myresearch

updated: Feb. 22, 2016

note: this is a very informal version of what can be viewed as my research statement.

note: this is not a completed version yet.. I will update it from time to time.

1. Overview

Just like for most mathematicians in the world, my mathematical research interests cannot be defined in a single-folded manner or in one sentence. I have several different branches of interests, which may someday turn out to be all connected to one another. Among them, there is a main theme, and there are also smaller projects that I have been working on.

1.1. Main interest

The main program which have been occupying my mind since the latter half of my graduate student years (i.e. since about July 2010) revolves around Edward Witten's idea [W07] on the (2+1)-dimensional gravity theory in mathemtical physics, which I do not yet have a clear understanding about. One physically natural setting is the so-called "asymptotically Anti-de Sitter (AdS)" theory on (2+1)-dimensional spacetime manifold with an 'asymptotic boundary'. Long story short, Witten claimed that this theory, in its simplest case, would be 'dual' to the 2-dimensional conformal field theory (CFT) model constructed by Frenkel-Lepowsky-Meurman [FLM88], called the 'Monstrous moonshine'.

As the 'Frenkel' in the Frenkel-Lepowsky-Meurman is my advisor Igor B. Frenkel, you see how I got introduced into this story. This suggestion of Witten's is briefly intoduced in the Wikipedia page [Wiki1] for Monstrous moonshine quite nicely, so readers can consult that source. In the present page, I will try to explain the basic setting, what can be done, and what I want to do about this.

Basically, my main program is ultimately to realize Witten's idea.

However, if Witten himself just hinted and suggested without constructing or claiming to prove, this means it is quite hard. So it is not even clear how much of his fantasy can be mathematically made into a reality.

Despite this possibly unconquerable difficulty of the ultimate goal, there are things that can be understood and can be 'done', in the very first stage toward it. My major works previously done so far are more or less of such kind:

  • [FK12] Quantum Teichmüller space from the quantum plane (joint with Igor Frenkel)

  • [K12] The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

  • [K16] Phase constants in the Fock-Goncharov quantization of cluster varieties (long and short versions)

Works in progress (i.e. not done yet) in this direction include:

  • Quantization of the 'globally hyperbolic' (2+1)-dimensional gravity (joint with Carlos Scarinci)

  • Relationship between the Chern-Simons formulation to (2+1)-dimensional gravity and the Teichmüller theory (discussing with Igor Frenkel, Carlos Scarinci and Jinsung Park, all separately)

So that was roughly about my 'main' research interest, or my big mathematical dream. Certainly it sounds quite tantalizing, although it might be just merely impossible to reach. But I think it is always good to have a big dream. I must note here that of course this is not my own idea; it is first brought out by Witten [W07], and delivered to me by Igor Frenkel.

Although I have been constantly gathering necessary knowledge around this story and thinking how to go about opening the first major door to the goal, this process has been very slow. So, I suspect that making a major progress, even the first such step, should be regarded as a long-term project.

1.2. Other interests

At the same time, there are several other places in mathematics where I have my foot(s) touched on ground, rather than just being in clouds in the sky. These include:

  • Uniform construction of finite simple groups in matrix groups over finite fields, using the computer algebra system MAGMA (joint with Gerhard O. Michler)

  • Denominators of R-matrices for finite dimensional representations of affine quantum groups (joint with Myungho Kim)

  • Quantum Duality between cluster A-variety and cluster X-variety (joint with Dylan Allegretti) [AK15]

  • Construction of quasi-phantom categories (joint with Kyoung-Seog Lee) [KKL15]

For the first project on the finite simple groups, there are some preprints in the arXiv and one in the proceedings of a sectional AMS meeting. The second is in progress now. There are some other projects which I have tried, but which haven't quite close up yet. Among them is:

  • Quantum higher Teichmüller space from the higher rank quantum groups (joint with Ivan Chi-ho Ip).

Regarding this last project, I had written a paper as its pre-requisite:

  • [K14] Ratio coordinates for higher Teichmüller spaces

I also tried some different things about the representations of braided Ptolemy-Thompson groups (joint with Louis Funar and Vlad Sergiescu), but I couldn't come to a sought-for conclusion yet. Besides, there are couple of things I tried with Igor Frenkel, but they are kept stuck at some point.

(*and there are some more things that I'm trying and writing, either by myself or with some other people, like Masahito Yamazaki at IPMU, Japan, which whom I am studying seed-trivial mutation sequences in cluster algebras)

1.3. Plan of this page

I do not know how much detail I would put into this page, nor when will I continue updating. For now, my plan is to explain in layman's term the basic setting and some history regarding my main research interest, introduced in section 1.1. Of course, by 'layman' I mean one in the mathematics community; perhaps, a reasonable graduate student in any area of mathematics will serve as a good criterion for this 'layman'.

2. Riemannian geometry

Instead of trying to fill in this section, at the moment let me just mention a very good source to learn about the subject. It is a book by Baez and Muniain:

  • [BM] John Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, 1994. (480pp.)

2.1. Basic notions

3. General Relativity, a theory on gravity

4. Teichmüller theory

5. Quantization

6. Conformal Field Theory

7. Other projects

References of other people

  • [W07] Edward Witten, Three-Dimensional Gravity Reconsidered, arXiv:0706.3349v1

  • [Wiki1] Wikipedia, Monstrous moonshine, http://en.wikipedia.org/wiki/Monstrous_moonshine

  • [FLM88] Igor B. Frenkel, James Lepowsky, Arne Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math. (Academic Press) 134 (1988)

My papers, published or put on arXiv (not a complete list yet)

  • [FK12] Igor B. Frenkel and Hyun Kyu Kim, Quantum Teichmüller space from the quantum plane, Duke Math. J. 161 no.2 (2012) 305-366.

  • [K12] Hyun Kyu Kim, The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization, to appear in Advances in Mathematics. arXiv:1211.4300

  • [K14] Hyun Kyu Kim, Ratio coordinates for higher Teichmüller spaces, to appear in Mathematische Zeitschrift. arXiv:1407.3074

  • [K16] Hyun Kyu Kim, Phase constants in the Fock-Goncharov quantization of cluster varieties, long version (for arXiv only) is arXiv:1602.00361 and short version (for journal submission) is arXiv:1602.00797

    • [AK15] Dylan Allegretti and Hyun Kyu Kim, A duality map for quantum cluster varieties from surfaces, submitted to a journal. arXiv:1509.01567

    • [KKL15] Hyun Kyu Kim, Yun-Hwan Kim, and Kyoung-Seog Lee, Quasiphantom categories on a family of surfaces isogenous to a higher product, arXiv:1511.02139