Lenhard Lee Ng | Scholars@Duke

Lenhard Lee Ng
Professor of Mathematics

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 2023:

Mondays 1:00-2:30
Thursdays 11:30-12:30

and by appointment.

Current Appointments & Affiliations

Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2016

Contact Information

120 Science Drive, Durham, NC 27708
216 Physics Building, Box 90320, Durham, NC 27708-0320
ng@math.duke.edu (919) 660-6972
https://services.math.duke.edu/~ng/

Background
Education, Training, & Certifications
- Ph.D., Massachusetts Institute of Technology 2001
Previous Appointments & Affiliations
- Eads Family Professor, Mathematics, Trinity College of Arts & Sciences 2016 - 2021
- Associate Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2011 - 2016
- Assistant Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2006 - 2011
Leadership & Clinical Positions at Duke
- Director of Undergraduate Research and Prizes. Duke Mathematics Department, 2019 to present.
Recognition
In the News
- NOV 8, 2018 Duke Today
  
  Duke Professor Named Fellow of the American Mathematical Society
- APR 18, 2017
  
  Five Faculty Named Bass Fellows for Excellence in Teaching and Research
Awards & Honors
Expertise
Subject Headings
Research
Selected Grants
- Holomorphic Invariants of Knots and Contact Manifolds awarded by National Science Foundation 2020 - 2023
- Holomorphic Invariants in Symplectic Topology awarded by National Science Foundation 2017 - 2021
- 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
Selected Publications
- Academic Articles
  - Casals, R., and L. Ng. “Braid loops with infinite monodromy on the Legendrian contact DGA.” Journal of Topology 15, no. 4 (December 1, 2022): 1927–2016. https://doi.org/10.1112/topo.12264.
    Full Text Open Access Copy
  - Ekholm, T., and L. Ng. “Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials.” Advances in Theoretical and Mathematical Physics 24, no. 8 (January 1, 2020): 2067–2145. https://doi.org/10.4310/ATMP.2020.V24.N8.A3.
    Full Text Open Access Copy
  - Chantraine, Baptiste, Lenhard Ng, and Steven Sivek. “Representations, sheaves and Legendrian (2,m) torus links.” Journal of the London Mathematical Society Second Series 100, no. 1 (August 1, 2019): 41–82. https://doi.org/10.1112/jlms.12204.
    Full Text Open Access Copy Link to Item
  - Ekholm, T., L. Ng, and V. Shende. “A complete knot invariant from contact homology.” Inventiones Mathematicae 211, no. 3 (March 1, 2018): 1149–1200. https://doi.org/10.1007/s00222-017-0761-1.
    Full Text Open Access Copy
  - Edgar, G. A., D. H. Ullman, D. B. West, P. Bracken, E. A. Brown, Z. Franco, C. Friesen, et al. “Problems and solutions.” American Mathematical Monthly 125, no. 1 (January 1, 2018): 81–89. https://doi.org/10.1080/00029890.2018.1397465.
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  - Edgar, G. A., D. H. Ullman, D. B. West, P. Bracken, E. A. Brown, Z. Franco, C. Friesen, et al. “Problems and solutions.” American Mathematical Monthly 124, no. 10 (December 1, 2017): 970–78. https://doi.org/10.4169/amer.math.monthly.124.10.970.
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  - Edgar, G. A., D. H. Ullman, D. B. West, P. Bracken, E. A. Brown, Z. Franco, C. Friesen, et al. “Problems and solutions.” American Mathematical Monthly 124, no. 8 (October 1, 2017): 754–62. https://doi.org/10.4169/amer.math.monthly.124.8.754.
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  - Edgar, G. A., D. H. Ullman, D. B. West, P. Bracken, E. A. Brown, Z. Franco, C. Friesen, et al. “Problems and solutions.” American Mathematical Monthly 124, no. 5 (May 1, 2017): 465–74. https://doi.org/10.4169/amer.math.monthly.124.5.465.
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  - Cieliebak, K., T. Ekholm, J. Latschev, and L. Ng. “Knot contact homology, string topology, and the cord algebra.” Journal De L’Ecole Polytechnique Mathematiques 4 (January 1, 2017): 661–780. https://doi.org/10.5802/jep.55.
    Full Text Open Access Copy
  - Ng, Lenhard, Dan Rutherford, Vivek Shende, and Steven Sivek. “The cardinality of the augmentation category of a Legendrian link.” Mathematical Research Letters 24, no. 6 (2017): 1845–74. https://doi.org/10.4310/MRL.2017.v24.n6.a14.
    Full Text Open Access Copy Link to Item
