|
Shana Yunsheng LiPh.D. Student atDepartment of Mathematics University of Illinois Urbana-Champaign Email: yl202  illinois.edu
Pronoun: She/Her |
|
Brief Introduction
Before my Ph.D. studies, I obtained my bachelor's and master's degrees both at Southern University of Science and Technology (SUSTech).During my master's studies, I was advised by Stavros Garoufalidis.
Generally speaking, my research interest can be best described by the following set of keywords:
Mathematical Physics; Low-dimensional Topology; Quantum Topology; Knot Theory.
Curriculum Vitae
in PDFPublications
| Title | Year |
| Patterns of the \(V_2\)-polynomial of knots | 2024 |
|
Abstract: Recently, Kashaev and the first author defined a sequence \(V_n\) of \(2\)-variable knot polynomials with integer coefficients,
coming from the \(R\)-matrix of a rank \(2\) Nichols algebra, the first polynomial been identified with the Links-Gould polynomial.
In this note we present the results of the computation of the \(V_n\) polynomials for \(n=1,2,3,4\) and discover applications and emerging patterns,
including unexpected Conway mutations that seem undetected by the \(V_n\)-polynomials as well as by Heegaard Floer Homology and Knot Floer Homology.
Joint work with Stavros Garoufalidis. arXiv:2409.03557 . Related Talk: Patterns or conspiracy theories of knots by Stavros Garoufalidis at Conference on Recent Developments in Topological Quantum Field Theory, BIMSA. |
|
| Algebraic aspects of holomorphic quantum modular forms | 2024 |
|
Abstract: Matrix-valued holomorphic quantum modular forms are intricate objects that arise
in successive refinements of the Volume Conjecture of knots and involve three
holomorphic, asymptotic and arithmetic objects. It is expected that the algebraic
properties of these objects can be deduced from the algebraic properties of descendant
state integrals, and we illustrate this for the case of the \((-2,3,7)\)-pretzel knot.
Joint work with Ni An and Stavros Garoufalidis. Research in Mathematical Sciences 11, 54 (2024). arXiv:2403.02880 . |
|
Notes
| Title | Year |
| Notes of Zelmanov's Algebraic Lectures | 2023 |
| Taken by me based on the contents of the lectures given by Prof. Efim Zelmanov in 2023 Spring at SUSTech. The topics include Gröebner-Shirshov bases theorem, Wreath product, Burnside's problems, rings of fractions and ultraproducts. | |
| A Review From Manifolds to Basic de Rham Theory | 2021 |
| A quick and compact review of basic notions of smooth manifolds and a beginning of de Rham theory. Served as a preliminary material in the reading seminar of Bott & Tu's Differential Forms in Algebraic Topology. | |
Thesis
| Title | Year |
| On the Quantum Modularity Conjecture for Knots | 2024 |
|
My master's thesis. Abstract: Quantum topology is considered to be initiated by the discovery of the Jones polynomial in 1984, followed with observations of numerous links to physics. In the late '80s, Atiyah, Segal, and Witten established an intrinsic definition of the Jones polynomial using \(\text{SU}(2)\) Chern-Simons theory, revealing the rich connections of the Jones polynomial with the physical world. Successive findings around the Jones polynomial emerged, including one famous conjecture that is the main topic of this thesis, the Quantum Modularity Conjecture. In 1995, R. Kashaev introduced a knot invariant using the quantum dilogarithm function, which for a hyperbolic knot K is conjectured to have an exponential growth rate, a conjecture known as the Volume Conjecture. In 2001, H. Murakami and J. Murakami discovered that Kashaev's invariant is equal to the value of the \(N\)-colored Jones polynomial at \(N\)-th roots of the unity. With this, D. Zagier observed a modular relation between the values of the \(N\)-colored Jones polynomial at different roots of the unity and extended the statement of Volume Conjecture to a modular relation of the functions. The extended statement is known as the Quantum Modularity Conjecture (QMC). More recently, J. E. Andersen and