# Research Area(s)

- Pure Mathematics
- Mathematical Analysis
- Functional Analysis
- Free Probability
- Probability

# Publications

### Selection of Publications (by Year)

- Hao-Wei Huang, JC Wang. "Regularity results for free Levy processes".
*Advances in Mathematics*402 (2022) - 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*2021, 23 (2021): 18037-18056. - Hao-Wei Huang, JC Wang. "Bi-free extreme values".
*Journal of Functional Analysis*278, 6 (2020): 1-39. - 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.

**Courses in Free Probability and Random Matrix:**

(2022-2023 Term 2) Math 872 Random Matrix

(2023-2024 Term 1) Math 872 Free Probability Part 1

(2023-2024 Term 1) Math 872 Random Matrices in Machine Learning

(2023-2024 Term 2) Math 872 Free Probability Part 2

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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.

**Free Probability and Random Matrix Seminar**:

**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**

**Time: March 23, 3-4 pm**

**Place: MCLH 242.1**

**Time: March 16, 3-4 pm**

**Place: MCLH 242.1**

**Time: March 9, 3-4 pm**

**Place: MCLH 242.1**

**Time: March 2, 3-4 pm**

**Place: MCLH 242.1**

this talk.

**Time: February 9, 3-4 pm**

**Place: MCLH 242.1**

talk.

**Time: February 2, 3:00 - 4:00 pm**

**Place: MCLH 242.1**

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**

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**

**Monday, 12 January, 3:00-4:00 pm**

**MCHL 242.1**