Research | Shashini Marasinghe | ශශිනි මාරසිංහ

Quantum low dimensional topology

Mon, 01 Jan 0001 00:00:00 +0000

The whole theory has been, to a great extent, inspired by ideas that arose in theoretical physics. Among the relevant areas of physics are the theory of exactly solvable models of statistical mechanics, the quantum inverse scattering method,the quantum theory of angular momentum, 2-dimensional conformal field theory,etc. The development of this subject shows once more that physics and mathematics intercommunicate and influence each other … - Vladimir G. Turaev

Quantum invariants, founded around 1980 on the ideas of quantum groups and quantum topology, have grown into a rich area of mathematics through the work of mathematicians such as M. Atiyah, L. Kauffman, N. Reshetikhin, V. Turaev, and others. Key developments include the Jones polynomial, introduced by V. Jones in 1984, which revolutionized knot theory, and its extension to 3-manifolds by Witten in 1988. Reshetikhin and Turaev then constructed invariants satisfying topological quantum field theory properties, and the Turaev-Viro invariants introduced in the early 1990s provide a combinatorial framework for computing 3-manifold invariants.

To learn more, see the references below—and keep reading to explore both the projects I have worked on and the exciting directions I plan to pursue next.

Project 1: Seifert fibered 3-manifolds and Turaev-Viro invariants volume conjecture

We study the large $r$ asymptotic behaviour of the Turaev-Viro invariants of oriented Seifert fibered 3-manifolds at the root $q=e^\frac{2\pi i}{r}$. As an application, we prove the volume conjecture for large families of oriented Seifert fibered 3-manifolds with empty or non-empty boundary.

Project 2: Colaborated work on Seifert cobordisms and the Chen-Yang volume conjecture (Renaud Detcherry, Efstratia Kalfagianni, Shashini Marasinghe)

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M;e^2πi?r)$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $∂M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

Future Directions: Topics of Interest

Teaching as Research

Mon, 01 Jan 0001 00:00:00 +0000

The secret of life, though, is to fall seven times and to get up eight times! Paulo Coelho

Project 1:

My first proposed FAST project will focus on how to integrate technological tools such as LATEX, Python, and LEAN to improve learning experiences and Mathematical writing. For example, LaTeX Beamer is more efficient and user-friendly when creating math-focused presentations than Microsoft PowerPoint. Moreover, theoretical subjects like quantum topology, which is not easy to visualize, can be easily visualized using Python. My second proposed project is very much related to the core tenet of my teaching philosophy; creating a motivating learning environment for students of all backgrounds and skill levels. Many students grapple with the fear of giving incorrect answers, which hinders their participation. I aim to develop strategies that reduce mathematics anxiety and encourage students to view failure as a stepping stone to success. I hope these two proposed projects will be excellent additions to the FAST community coming from a fellow mathematics teacher.

In theoretical advanced mathematics subjects, technological tools are often underutilized resources. Through FAST, I aim to interact with participants in other STEM disciplines and exchange knowledge to explore innovative approaches to improve mathematical learning. The FAST fellowship offers ample time to learn and implement these projects and provides opportunities for one-on-one feedback and effective peer mentoring in a structured environment. This supportive environment will help me refine my teaching methodologies, improve project outcomes, and strengthen my ability to effectively incorporate technology into mathematics education.

Future Directions: Topics of Interest

Here you discuss what you are interested in exploring next. For example:

Applying quantum invariants to low-dimensional topology problems.
Connections between TQFT and quantum computing.
New combinatorial methods for computing invariants efficiently.

Technology Resources

Mon, 01 Jan 0001 00:00:00 +0000