  - “Problems and Solutions.” The American Mathematical Monthly 124, no. 4 (2017): 369–369. https://doi.org/10.4169/amer.math.monthly.124.4.369.
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  - Cornwell, Christopher, Lenhard Ng, and Steven Sivek. “Obstructions to Lagrangian concordance.” Algebraic &Amp; Geometric Topology 16, no. 2 (April 26, 2016): 797–824. https://doi.org/10.2140/agt.2016.16.797.
    Full Text Open Access Copy
  - Lipshitz, R., L. Ng, and S. Sarkar. “On transverse invariants from Khovanov homology.” Quantum Topology 6, no. 3 (September 9, 2015): 475–513. https://doi.org/10.4171/QT/69.
    Full Text Open Access Copy
  - Ekholm, T., and L. Ng. “Legendrian contact homology in the boundary of a subcritical weinstein 4-Manifold.” Journal of Differential Geometry 101, no. 1 (September 1, 2015): 67–157. https://doi.org/10.4310/jdg/1433975484.
    Full Text Open Access Copy
  - Aganagic, M., T. Ekholm, L. Ng, and C. Vafa. “Topological strings, D-model, and knot contact homology.” Advances in Theoretical and Mathematical Physics 18, no. 4 (January 1, 2014): 827–956. https://doi.org/10.4310/ATMP.2014.v18.n4.a3.
    Full Text Open Access Copy
  - Ng, L. “A topological introduction to knot contact homology.” Bolyai Society Mathematical Studies 26 (January 1, 2014): 485–530. https://doi.org/10.1007/978-3-319-02036-5_10.
    Full Text Open Access Copy
  - Ng, L., and D. Rutherford. “Satellites of Legendrian knots and representations of the Chekanov-Eliashberg Algebra.” Algebraic and Geometric Topology 13, no. 5 (August 1, 2013): 3047–97. https://doi.org/10.2140/agt.2013.13.3047.
    Full Text Open Access Copy
  - Ekholm, T., J. B. Etnyre, L. Ng, and M. G. Sullivan. “Knot contact homology.” Geometry and Topology 17, no. 2 (May 13, 2013): 975–1112. https://doi.org/10.2140/gt.2013.17.975.
    Full Text Open Access Copy
  - Ekholm, T., J. Etnyre, L. Ng, and M. Sullivan. “Filtrations on the knot contact homology of transverse knots.” Mathematische Annalen 355, no. 4 (April 1, 2013): 1561–91. https://doi.org/10.1007/s00208-012-0832-y.
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  - 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. https://doi.org/10.1016/j.aim.2011.04.014.
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  - Chongchitmate, W., and L. Ng. “An atlas of Legendrian knots (Accepted).” Experimental Mathematics, 2011.
  - Etnyre, J., L. Ng, and V. Vertesi. “Legendrian and transverse twist knots (Accepted).” Journal of the European Mathematical Society, 2011.
  - Ng, L. “Rational symplectic field theory for Legendrian knots.” Inventiones Mathematicae 182, no. 3 (December 1, 2010): 451–512. https://doi.org/10.1007/s00222-010-0265-8.
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  - Khandhawit, T., and L. Ng. “A family of transversely nonsimple knots.” Algebraic and Geometric Topology 10, no. 1 (July 28, 2010): 293–314. https://doi.org/10.2140/agt.2010.10.293.
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  - Ng, Lenhard, and Dylan Thurston. “Grid Diagrams, Braids, and Contact Geometry.” Proceedings of Gokova Geometry Topology Conference 2008, 120 136, December 19, 2008.
    Link to Item
  - Lenhard, N. G. “A skein approach to bennequin-type inequalities.” International Mathematics Research Notices 2008, no. 1 (December 1, 2008). https://doi.org/10.1093/imrn/rnn116.
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  - Ng, L. “Framed knot contact homology.” Duke Mathematical Journal 141, no. 2 (February 1, 2008): 365–406. https://doi.org/10.1215/S0012-7094-08-14125-0.
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  - Ng, L., P. Ozsvath, and D. Thurston. “Transverse knots distinguished by knot Floer homology.” Journal of Symplectic Geometry 6, no. 4 (2008): 461–90.
  - Ng, Lenhard. “Conormal bundles, contact homology and knot invariants.” Geometry and Topology Monographs, April 22, 2006. https://doi.org/10.2140/gtm.2006.8.129.
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  - NG, L. L., and J. M. Sabloff. “The correspondence between augmentations and rulings for legendrian knots.” Pacific Journal of Mathematics 224, no. 1 (March 1, 2006): 141–50. https://doi.org/10.2140/pjm.2006.224.141.
