Thang T. Q. Le
(Lê Tự Quốc Thắng)
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Professor
School of Mathematics
Georgia Institute of Technology
FAX: (404)894-4409
Email: letu@math.gatech.edu
Research Interests: Differential topology, 3-manifolds, knot theory, quasicrystals.
Teaching:
+ Please visit
Canvas for course syllabus and
information.
Services:
+ Editor, Quantum Topology. Papers are submitted to The European Mathemathical Society Publishing House at this site.
+ Editor for The Journal of Knot theory and its Ramifications. Papers are submitted to World Scientific at this site.
+ Editor for Acta Mathematica Vietnamica. Papers are submitted to Springer at this site.
Publications
Research papers
1. Frobenius
homomorphisms for stated SLn-skein modules, with Hyun Kyu Kim
and Zhihao Wang.
Preprint arXiv:2504.08657.
2. Sliced skein
algebras and geometric Kauffman bracket, with C. Frohman and J. Kania-Bartoszynska. Adv. Math. 463 (2025), Paper No.
110118, 65 pp. Preprint arXiv:2310.06189.
3. Degenerations Of
Skein Algebras and Quantum Traces, with
W. Bloomquist and H. Karuo, Trans. Amer. Math.
Soc., to appear. Preprint arXiv:2308.16702.
4.
From
3-dimensional skein theory to functions near Q, with S. Garoufalidis, Annales de l'Institut
Fourier, to appear. Preprint arXiv:2307.09135.
5.
Skein algebras of
three-manifolds at 4th roots of unity, with C. Frohman and J.
Kania-Bartoszynska, New York J. Math. 29 (2023),
1075–1096. Preprint arXiv:2212.04558.
6.
Quantum traces for SL_n-skein algebras, with
T. Yu, Preprint ArXiv:2303.08082
7.
Stated skein modules of 3-manifolds and TQFT, With
F. Costantino, Journal
of the Institute of Mathematics of Jussieu, to appear. Prerint arXiv:2206.10906
8.
Stated
SL(n)-Skein Modules and Algebras, with A. Sikora, J. Topol. 17 (2024), no. 3, Paper No. e12350, 93 pp. Preprint arXiv
2201.00045.
9.
Root of unity quantum cluster algebras and Cayley-Hamilton
algebras, with S Huang and M.
Yakimov, Trans. Amer. Math. Soc. 376 (2023), no. 10,
6767–6798. Preprint arXiv:2107.11926.
10. Faithfulness of
geometric action of skein algebras. Acta
Math. Vietnam. 47 (2022), no. 1, 279--291. Preprint arXiv:2103.11532
11. Quantum traces
and embeddings of stated skein algebras into quantum tori, with T. Yu, Selecta
Math. (N.S.) 28 (2022), no. 4, Paper No. 66, 48 pp. Preprint arXiv:2012.15272.
12. The
Chebyshev-Frobenius Homomorphism for stated skein modules of 3-manifolds, with W. Bloomquist, Math. Z.
301 (2022), no. 1, 1063-1105.
Preprint arXiv:2011.02130.
13. SL_2 quantum trace in quantum Teichmüller
theory via writhe, With H. K. Kim and M. Son, Algebr.
Geom. Topol, 23 (2023), no. 1, 339–418.. Preprint Arxiv:1812.11628.
(new, updated version).
14. Stated skein
modules of marked 3-manifolds/surfaces, a survey, With T. Yu, Acta Math. Vietnam. 46 (2021), no. 2, 265–287. pdf
15. Lower and upper
bounds for positive bases of skein algebras.
With D. Thurston and T. Yu. Int. Math. Res. Not. IMRN 2021, no. 4, 3186-3202. pdffile
16. Stated
skein algebras of surfaces. With F. Costantino. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 12,
4063--4142. Preprint arXiv:1907.11400.
17. Dimension
and Trace of the Kauffman Bracket Skein Algebra, With C. Frohman and J.
