Picture of Jiun-Chau (JC) Wang

Jiun-Chau (JC) Wang Ph.D. Assistant Professor (with Tenure)

Office
McLean Hall 208

Research Area(s)

  • Pure Mathematics
  • Mathematical Analysis
  • Functional Analysis
  • Free Probability
  • Probability

Publications

Selection of Publications (by Year)

  • Hao-Wei Huang, JC Wang. "Bi-free extreme values". Journal of Functional Analysis In Press
  • Octavio Arizmendi, Mauricio Salazar, JC Wang. "Berry–Esseen Type Estimate and Return Sequence for Parabolic Iteration in the Upper Half-Plane". International Mathematics Research Notices In Press
  • Takahiro Hasebe, Hao-Wei Huang, JC Wang. "Limit theorems in bi-free probability theory". Probability Theory and Related Fields 172, 3 (2018): 1081-1119.
  • Hao-Wei Huang, JC Wang. "Harmonic analysis for the bi-free partial S-transform". Journal of Functional Analysis 274, 5 (2018): 1306-1344.
  • Hari Bercovici, JC Wang, Ping Zhong. "Superconvergence to freely infinitely divisible distributions". Pacific Journal of Mathematics 292, 2 (2017): 273-291.
  • Hao-Wei Huang, JC Wang. "Analytic aspects of the bi-free partial R-transform". Journal of Functional Analysis 271, 4 (2016): 922-957.
  • Michael Anshelevich, JC Wang, Ping Zhong. "Local limit theorems for multiplicative free convolutions". Journal of Functional Analysis 267, 9 (2014): 3469-3499.
  • JC Wang. "The central limit theorem for monotone convolution with applications to free Lévy processes and infinite ergodic theory". Indiana University Mathematics Journal 63, 2 (2014): 303-327.
  • JC Wang. "Strict limit types for monotone convolution". Journal of Functional Analysis 262, 1 (2012): 35-58.
  • Mihai Popa, JC Wang. "On multiplicative conditionally free convolution". Transactions of the American Mathematical Society 363, 12 (2011): 6309-6335.
  • Jiun-Chau Wang. "Local limit theorems in free probability theory". Annals of Probability 38, 4 (2010): 1492-1506.

Research

Free convolution Free probability Mathematics functional analysis probability

Research Field: Free Probability

Brief Description:

Free probability theory studies non-commutative objects (called free random variables) using probabilistic methods. It was invented by Dan Voiculescu (UC Berkeley) in 1985 in order to study problems in pure mathematics. Voiculescu later found that certain matrices with randomly sampled entries, the so-called random matrices, behave like free random variables as the size of the matrices tends to infinity. Hence, free probability can be used to study the asymptotic behavior of these random matrices. This remarkable discovery changed free probability and brought new tools and concepts into the random matrix theory. Here in Saskatoon, our research in free probability focuses on the theory of free convolution and its applications.  

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2018 Summer Free Probability Seminar:

 
Time: Every Friday, 10:00-11:30 am 
Place: McLean Hall 242.1
Speaker: JC Wang 

Talk Titles: 

May 25: Gaussian random matrices

June 1: Genus expansion for the GUE 

June 8: Asymptotic freeness of independent GUE's


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2017 Free Probability and Random Matrix Seminar:

 
Time: August 25, 3:30-4:30 pm (Department Colloquium)
Place: ARTS 105
Speaker: Mihai Popa (U of Texas at San Antonio)

Title: Asymptotic independence relations and permutation on entries for some classes of random matrices

Abstract:
Almost three decades ago, D.-V. Voiculescu, in his attempt to prove the free factors conjecture, proved and used the fact that 2 Gaussian random matrices with independent entries are asymptotically free. This result was much improved since then, notably by R. Speicher, J. Mingo, B. Collins (all based in Canada at some times) and O. Ryan. The result gave rise to the idea that `freeness' is a asymptotic relation resulting from entry independence and large dimension of random matrices. Recently (2013), we found the surprising result that unitarily invariant random matrices are asymptotically free from their transposes. Since then, new properties are showing that asymptotic freeness can also be induced by various permutation of entries of relevant classes of random matrices.


