**Contact Information**

- mcheng$\color{red}{@}$gate.sinica.edu.tw
- +886 2 2368-5999 ext. 709
- +886 2 2368-9771

**Research Interests**

- Mathematical Physics

**Education**

- PhD University of Amsterdam (2003/9 - 2008/7)
- MSc in Theoretical Physics Utrecht University (2001/9 - 2003/7)
- BSc in Physics National Taiwan University (1997/9 - 2001/7)

**Links**

**Work Experience**

- Associate Professor Institute of Physics and Korteweg-de-Vries Institute for Mathematics, Universiteit van Amsterdam 2017/6 - currently
- Assistant Professor Institute of Physics and Korteweg-de-Vries Institute for Mathematics, Universiteit van Amsterdam 2014/1 - 2017/6
- Long Term Visitor Department of Applied Mathematics and Theoretical Physics, University of Cambridge 2013 - 2016
- CNRS Researcher (CR) Institut de math ́ematiques de Jussieu, U. Paris 7 2012/1 - currently on leave
- Postdoctoral Fellow Mathematics Department, Harvard University 2010/9 - 2011/12
- Visiting Fellow IAS, Princeton University 2010/9 - 2010/12
- Postdoctoral Fellow Physics Department, Harvard University 2008/8 - 2010/8

**Research Descriptions**

My main research area is mathematical physics. I have a solid background in both mathematics and

physics and enjoy employing the tools and insights in both disciplines to solve interesting and meaningful problems in the realm of mathematical physics. My research programme often fits nicely into the

flourishing and productive subfield of "String-Math" (see the annual conference since 2011).

One of the main focus of my research since 2010 is the topic of moonshine. Highlights in the past

few years include the following. 1) The discovery of umbral moonshine. 2) Uncovering the relation between umbral moonshine and string theory in the background of K3 surfaces. The recent construction

of the umbral moonshine modules for 5 of the 23 cases of umbral moonshine. 4) The classification of

mock forms of the umbral moonshine. 5) The very recent discovery of a few examples of a novel type

of connection between modular-type functions and finite group representations.

Another recent interest is to apply the theory and techniques of **modular forms** to help understand questions in other topics in mathematics and physics. Examples include to put constraints on

the spectrum of 2d conformal field theory admitting a weakly coupled gravity dual description (see

my paper Elliptic Genera and 3d Gravity), and a new method to compute the four-point conformal

blocks of vertex operator algebras and 2-dimensional CFTs (see my paper Modular Exercises for Four-Point Blocks - I). Another important theme of my recent research is to investigate the relation between

modular-type functions, in particular **quantum modular forms**,** logarithmic vertex algebras**, and

**three-dimensional topology**. I believe that this is a rich and multi-disciplinary topic which will lead

to progress in various fields including topology, algebra, number theory, and string theory. See my

paper 3d Modularity, Three-Manifold Quantum Invariants and Mock Theta Functions and a related

paper https://arxiv.org/abs/1904.06057 by Gukov and Manolescu.

Finally, my latest new interest lies in the application of mathematics and theoretical physics in **artificial intelligence**. See for instance the publication Covariance in Physics and Convolutional Neural

Networks.

**Selected Publications**

- (with J. Duncan, M. Mertens) "Class Numbers, Cyclic Simple Groups and Arithmetic" , 2023. (https://arxiv.org/abs/2204.00775)
- (with R. Bondesan, M. Gerdes, P. de Haan, C. Rainone,) "Learning Lattice Quantum Field Theories with Equivariant Continuous Flows" , 2023.
- (with I. Coman, D. Passaro, G. Sgroi) "Quantum Modular $\hat Z^G$-Invariants" , 2023.
- (with G. W. Moore and N. M. Paquette) "Flux vacua: a voluminous recount"
,
*Communications in Number Theory and Physics*, 16 (4), 761-800, 2022-10. - (with Gabriele Sgroi) "Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules"
,
*https://arxiv.org/abs/2203.03052*, 2022-03. - (with Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah M. Harrison, Davide Passaro) "3-Manifolds and VOA Characters"
,
*https://arxiv.org/abs/2201.04640*, 2022-01. - (with V. Anagiannis, J. Duncan and R. Volpato) "Vertex operator superalgebra/sigma model correspondences: The four-torus case"
,
*Progress of Theoretical and Experimental Physics*, 2021 (8), 2021-8. - (with Pim de Haan, Corrado Rainone, Roberto Bondesan) "Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flows"
,
*Fourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021)*, 2021-10. - (with Vassilis Anagiannis) "Entangled q-convolutional neural nets"
,
*Mach.Learn.Sci.Tech.*, 2 (4), 2021. - (with V. Anagiannis, M. Weiler, P. de Haan, T. S. Cohen, M. Welling) "Covariance in Physics and Convolutional Neural Networks"
,
*https://arxiv.org/abs/1906.02481*, 2021. (https://arxiv.org/abs/1906.02481) - (with T. Gannon, G. Lockhart) "Modular Exercises for Four-Point Blocks -- I"
,
*https://arxiv.org/abs/2002.11125*, 2021. (https://arxiv.org/abs/2002.11125) - (with John F.R. Duncan) "Optimal mock Jacobi theta functions"
,
*Advances in Mathematics*, 372, 2020-10. - (with F. Ferrari and G. Sgroi) "Three-Manifold Quantum Invariants and Mock Theta Functions"
,
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, 378 (2163), 2019. (https://arxiv.org/abs/1912.07997) - (with S. Chun, F. Ferrari, S. Gukov, S. M. Harrison) "3d modularity"
,
*Journal of High Energy Physics*, 2019. (https://arxiv.org/abs/1809.10148)