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  - Muratori, M. C., J. C. Stanley, L. Ng, J. Ng, M. U. M. Gross, T. Tao, and B. Tao. “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): 307–24. https://doi.org/10.1177/001698620605000404.
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  - Lenhard, N. G., and S. Gadgil. “Knot and braid invariants from contact homology II.” Geometry and Topology 9 (August 26, 2005).
  - Ng, Lenhard. “Plane curves and contact geometry.” Proceedings of G\~A{\P}Kova Geometry Topology Conference 2005 (2006), 165 174, March 8, 2005.
    Link to Item
  - Ng, L. “Knot and braid invariants from contact homology I.” Geometry and Topology 9 (January 26, 2005): 247–97. https://doi.org/10.2140/gt.2005.9.247.
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  - Ng, L. “A Legendrian Thurston–Bennequin bound from Khovanov homology.” Algebraic & Geometric Topology 5 (2005): 1637–53.
  - Ng, L., and L. Traynor. “Legendrian solid-torus links.” Journal of Symplectic Geometry 2, no. 3 (2005): 411–43.
  - Ng, L. L. “Computable Legendrian invariants.” Topology 42, no. 1 (January 1, 2003): 55–82. https://doi.org/10.1016/S0040-9383(02)00010-1.
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  - Ng, L., J. Etnyre, and J. Sabloff. “Invariants of Legendrian links and coherent orientations.” J. Symplectic Geom. 1, no. 2 (2002): 321–67.
  - Ng, Lenhard L. “The Legendrian satellite construction.” Appears as Part Of"Legendrian Solid Torus Links", J. Symplectic Geom. 2 (2005), No. 3, 411 443, December 11, 2001.
    Link to Item
  - Ng, L. “Maximal Thurston–Bennequin number of two-bridge links.” Algebr. Geom. Topol. 1 (2001): 427–34.
  - Ng, Lenhard L. “Legendrian mirrors and Legendrian isotopy,” August 28, 2000.
    Link to Item
  - Ng, L. L. “Hamiltonian Decomposition of Lexicographic Products of Digraphs.” Journal of Combinatorial Theory. Series B 73, no. 2 (July 1, 1998): 119–29. https://doi.org/10.1006/jctb.1998.1816.
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  - Kedlaya, K. S., and L. L. Ng. “The rook on the half-chessboard, or how not to diagonalize a matrix.” American Mathematical Monthly 105, no. 9 (January 1, 1998): 819–24. https://doi.org/10.2307/2589210.
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  - Ng, L. L. “Hamiltonian decomposition of complete regular multipartite digraphs.” Discrete Mathematics 177, no. 1–3 (December 1, 1997): 279–85. https://doi.org/10.1016/S0012-365X(97)00017-4.
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  - Ng, L., and M. Schultz. “k-Ordered Hamiltonian Graphs.” Journal of Graph Theory 24, no. 1 (January 1, 1997): 45–57. https://doi.org/10.1002/(SICI)1097-0118(199701)24:1<45::AID-JGT6>3.0.CO;2-J.
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  - Ekholm, Tobias, and Lenhard Ng. “Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials (Submitted),” n.d.
    Open Access Copy Link to Item
  - Etnyre, John B., and Lenhard L. Ng. “Legendrian contact homology in R^3 (Submitted),” n.d.
    Open Access Copy Link to Item
  - Ng, L., D. Rutherford, V. Shende, S. Sivek, and E. Zaslow. “Augmentations are Sheaves (Accepted).” Geometry and Topology, n.d.
    Open Access Copy Link to Item
Teaching & Mentoring
Recent Courses
- MATH 401: Introduction to Abstract Algebra 2023
- MATH 411: Topology 2023
- MATH 701: Introduction to Abstract Algebra 2023
- MATH 711: Topology 2023
- MATH 281S: Problem Solving Seminar 2022
- MATH 391: Independent Study 2022
- MATH 612: Algebraic Topology II 2022
- MATH 790-90: Minicourse in Advanced Topics 2022
- MATH 611: Algebraic Topology I 2021
- MATH 633: Complex Analysis 2021
- MATH 790-90: Minicourse in Advanced Topics 2021
Advising & Mentoring
- Current Ph.D. advisee:
  Chindu Mohanakumar
  
  Current Ph.D. mentor:
  Jason Xu
Scholarly, Clinical, & Service Activities
Presentations & Appearances
Service to the Profession
Service to Duke

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Education, Training, & Certifications

Previous Appointments & Affiliations

Leadership & Clinical Positions at Duke

In the News

Awards & Honors

Subject Headings

Selected Grants

Selected Publications

Academic Articles

Recent Courses

Advising & Mentoring

Presentations & Appearances

Service to the Profession

Service to Duke