 
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Talks in 2014-2015 Term 2 and Summer
 
Free Probability and Random Matrix Seminar:
The focus of this seminar in Term 2 will be on GUE/GOE. In particular, we will discuss how to compute correlation functions for the Gaussian Unitary Ensemble using Hermite polynomials, then how to compute probabilities using Fredholm determinants and we will look at the asymptotic results when the size of matrices goes to infinity. The first two introductory lectures will be on the basic setup of random matrix theory and, in the second lecture, on Painleve equations. Painleve equations show up in the asymptotic regime of Random Matrix Theory, but they are of independent interest. One can think of them as special functions of canonical nonlinear ordinary differential equations, in the same way as most of special functions are coming from ordinary differential equations of second order with regular singular points.  

 
Time: September 25, 10:30-11:30 am
Place: MCLH 242.2
Speaker: Allen Tseng (U of S)
Title: Gaudin–Mehta's theorem for GUE
Abstract:
We recall that the N eigenvalues of the GUE are spread out on an interval of width roughly equal to 4/\sqrt(N), and hence the spacing between adjacent eigenvalues is expected to be of order 1/\sqrt(N). We present a preliminary result of this sort, due to Gaudin and Mehta.



Time: September 18, 10:30-11:30 am
Place: MCLH 242.2
Speaker: Victor Vinnikov (Ben-Gurion University of Negev, Israel)
Title: Non-commutative functions and free infinite divisibility
Abstract:
This is an expository talk on the theory of nc (non-commutative) functions and its application to the Levy-Khintchine formula in free probability.



Time: August 5, 10-11 am

Place: MCLH 242.1
Speaker: Mihai Popa (University of Texas at San Antonio)
Title: Asymptotic Freeness and Matrix Transpose for Wishart and Unitarily Invariant Ensembles of Random Matrices
Abstract:
From the beginning of Free Probability Theory, free independence was connected to the asymptotic behavior of various classes of independent random matrices. Motivated by some questions concerning fluctuations moments of unitarily and orthogonally invariant random matrices, we obtained the surprising result that unitarily invariant random matrices are asymptotically free from their transposes. Some more general results are shown for the the cases of GUE, GOE and Wishart ensembles. Joint work with J. A. Mingo.
 

Time: April 6, 3-4 pm
Place: MCLH 242.1
Speaker: Allen Tseng (Department of Mathematics and Statistics) 
Abstract: Fredholm determinants Part II
 
Time: March 23, 3-4 pm
Place: MCLH 242.1
Speaker: Allen Tseng (Department of Mathematics and Statistics) 
Abstract: Fredholm determinants
 
Time: March 16, 3-4 pm
Place: MCLH 242.1
Speaker: Allen Tseng (Department of Mathematics and Statistics) 
Abstract: Wigner's Law revisited. Part II
 
Time: March 9, 3-4 pm
Place: MCLH 242.1
Speaker: Allen Tseng (Department of Mathematics and Statistics) 
Abstract: Wigner's Law revisited.
 
Time: March 2, 3-4 pm
Place: MCLH 242.1
Speaker: Xiangke Chang /Allen Tseng (Department of Mathematics and Statistics) 
Abstract: Conclusions of the proof of  the Dyson-Mehta theorem will be presented in
this talk.
 
Time: February 9, 3-4 pm
Place: MCLH 242.1
Speaker: Xiangke Chang (Department of Mathematics and Statistics)
Abstract: The detailed proof of  the Dyson-Mehta theorem will be presented in this
talk. 
 
