Research Interests: My research concerns the quantum topology of low dimensional manifolds. I have pursued research on the construction of links from Thompson's groups initiated by Vaughan Jones in 2014, and how it relates to link homology theories. I've also studied number-theoretic aspects of quantum 3-manifold invariants, focusing on a recently developed family of two-variable series.
Videos
Infinite Families of Quantum Modular 3-Manifold Invariants, Conference on Modern Tools in Low Dimensional Topology, June 2025, ICTP
Thompson's Groups, Annular Links, and Tangles, Workshop on Interactions of Geometric and Quantum Topology, April 2025, BIRS
Papers
(t,q)-series invariants of Seifert manifolds. Preprint: arxiv:2410.10561
Kruskhal, V., Liles, L., and Luo, Y. Thompson's Group F, Tangles, and Link Homology. Preprint: arxiv:2403.16838
Liles, L. Annular Links from Thompson's Group T. To appear in Algebraic and Geometric Topology. Preprint: arxiv:2401.12065
Liles, L. and McSpirit, E. Infinite Families of Quantum-Modular 3-Manifold Invariants. Communications in Number Theory and Physics, Vol. 18 no. 1, 237-260, 2024. arxiv:2306.14765
Seminar Talks
Geometry and Topology Seminar, Virginia Tech, Jan. 2025
Topology Seminar, Texas A&M University, Nov. 2024
Graduate Students Seminar, University of Virginia, Sept. 2024
Geometry Seminar, Virginia Commonwealth University, Aug. 2024
Geometry and Topology Seminar, Virginia Tech, Mar. 2024
Geometry Seminar, University of Virginia, Jan. 2024
Geometry Seminar, Virginia Commonwealth University, Sept. 2023 slides
Math is FUNdamental Seminar at UVA Biocomplexity Institute, Spring 2023) slides
UVA Graduate Students Seminar, Sept. 2022
Conference Talks
ICTP Conference on Modern Tools in Low-Dimensional Topology, June 2025. video recording
Interactions of Geometric and Quantum Topology, Banff International Research Station (BIRS), April 2025. video recording
Knots in Washington 51, Distinguished Graduate Student Speaker, Apr. 2025
Joint Mathematics: Contributed Session on Manifolds and Cell Complexes, Jan. 2025
Knots in Washington 50, The George Washington University, Dec. 2024
AMS Fall Eastern Sectional, Special Session on Invariants of Knots, Links, and Low Dimensional Manifolds, Albany, NY, Oct. 2024
New Structures in Low-Dimensional Topology, Renyi Institute, lightning talk, July 2024
Trisectors Workshop, University of Nebraska Lincoln, lightning talk, June 2024
Georgia Topology Summer School and Conference, lightning talk, May 2024
Gender Minorities in Topology and Related Areas, University of Pennsylvania, April 2024
American Mathematical Society Spring Eastern Sectional, Howard University, Special Session on Skein Modules in Low-Dimensional Topology, April 2024
Mini-Workshop on Plumbings, Spectra, and Resurgence, California Institute of Technology, March 2024
Tech Topology Conference, lightning talk, Georgia Institute of Technology, Dec. 2023 slides
Bucknell University Aspiring PUI Faculty Workshop, Sept. 2023
Graduate Student Topology and Geometry Conference, Harvard University, July 2023, slides
ICERM Tangled in Knot Theory Conference, lightning talk, May 2023
Binghamton University Graduate Combinatorics, Algebra and Topology Conference, Nov. 2022 slides
Poster Presentations
Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting, March 2025 (poster below)
Richmond Geometry Meeting, June 2023