R. Kashaev introduced the Teichmuller TQFT based on Chern-Simons theory with infinite dimensional gauge groups, promoting the quantum Teichmuller theory to a TQFT of categroids. On further investigation into values of the Teichmuller TQFT on knot complements of the \(4_1\) knot and the \(5_2\) knot, S. Garoufalidis and D. Zagier discovered phenomena suggesting deep relationships between the state integral from the Teichmuller TQFT and QMC. Furthermore, their observation also suggested rich connections with several other topics, such as the Dimofte-Gaiotto-Gukov index and the quantum spin network. This thesis will mainly focus on introducing the construction of the Teichmuller TQFT and the contents of QMC, and demonstrate their connections by listing the observations made by S. Garoufalidis and D. Zagier and more recent results along with elementary proofs for some of them from joint work of the author, N. An and S. Garoufalidis. |
|
| Foundation of Supergeometry and Its Application in Mathematical Physics | 2022 |
|
My bachelor's thesis. Abstract: Supergeometry is a natural extension of the theory of differential geometry, which enjoys values on its own right as a purely mathematical object, and also turns out to be useful in physics: it gives a model of spacetime that unifies quantum science and gravity, the string theory. The first part of this thesis gives a detailed and mathematically strict introduction to supergeometry, rearranged from the lecture notes by Covolo and Poncin, the paper of Leites and the notes by Deligne and Morgan. The second part focuses on an explicit discussion on the important example \(\underline{\text{SMan}}(R^{0|\delta},X)\) and outlines a corresponding proof of the Chern-Gauss-Bonnet theorem, following Berwick-Evans’ work. |
|
Seminars and Teachings
| Event | Year |
| Seminar talk at the
Graduate Topology Seminar (SUSTech)
.
Topic: The Teichmüller TQFT and the Quantum Modularity Conjecture. |
2024 |
| Teaching assistant of MA109 (Advanced Linear Algebra). | 2022 |
| Reading seminar of Bott & Tu's Differential Forms in Algebraic Topology. Homepage | 2021 |
| Reading seminar of Riehl's Category Theory in Context. | 2019 |
Outreach
Popular Science PostsI have several Chinese posts on Zhihu, leaning towards popular science aspects of math. Some of them are extracted from my notes and thesis.
| Post | Remark |
为什么要用层重新定义微分流形?
(Why do we use sheaves to re-define differential manifolds?) |
Explaining the advantages of sheaf language in the study of geometric objects, for beginners in differential manifolds. |
|
我们为什么要定义微分流形这个概念，motivation是什么?
(Why do we define the concept of differential manifolds, what is the motivation?) |
Explaining the common partern of development of mathematics, for junior undergraduate students. |
|
为什么四面体的体积公式不存在初等证明？- 希尔伯特第三问题
(Why there does not exist an elementary proof for the volume formula of tetrahedrons? - Hilbert's Third Problem) |
Extracted from my Notes of Zelmanov's Algebraic Lectures, Lecture 17. |
|
拓扑是如何刻画连续性的？
(How does topology characterize continuity?) |
Short article for beginners in topology. |
|
流形切丛上光滑函数环的泛性质
(Universal property of rings of smooth functions on tangent bundles of manifolds) |
Extracted from my Undergraduate Thesis , Appendix B. |
|
对于当今数学来说，「几何」到底是什么？
(What is "geometry" in modern mathematics?) |
Very short article depicting the "big picture" of geometry (to my understanding at that time), for senior undergraduate students. |
Piano
I'm also semi-professional in piano performance. During my life at SUSTech, I've participated in multiple concerts and masterclasses. Some of them are listed below:
| Piano Performance | Year |
| Debussy: Rêverie, L. 68 at Steinway Garden in Shenzhen Concert Hall | 2023 |
| Zhangshuai: Three Preludes, No.1 at Shenzhen Concert Hall | 2022 |
| Zhangshuai: Three Preludes, No.1 at Steinway Garden in Shenzhen Concert Hall | 2021 |
| Lang Lang's Piano Masterclass at SUSTech 10th Anniversary | 2020 |
| Haochen Zhang's Piano Masterclass | 2019 |
