Lenhard Lee Ng
Eads Family Professor
My research mainly focuses on symplectic topology and low-dimensional topology. I am interested in studying structures in symplectic and contact geometry (Weinstein manifolds, contact manifolds, Legendrian and transverse knots), especially through holomorphic-curve 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, low-dimensional topology, knot theory, topological string theory.
Office Hours
Spring 2019 (tentative):
- Mondays 2:00-3:00
- Thursdays 2:45-3:45
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 27708-0320
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ng@math.duke.edu
(919) 660-6972
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http://www.math.duke.edu/~ng/
- Background
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Education, Training, & Certifications
- Ph.D., Massachusetts Institute of Technology 2001
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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
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In the News
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NOV 8, 2018 Duke Today -
APR 18, 2017
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Awards & Honors
- Expertise
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Subject Headings
- Research
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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 Low-Dimensional Topology awarded by National Science Foundation 2009 - 2015
- Holomorphic Curves and Low-Dimensional Topology awarded by National Science Foundation 2007 - 2010
- Publications & Artistic Works
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Selected Publications
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Academic Articles
- Ekholm, T, Ng, L, and Shende, V. "A complete knot invariant from contact homology." Inventiones Mathematicae 211, no. 3 (March 1, 2018): 1149-1200. 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): 661-780. Full Text Open Access Copy
- 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): 1845-1874. Full Text Open Access Copy
- Cornwell, C, Ng, L, and Sivek, S. "Obstructions to Lagrangian concordance." Algebraic & Geometric Topology 16, no. 2 (April 26, 2016): 797-824. Full Text Open Access Copy
- Ekholm, T, and Ng, L. "Legendrian contact homology in the boundary of a subcritical weinstein 4-Manifold." Journal of Differential Geometry 101, no. 1 (September 1, 2015): 67-157. Full Text Open Access Copy
- Lipshitz, R, Ng, L, and Sarkar, S. "On transverse invariants from Khovanov homology." Quantum Topology 6, no. 3 (January 1, 2015): 475-513. Full Text Open Access Copy
- Aganagic, M, Ekholm, T, Ng, L, and Vafa, C. "Topological strings, D-model, and knot contact homology." Advances in Theoretical and Mathematical Physics 18, no. 4 (January 1, 2014): 827-956. Full Text Open Access Copy
- Ng, L. "A topological introduction to knot contact homology." Bolyai Society Mathematical Studies 26 (January 1, 2014): 485-530. Full Text Open Access Copy
- Ng, L, and Rutherford, D. "Satellites of Legendrian knots and representations of the Chekanov-Eliashberg Algebra." Algebraic and Geometric Topology 13, no. 5 (August 1, 2013): 3047-3097. Full Text Open Access Copy
- Ekholm, T, Etnyre, JB, Ng, L, and Sullivan, MG. "Knot contact homology." Geometry and Topology 17, no. 2 (May 13, 2013): 975-1112. Full Text Open Access Copy
- Chongchitmate, W, and Ng, L. "An atlas of Legendrian knots." Experimental Mathematics 22, no. 1 (January 1, 2013): 26-37. 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 (January 1, 2013): 1561-1591. 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): 1561-1591. Full Text
- Etnyre, J, Ng, L, and Vértesi, V. "Legendrian and transverse twist knots." Journal of the European Mathematical Society 15, no. 3 (2013): 969-995. Full Text
- Ng, L. "On arc index and maximal Thurston-Bennequin number." Journal of Knot Theory and Its Ramifications 21, no. 4 (2012). Open Access Copy
- Ng, L. "Combinatorial knot contact homology and transverse knots." Advances in Mathematics 227, no. 6 (August 20, 2011): 2189-2219. Full Text
- Ng, L. "Rational symplectic field theory for Legendrian knots." Inventiones Mathematicae 182, no. 3 (December 1, 2010): 451-512. Full Text
- Khandhawit, T, and Ng, L. "A family of transversely nonsimple knots." Algebraic and Geometric Topology 10, no. 1 (July 28, 2010): 293-314. Full Text
- Lenhard, NG. "A skein approach to bennequin-type inequalities." International Mathematics Research Notices 2008, no. 1 (December 1, 2008). Full Text
- Ng, L. "Framed knot contact homology." Duke Mathematical Journal 141, no. 2 (February 2008): 365-406. Full Text
- Ng, L, Ozsváth, P, and Thurston, D. "Transverse knots distinguished by knot Floer homology." Journal of Symplectic Geometry 6, no. 4 (2008): 461-490. Full Text
- Ng, L. "Conormal bundles, contact homology and knot invariants." The Interaction of Finite Type and Gromov Witten Invariants (Birs 2003) (April 22, 2006). Full Text
- NG, LL, and Sabloff, JM. "The correspondence between augmentations and rulings for legendrian knots." Pacific Journal of Mathematics 224, no. 1 (March 1, 2006): 141-150. 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 (January 1, 2006): 307-324. Full Text
- Ng, L. "A Legendrian Thurston–Bennequin bound from Khovanov homology." Algebraic & Geometric Topology 5, no. 4 (November 25, 2005): 1637-1653. Full Text
- Lenhard, NG, and Gadgil, S. "Knot and braid invariants from contact homology II." Geometry and Topology 9 (August 26, 2005).