Time: February 2, 3:00 - 4:00 pm
Place: MCLH 242.1
Speaker: Xiangke Chang (Department of Mathematics and Statistics)
Abstract: I’ll introduce the detailed proof to the Dyson-Mehta theorem
in random matrix theory, which connects the k-point correlation function
and the kernel polynomials of orthogonal polynomials. I’ll focus on the
Gaussian Unitary Ensemble, which involves Hermite polynomials.
 
Monday January 26, 3:00 - 4:00 pm
MCLH 242.1
Speaker: Xiangke Chang (Department of Mathematics and Statistics)
Abstract: I’ll introduce the detailed proof to the Dyson-Mehta theorem in random
matrix theory, which connects the k-point correlation function and the kernel
polynomials of orthogonal polynomials. I’ll focus on the Gaussian Unitary Ensemble,
which involves Hermite polynomials. Some facts on Hermite polynomials will also be
introduced. There will be two talks on this topic.
 
Monday, 19 January, 3:00-4:00 pm
MCHL 242.1
Speaker: Jacek Szmigielski (Department of Mathematics and Statistics) 
Abstract: I will continue an elementary review of the basic setup of Random Matrix theory, concentrating on the Gaussian Unitary Ensemble and the Wishart Ensemble.  I will explain why Hermite and Laguerre polynomials play a fundamental role in these two ensembles. This is a preview of forthcoming attractions. The actual details will be unfolded by other speakers in subsequent talks. I will give one more talk on Painleve equations.  
 
 
Monday, 12 January, 3:00-4:00 pm
MCHL 242.1
Speaker: Jacek Szmigielski (Department of Mathematics and Statistics)
Abstract: I will review the basic setup of Random Matrix theory, concentrating on the Gaussian Unitary Ensemble and Wishart Ensemble.  I will explain why Hermite and Laguerre polynomials play a fundamental role in these two ensembles.  This will be a non-technical talk. The following talk (a week from the coming Monday) on the Painleve equations will also be elementary. The technical details will be discussed later by Dr.  Xiangke Chang and, hopefully, others.
 
 
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Some Talks in Term 1:
 
Free Probability and Random Matrix Seminar:
Time: 3 pm, Monday Sep. 22, 2014
Location: McLean Hall 242.2
Title: Wigner matrix and convergence to the semicircle law 
Speaker: JC Wang (U of S)
 
Free Probability and Random Matrix Seminar:
Time: 3 pm, Monday Oct. 6, 2014
Location: McLean Hall 242.2
Title: Wigner's semicircle law: The method of moments
Speaker: JC Wang (U of S)
 
Free Probability and Random Matrix Seminar:
Time: 3-4 pm, Tuesday Oct. 14, 2014
Location: McLean Hall 242.1
Title: Wigner's semicircle law: The method of Stieltjes transform
Speaker: JC Wang (U of S)
 
Free Probability and Random Matrix Seminar:
Time: 3-4 pm, Monday Oct. 20, 2014
Location: McLean Hall 242.2
Title: The combinatorial proof of Wigner's semicircle law
Speaker: Pei-Lun (Allen) Tseng (U of S)
 
Free Probability and Random Matrix Seminar:
Time: 3-4 pm, Tuesday Nov. 4, 2014
Location: McLean Hall 242.2
Title: The combinatorial proof of Wigner's semicircle law Part II
Speaker: Allen Tseng (U of S)
 
Free Probability and Random Matrix Seminar:
Time: 3-4 pm, Monday Nov. 10, 2014
Location: McLean 242.2
Title: Asymptotic expansions in free limit theorems
Speaker: Anna Reshetenko (University of Bielefeld, Germany)
 
Free Probability and Random Matrix Seminar:
Time: 3-4 pm, Tuesday Nov. 24, 2014
Location: McLean Hall 242.1
Title: Joint distribution of eigenvalues of GUE and that of GOE
Speaker: Allen Tseng (U of S)
 
Free Probability and Random Matrix Seminar:
Time: 3-4 pm, Tuesday Dec. 1, 2014
Location: McLean Hall 242.1
Title: Selberg integral formula
Speaker: Allen Tseng (U of S)