Renjie Feng                   

I received my Bachelor's degree from University of Science and Technology of China in 2006, Master degree at Johns Hopkins University in 2009 and Ph. D at Northwestern University with S. Zelditch in 2012.  I was a postdoc at McGill University (2012-2013) and Maryland University Park College (2013-2015). Now I am an assistant professor at BICMR, Peking University.

 

 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.        Extreme spacing:

With D. Wei, we study the small gaps for the circular beta-ensemble and large gaps for CUE and GUE, in all cases, we derived the rescaling limits of extreme gaps based on the proof that the extreme gaps are asymptotic to the Poisson distribution after rescaling.

 

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 there is a lower bound of critical radius of the embedding/immersion of the sphere to higher dimensional sphere by the spherical harmonics, although the image become more and more twisted.

 

  Random matrices

1.Large gaps of CUE and GUE (with D. Wei)

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

3.Spectrum of SYK model (with G. Tian and D. Wei)

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

5.Spectrum of SYK model III: Large deviations and concentration of measures (with G. Tian and D. Wei)

 

Random geometry

1.Critical radius and supremum of random spherical harmonics (with R. Adler), to appear in Annals of Probability.

2.Critical radius and supremum of random spherical harmonics II (with X. Xu and R. Adler), to appear in Electronic Communications in Probability.  

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

4.Critical values of random analytic functions on complex manifolds    

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

5.Median and mean of the Supremum of L^2 normalized random   

holomorphic fields (withS. 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, arXiv:1604.07693, to appear in Trans. Amer. Math. Soc.

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

 

Geometric analysis and PDEs

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

2.The global convergence of the Calabi flow on Abelian varieties

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

3.Bergman metrics and geodesics in the space of Kahler metrics on

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

4.Szasz analytic functions and noncompact Kahler toric manifolds,

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

 

 Email: renjie@math.pku.edu.cn