# Renjie Feng

University of Science and Technology of China, Bachelor, 2002-2006

Johns Hopkins University, Master, 2006-2009

Northwestern University, Ph. D (advisor: S. Zelditch), 2009-2012

McGill University, postdoc, 2012-2013

Maryland University College Park, Brin postdoc, 2013-2015

BICMR, Peking University, assistant professor, 2015-

Recent research

For the last two years, I am interested in the following three topics,

1.        SYK model:

With Gang Tian and Dongyi Wei, we have three papers on the global distribution of eigenvalues of SYK model such as the global density of eigenvalues, the central limit theorem and the concentration of measure theorem. The SYK model a random matrix model which is a simple model of the black hole and it's very topical in physics recently, and it's also a quantum spin glass model. There are still many interesting open problems in this new emerging area, such as the behaviors of partition function (recall the Parisi formula in the spin glass) and the largest eigenvalue.

2.        Random matrices:

a.        We study the smallest gaps for the circular beta-ensemble and GOE, and largest gaps for CUE and GUE, in all cases, we proved that the extreme gaps are asymptotic to the Poisson distribution after rescaling, as a consequence, we derived the rescaling limits of extreme gaps.

b.        We derive the Berry-Esseen theorem for the number counting function of circular beta-ensemble, which implies the central limit theorem of number of points in arcs. We also derived the uniform bound for the variance.

3.        Random waves:

With R. Adler, we get an explicit formula for the supremum of random spherical harmonics by Weyl's tube formula, where we proved that unexpectedly the critical radius of the embedding/immersion of the sphere to higher dimensional sphere by the spherical harmonics has an explicit limit given by the Bessel function, although the image become more and more twisted. The result is universal, i.e., true for more general Riemannian manifolds with assumptions.

Random matrix theory

with D. Wei)

2.Small gaps of circular beta-ensemble(with D. Wei)

3.Small gaps of GOE (with G. Tian and D. Wei), to appear in GAFA

4.Normality of circular beta-ensemble(with G. Tian and D. Wei)

5.Spectrum of SYK model (with G. Tian and D. Wei), Peking Mathematical J (2019) 2:41-70.

6.Spectrum of SYK model II: Central limit theorem (with G. Tian and   D.Wei)

7.Spectrum of SYK model III: Large deviations and concentration of measures (with G. Tian and D. Wei), to appear in Random matrices: Theory and Applications.

Random geometry

1.Critical radius and supremum of random spherical harmonics (with R. Adler),  Annals of Probability, 2019, Vol.47, No.2, 1162-1184.

(with X. Xu and R. Adler), Electronic Communications in Probability, Volume 23 (2018), paper no. 50, 11 pp.

3.Large deviations for zeros of P(\phi)_2  random polynomials (with S. Zelditch), J. Stat. Phys (2011) 143: 619-635.

(with S. Zelditch), Indiana Univ. Math. J.63. (2014). no.3.,651-686.

holomorphic fields (with S. Zelditch), J. Funct. Anal. 266 (2014), no. 8, 5085-5107.

6.Random Riesz energies on compact Kahler manifolds (with S. Zelditch), Trans. Amer. Math. Soc., Vol 365, no. 10, 5579-5604, (2013).

7.Critical values of Gaussian SU(2) random polynomials (with Z. Wang), Proceeding of AMS, Vol 144, no 2, 2016, 487-502.

8.Correlations between zeros and critical points of random analytic functions, Trans. Amer. Math. Soc.371 (2019), no. 8, 5247-5265.

9.Conditional expectations of random holomorphic fields on Riemann surfaces,  IMRN, Volume 2017, Issue 14, 4406-4434.

10. Zeros of repeated derivatives of random polynomials (with D. Yao),  Anal. PDE 12 (2019), no. 6, 1489-1512.

Geometric analysis and PDEs

1.Periodic solutions of Abreu's equation (with G. Szekelyhidi), Math. Res. Lett. 18 (2011), no. 6, 1271-1279.

(with H. Huang), J. Funct. Anal. 263 (2012), no. 4, 1129-1146.

principally polarized Abelian varieties, Journal of the Institute of Mathematics of Jussieu (2012) Volume 11, Issue 01, 1-25.

Journal of Geometric Analysis (2012), Volume 22, Number 1, 107-131.

Email: renjie@math.pku.edu.cn