Kania-Bartoszynska. Trans. Amer.
Math. Soc. Ser. B 8 (2021), 510–547. Link to pdf.
18. On
Kauffman bracket skein modules of marked 3-manifolds and the
Chebyshev-Frobenius homomorphism. With J. Paprocki. Algebr. Geom.
Topol. 19 (2019), no. 7, 3453-3509. Preprint arXiv:1804.09303,
2018. pdf.
19. Unicity
for Representations of the Kauffman bracket Skein Algebra. With
C. Frohman and J. Kania-Bartoszynska. Invent. Math. 215 (2019), no. 2, 609-650. pdf.
20. Triangular decomposition of skein algebras, Quantum Topology, 9 (2018), 591-632. pdf.
21. The colored Homflypt function is q-holonomic. With S. Garoufalidis and A. Lauda. pdf. Duke Math. Journal, 167, Number 3 (2018), 397-447.
22. On positivity of Kauffman bracket skein algebras of surfaces, International Mathematics Research Notices, 2018, no. 5, 1314-1328. pdf.
23. A survey of q-holonomic functions. With S. Garoufalidis. Preprint arXiv:1601.07487, 2016, pdf. L'Enseignement Mathematique, 62 (2016), no. 3-4, 501-525.
24. Quantum Teichmuller spaces and quantum trace map, preprint arXiv:1511.06054, 2015, pdf. Journal of the Institute of Mathematics of Jussieu, 18 (2019), no. 2, 249–291.
25. Unified quantum invariants for integral homology spheres associated with simple Lie algebras. With K. Habiro. Geometry and Topology 20 (2016), 2687-2835. DOI 10.2140/gt.2016.20.2687. pdf
26. Character varieties, A-polynomials, and the AJ Conjecture. With Xingru Zhang. pdf. Algebraic & Geometric Topology, 17 (2017), no. 1, 157-188.
27. The colored Jones polynomial and the AJ conjecture. Lecture notes from "Etats de la Recherche : Geometric and quantum topology in dimension 3" at the CIRM in Luminy, July 2014. pdf. Appeared in in ``Lectures on quantum topology in dimension three" (by T. Le, C. Lescop, R. Lipshitz, P. Turner), Panoramas et Syntheses, N 48 (2016), Soc. Math. France, pp 33-90.
28. Growth of homology torsion in finite coverings and hyperbolic volume. Annales de l'Institut Fourier, 68 (2018), no. 2, 611--645. Preprint arXiv:1412.7758.
29. On Kauffman Bracket Skein Modules at Root of Unity. Algebraic & Geometric Topology, 15 (2015), no. 2, 1093-1117. pdf
30. Nahm sums, stability and the colored Jones polynomial. With S. Garoufalidis. Research in Mathematical Sciences, 2 (2015), Art. 1, 55 pp. pdf
31. On the AJ conjecture for knots. With Anh Tran. Indiana Univ. Math. J. 64 (2015), no. 4, 1103-1151. pdf
32. Homology torsion growth and Mahler measure. Comment. Math. Helv. 89 (2014), no. 3, 719-757. pdf
33. On the integrality of the Witten-Reshetikhin-Turaev 3-manifold invariants. With A. Beliakova and Q. Chen. Quantum Topology 5 (2014), 99-141 .pdf
34. The skein module of two-bridge links. With Anh Tran. Proceedings Amer. Math. Soc. 142 (2014), 1045-1056. pdf
35. Growth of regulators in finite abelian coverings. Algebraic & Geometric Topology 13 (2013), 2383-2404. pdf
36. The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant. With T. Kuriya and T. Ohtsuki. J. Topology 5 (2012), 458--484. pdf
37. On the unification of quantum 3-manifold invariants. With A. Beliakova, in "Introductory lectures on knot theory", Ser. Knots Everything 46, World Sci. Publ., Hackensack, NJ, 2012, 1--21, .pdf
