Nghiên cứu - Research
Lĩnh vực quan tâm nghiên cứu hiện nay của tôi là hình học và tôpô, đặc biệt là tôpô số chiều thấp.
My current research interests are in geometry and topology, in particular low dimensional topology.
Seminars
Tôi đang tổ chức một seminar toán, thường vào các buổi chiều Thứ sáu tại Trường Đại học Khoa học Tự nhiên, 227 Nguyễn Văn Cừ Quận 5. Để nhận thông báo về các buổi seminar này hãy đăng kí địa chỉ email ở đây hoặc trực tiếp ở đây: https://groups.google.com/g/friday-afternoon-math-seminar/about. Ai muốn trình bày trong seminar này hãy gởi tôi một email.
I am running a mathematics seminar, often on Friday afternoon. If you are interested in giving a talk, please drop me an email.
Derivatives, products, and pullbacks in Forman's combinatorial differential forms
With Thach Phu Nguyen (Nguyễn Phú Thạch), Phuong Van Phan (Phan Văn Phương), Journal of Nepal Mathematical Society, Volume 3, Issue 1, 2020, p. 7--16.
We study derivatives, closedness, and exactness of 0-forms and 1-forms in the theory of combinatorial differential forms constructed by Robin Forman. We give an example of a closed but not exact 1-form on a non-simply connected domain. We give a sufficient condition on the domain for a closed 1-form to be exact. We show that the product of forms proposed by Forman is not anti-commutative. We propose a definition of pullbacks of forms and show that this operation has several properties analogous to pullbacks on smooth forms.
Slides of the opening talk for a seminar course on Computational Topology.
Bài viết ngắn giới thiệu bất biến Alexander xoắn.
Runs on the free computer algebra system Maxima.
With Jérôme Dubois and Yoshikazu Yamaguchi, Journal of Knot Theory and Its Ramifications, vol 18, (2009), no 3, 303 - 341. The published version is shortened from the preprint version on the arXiv.
This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion in the cases of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.
With Thang T. Q. Le. Part of this article is included as an appendix of this paper.
We show that the A-polynomial A(L,M) of a 2-bridge knot b(p,q) is irreducible if p is prime, and if (p-1)/2 is also prime and q\neq 1 then the L-degree of A(L,M) is (p-1)/2. This shows that the AJ conjecture relating the A-polynomial and the colored Jones polynomial holds true for these knots, according to work of the second author. We also study relationships between the A-polynomial of a 2-bridge knot and a twisted Alexander polynomial associated with the adjoint representation of the fundamental group of the knot complement. We show that for twist knots the A-polynomial is a factor of the twisted Alexander polynomial.
With Thang T. Q. Le, Journal of Knot Theory and Its Ramifications, vol 17 (2008), no 4, 411- 438. arXiv
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
Ph. D. dissertation.
With Thang T. Q. Le, Fundamental and Applied Mathematics, vol 11 (2005), no 5, 57--78. Also appeared in Journal of Mathematical Sciences, vol 146, no 1, 5490-5504, 2007. arXiv
We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the q-Weyl algebra of q-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture.
Notes for a graduate student seminar.
Senior thesis.