- Ng, L. "Knot and braid invariants from contact homology I." Geometry and Topology 9 (January 26, 2005): 247-297. Full Text
- Ng, L, and Traynor, L. "Legendrian solid-torus links." Journal of Symplectic Geometry 2, no. 3 (2005): 411-443. (Academic Article)
- Ng, LL. "Computable Legendrian invariants." Topology 42, no. 1 (January 1, 2003): 55-82. (Review) Full Text
- Ng, L, Etnyre, J, and Sabloff, J. "Invariants of Legendrian links and coherent orientations." J. Symplectic Geom. 1, no. 2 (2002): 321-367. (Academic Article)
- Ng, L. "Maximal Thurston–Bennequin number of two-bridge links." Algebraic & Geometric Topology 1, no. 1 (July 31, 2001): 427-434. Full Text
- Ng, LL. "Hamiltonian Decomposition of Lexicographic Products of Digraphs." Journal of Combinatorial Theory, Series B 73, no. 2 (July 1998): 119-129. Full Text
- Kedlaya, KS, and Ng, LL. "The rook on the half-chessboard, or how not to diagonalize a matrix." American Mathematical Monthly 105, no. 9 (January 1, 1998): 819-824. Full Text
- Ng, LL. "Hamiltonian decomposition of complete regular multipartite digraphs." Discrete Mathematics 177, no. 1-3 (December 1, 1997): 279-285. Full Text
- Ng, L, and Schultz, M. "k-Ordered Hamiltonian Graphs." Journal of Graph Theory 24, no. 1 (January 1, 1997): 45-57. Full Text
- Chantraine, B, Ng, L, and Sivek, S. "Representations, sheaves, and Legendrian $(2,m)$ torus links(Published online)." Journal of the London Mathematical Society . Full Text Open Access Copy Link to Item
- Ekholm, T, and Ng, L. "Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials(Submitted)." . Open Access Copy Link to Item
- Etnyre, JB, and Ng, LL. "Legendrian contact homology in R^3(Submitted)." . Open Access Copy 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 Geometry Topology Conference 2008, 120 136 . Link to Item
- Ng, L. "Plane curves and contact geometry." Proceedings of G\~A{\P}Kova Geometry Topology Conference 2005 (2006), 165 174 . Link to Item
- Ng, LL. "The Legendrian satellite construction." Appears as Part Of"Legendrian Solid Torus Links", J. Symplectic Geom. 2 (2005), No. 3, 411 443 . Link to Item
- Ng, LL. "Legendrian mirrors and Legendrian isotopy." . Link to Item
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- Teaching & Mentoring
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Recent Courses
- MATH 112L: Laboratory Calculus II 2019
- MATH 393: Research Independent Study 2019
- MATH 611: Algebraic Topology I 2019
- MATH 612: Algebraic Topology II 2019
- MATH 89S: First-Year Seminar 2018
- MATH 621: Differential Geometry 2018
- MATH 790-90: Minicourse in Advanced Topics 2018
- MATH 391: Independent Study 2017
- MATH 411: Topology 2017
- MATH 431: Advanced Calculus I 2017
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Advising & Mentoring
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Current Ph.D. advisees:
Chester Lian
Orsola Capovilla-Searle
Dmitry Vagner (co-advising with Ezra Miller)
Current postdoctoral fellows mentored:
Ziva Myer, Elliot Assistant Research Professor
Faramarz Vafaee, Griffiths Assistant Research Professor
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Current Ph.D. advisees:
- Scholarly, Clinical, & Service Activities
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Presentations & Appearances
- Augmentations and a contact Fukaya category. Mini-workshop on Legendrian submanifolds. Université du Québec à Montréal. November 10, 2018 2018
- A contact Fukaya category. Seminar in Topology and Symplectic Geometry. Stony Brook University. October 11, 2018 2018
- Studying knots through the cotangent bundle and symplectic geometry. Colloquium. Stony Brook University. October 11, 2018 2018
- A contact Fukaya category. Joint GT-UGA Topology Seminar. Georgia Tech. October 1, 2018 2018
- Knot contact homology and colored HOMFLY recursion. Geometric methods in symplectic and contact topology. Stanford University. August 21, 2018 2018
- Conormals and knot invariants. Symplectic geometry and its interactions with low-dimensional topology mini-conference. Princeton University. June 28, 2018 2018
- Knot contact homology and colored HOMFLY-PT recursion. Categorification in Mathematical Physics conference. Simons Center for Geometry and Physics. April 11, 2018 2018
- Knot contact homology, string topology, and topological strings (minicourse). Mirror Symmetry and Related Topics conference. University of Miami. January 29, 2018 - January 31, 2018 2018
- 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 Mittag-Leffler. 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 Mittag-Leffler. 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 "Low-dimensional 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 Low-Dimensional 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
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Service to the Profession
- Co-organizer. Fourth Duke Mathematical Journal conference. Mathematics Department. April 2018 2018
- Collaborating editor. American Mathematical Monthly problems section. Mathematical Association of America. 2017 - 2019 2017 - 2019
- Special session organizer, "Low dimensional topology". American Mathematical Society Southeastern Section meeting. November 12, 2016 - November 13, 2016 2016
- Organizer. Triangle Topology Seminar. 2016 - 2019 2016 - 2019
- Conference organizer. Symplectic Geometry and Topology. Uppsala University. September 21, 2015 - September 25, 2015 2015
- Program organizer. Symplectic Geometry and Topology. Institut Mittag-Leffler. 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 - 2019 2009 - 2019
- MAA Subcommittee for the United States Mathematical Olympiad. 2008 - 2014 2008 - 2014
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Service to Duke
- Duke University. Vice Provost for Undergraduate Education search committee. 2018 - 2018 2018
- Co-director. Duke Opportunities in Mathematics (DOmath). Mathematics Department, Duke University. 2017 - 2019 2017 - 2019
- Duke University. Arts & Sciences Research Grants Committee. 2017 - 2018 2017 - 2018
- Duke University. Provost's Advisory Committee on Bass Professorships. 2017 - 2019 2017 - 2019
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