38. Asymptotics of the colored Jones function of a knot. With S. Garoufalidis. Geometry and Topology, 15 (2011), 2135-2180. .pdf
39. Unified SO(3) Quantum Invariants for Rational Homology 3-Spheres. With A. Beliakova and I. Buehler, Invent. Math. 185 (2011), 121-174. .pdf
40. On the Volume Conjecture for Cables of Knots. With Anh Tran. J. Knot Theory Ramifications, 19 (2010), 1673--1691. .pdf
41. Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion. With A. Beliakova and C. Blanchet. Fund. Math. 201 (2008), 217-239. .pdf
42. Analyticity of the free energy of a closed 3-manifold. With S. Garoufalidis and M. Marino. Symmetry, Integrability and Geometry: Methods and Applications, 4 (2008), 080, 20 pages (electronic). .pdf
43. Strong Integrality of Quantum Invariants of 3-manifolds. Trans. Amer. Math. Soc. 360 (2008), 2941-2963. .pdf
44. Twisted Alexander Polynomial of links in the projective space. With Vu Huynh. Knot Theory Ramifications 17 (2008), 411-438. pdf
45. Gevrey series in quantum topology. With S. Garoufalidis. J. Reine Angew. Math. 618 (2008), 169-195. .pdf
46. Integrality of quantum 3-manifold invariants and a rational surgery formula. With A. Beliakova. Compositio Mathematica, 143 (2007), 1593-1612. .pdf
47. A TQFT associated to the LMO invariant of three-dimensional manifolds. With D. Cheptea. Comm. Math. Phys. 272 (2007), 601-634. pdf
48. Is the Jones polynomial of a knot really a polynomial? With S. Garoufalidis. J. Knot Theory and its Ramification 15 (2006), 983--1000. pdf
49. The Colored Jones Polynomial and the A-Polynomial of Knots. Adv. in Math. 207 (2006), 782--804 .pdf . Corrections.
50. The quantum MacMahon Master Theorem. With S. Garoufalidis and D. Zeiberger. Proceedings of the Nat. Acad. Sci. 103 (2006), N 38, 13928--13931. pdf.
51. On the Colored Jones Polynomial and the Kashaev invariant. With Vu Huynh. Fundam. Prikl. Mat. 11 (2005), no. 5, 57-78. pdf
52. Almost integral TQFTs from simple Lie algebras. With Q. Chen. Algebr. Geom. Topol. 5 (2005), 1291--1314. pdf
53. The colored Jones function is q-holonomic. With S. Garoufalidis. Geom. Topol. 9 (2005), 1253--1293. pdf
54. Quantum Invariants of Periodic Links and Periodic 3-Manifolds. With Q. Chen. Fund. Math. 184 (2004), 55--71. pdf
55. Quantum groups and ribbon G-categories. With V. Turaev. J. Applied and Pure Algebra 178 (2003), 169--185. pdf
56. Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. With D. Bar-Natan and D. Thurston. Geometry and Topology, 7 (2003), 1-31. pdf
57. Quantum 3-manifolds invariants: integrality, splitting, and perturbative expansion. Topology and its applications, 127 (2003), 125-152. .pdf
58. Integrality and symmetry of quantum link invariants. Duke Math. J. 102 (2000), no. 2, 273--306. pdf
59. On the perturbative PSU(n) invariants of rational homology 3-spheres. Topology 39 (2000), no. 4, 813-849. .pdf
60. 3-manifold invariant derived from the Kontsevich integral. With J. Murakami, H. Murakami, T. Ohtsuki. Osaka J. Math. 36 (1999), no. 2, 365--395.
