Daniel Tan Maths

Hi! I'm a PhD student at Rutgers University in the Department of Mathematics, where I study vertex operator algebra theory.

I was born and raised in South-East Melbourne, Australia (a proud Frankstonian), and went to Melbourne Uni to study physics as an undergraduate. Falling into the temptation of rigor and abstraction, I then did a master's degree in pure mathematics, also at Melbourne. During this period, I learnt about conformal field theory, categorified algebraic structures, various delicious flavours of representation theory, and ... the beautiful structure of vertex operator algebras. I can thank David Ridout for the amazing guidance.

Hungry for more, I wanted to go to Rutgers University to study vertex operator algebra theory straight from the source, Yi-Zhi Huang and James Lepowsky. I was lucky enough for this to happen! (It is now customary to make a joke about how it is actually unlucky to have to move to New Jersey, but I'm really quite fortunate to live here.)

My thesis will be on mathematical orbifold conformal field theory. (On this site, I have short explanations about conformal field theory and vertex operator algebra theory.) Orbifold conformal field theories are built out of "twisted modules" for a given vertex operator algebra. "Twisted intertwining operators" are maps that appropriately define a multiplication among these twisted modules. Roughly speaking, these twisted intertwining operators are quantum fields! So, in a good model, it is expected that they should converge in an appropriate sense. The results in my thesis investigate this problem.

The tools I use to do this mostly consist of the (twisted) representation theory of vertex operator algebras, complex analysis, differential equations, and some common algebraic techniques. I also get a lot of intuition from reading physics literature, specifically in conformal field theory and string theory. Though it might not seem like it, one of my main motivations for studying this stuff is the application to tensor category theory. Mathematical orbifold conformal field theory predicts that the (twisted) representation theory for vertex operator algebras can be used to build G-crossed (modular) tensor categories. The convergence I study is imperative for the structural natural isomorphisms in these categorical structures.

Here's an old version of me from back in Melbourne 2020.

Hello! I'm Daniel Tan (usually shortened to Dan Tan by those who admire symmetry). I've recently finished my master's degree in mathematics at The University of Melbourne, specialising in pure mathematics and mathematical physics.

My current interests include the representation theories of vertex operator algebras and quantum groups and how they produce tensor categories. Here, I use “tensor categories” loosely to mean monoidal categories and other categorical structures with an underlying monoidal structure. These include braided monoidal categories, monoidal categories with a compatible abelian structure, modular tensor categories, etc.

I have written a master's thesis on these topics, completed in October 2020. It includes introductory material on vertex operator algebras, modular tensor categories and how representations of vertex operator algebras can be used, together with the fusion product, to produce braided monoidal categories. In a later chapter, I demonstrate how to compute modular tensor categorical data for lattice vertex operator algebras. In the final chapter, I detail an example of a Kazhdan-Lusztig correspondence at a non-integral level, producing modular tensor categories from a vertex operator algebra and a quantum group, then constructing an equivalence of modular tensor categories.

For now, I’ll still be doing maths in The University of Melbourne’s School of Mathematics and Statistics. But, by the end of 2021, I’m hoping to have found another math(s) department to call home.

(When I'm not doing maths, I may be spotted eating food or drinking coffee just outside of the maths building.)

(Cape Schanck. One of my favourite trail running spots.)