Symplectic Geometry and Topology. Institut MittagLeffler. September 2015  December 2015 2015
 Organizer, Duke/UNC Topology Seminar. 2013  2014 2013  2014
 Organizer : 27th Annual Geometry Festival. April 27, 2012  April 29, 2012 2012
 Organizer, workshop "Algebraic structures in the theory of holomorphic curves". November 2009 2009
 Organizer, workshop "Cyclic homology and symplectic topology". November 2009 2009
 Editor, Quantum Topology. 2009 2009
 MAA Subcommittee for the United States Mathematical Olympiad. 2008  2014 2008  2014

Service to Duke
 Duke University. Vice Provost for Undergraduate Education search committee. 2018 2018
 Codirector. Duke Opportunities in Mathematics (DOmath). Mathematics Department, Duke University. 2017  2018 2017  2018
 Duke University. Arts & Sciences Research Grants Committee. 2017  2018 2017  2018
 Duke University. Provost's Advisory Committee on Bass Professorships. 2017  2018 2017  2018
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