61. On the denominators of the Kontsevich integral and the universal perturbative invariant. Invent. Math. 135 (1999), no. 3, 689-722. .pdf
62. A universal perturbative invariant of 3-manifolds. With J. Murakami and T. Ohtsuki. Topology, 37 (1998), 539--574. pdf
63. The parallel version of the Kontsevich integral. With J. Murakami.J. Pure and Applied Algebra, 121 (1997), 271--291. .pdf
64. An invariant of integral homology 3-spheres which is universal for all finite types invariants. In ``Soliton, Geometry, and Topology: On the Crossroad'', AMS Translations series 2, 179 (1997) Eds. V. Buchstaber and S. Novikov, pp. 75-100. .pdf
65. Local rules for quasiperiodic tilings. In "The mathematics of long-range aperiodic order", NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 489, Kluwer Acad. Publ., Dordrecht, 1997. pp 331-366. pdf
66. A quantum $sl_2$-invariant of 3-manifolds which contains all the Witten-Reshetikhin-Turaev invariants. Lect. Notes in Pure and Appl. Math. 184, Marcel Dekker Inc. 1996. Eds. J. E. Andersen, J. Dupont, H. Pedersen and A.Swann, 399-409. .pdf
67. The Universal Vassiliev-Kontsevich Invariant for framed oriented links. With J. Murakami. Compositio Mathematics 102 (1996), pp. 41--64. pdf
68. The Kontsevich integral for Kauffman's polynomials. With J. Murakami. Nagoya Mathematical Journal, {\bf 142} (1996), pp. 39--66. pdf
69. A 3-manifold invariant derived from the universal Vassiliev-Kontsevich invariant. With J. Murakami, H. Murakami, T. Otsuki. Proceeding of the Japan Academy, series A, Mathematical Sciences LXXI (1995), ser. A, N 6, pp. 125--127.
70. Representation of the category of Tangles by Kontsevich iterated integrals. With J. Murakami. Commun. Math. Physics 168 (1995), 535-562. .pdf
71. Kontsevich's integrals for Homfly polynomials and relations between multiple zeta values. With J. Murakami. Topology and its Applications, 62 (1995), 193--206. pdf
72. Local rules for pentagonal quasiperiodic tilings. Discrete and Computational Geometry, 14 (1995), 31-70. pdf
73. Local rules for multidimensional quasiperiodic tilings. With S. Piunikhin. Differential Geometry and its applications, 5 (1995), 13-31. pdf
74. The geometry of quasicrystals. With S. Piunikhin and V. Sadov. (Russian) Uspekhi Mat. Nauk 48 (1993) , 41-102; translation in Russian Math. Surveys 48 (1993), no. 1, 37-100.
75. Varieties of representations and their subvarieties of cohomology jumps for knot groups. (Russian) Mat. Sb. 184 (1993), no. 2, 57--82; translation in Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 187-209. pdf
76. Local rules for quasiperiodic tilings of quadratic 2-planes in R^4. With S. Piunikhin and V. Sadov. Commun. Math. Phys. 150 (1992), 23-44. pdf
77. Varieties of representations and their subvarieties of cohomology jumps for certain knot groups. (Russian) Uspekhi Mat. Nauk 46 (1991), no. 2(278), 223-224; translation in Russian Math. Surveys 46 (1991), no. 2, 250--251.
78. Novikov's numbers. (Russian) Mat. Zametki 47 (1990), no. 1, 98-104; translation in Math. Notes 47 (1990), no. 1-2, 64-68.
79. Structure of level surfaces of closed 1-forms. (Russian) Mat. Zametki 44 (1988), no. 1, 124--133; translation in Math. Notes 44 (1988), no. 1-2, 556-562.
a.
Others
80. Finite type invariants of 3-manifolds. In ``Encyclopedia of Mathematical Physics", pp 384-356, Elsevier 2006. pdf
81. Lectures on the LMO invariant. Lectures given at the Grenoble 1999 Summer School on invariants of 3-manifolds. pdf
82. Aktueller Forschungsschwerpunkt: Invarianten von Knoten. With M. Kontsevich. In ``Max-Planck-Gesellschaft Jahrbuch 1994", pp 451--455. Max-Planck-Gesellschaft. pdf