Lec 1  05 Jan 
Introduction Gaussian and Semicircle laws 

Notes 
Lec 2  10 Jan 
Gaussian and semicircle laws, Catalan numbers 
Stanley's exercise


Lec 3  12 Jan 
Convergence of measures. Wigner matrices. ESD of a matrix. Statement of Wigner's semicircle law 


Lec 4  17 Jan 
Method of moments and WSL for expected ESD of GOE matrix 


Lec 5  19 Jan 
Expected ESD of GOE matrix. Connection to maps on surfaces.


Notes 
Lec 6  24 Jan 
Continuity properties of eigenvalues of Hermitian matrices
Rank inequality and HoffmanWielandt (von Neumann)



Lec 7  25 Jan 
Method of moments to prove WSL for general Wigner matrices


Notes 
Lec 8  07 Feb 
Stieltjes tranforms, basic properties, inversion formula and continuity theorem



Lec 9  09 Feb 
Wigner's semicircle law by Stieltjes' transform method, under fourth moment assumption



Lec 10  14 Feb 
Stieltjes' transform proof (cont'd). Remarks on fourth moment assumption.


Notes 
Lec 11  16 Feb 
Chatterjee's invariance principle. Proof of Wigner's semicircle law under Pastur's condition



Lec 12  07 Mar 
Tridiagonalization of GUE and GOE matrices.



Lec 13  09 Mar 
Tridiagonal matrices and probability measures on the line.


Notes 
Lec 14  14 Mar 
Eigenvalue density for βtridiagonal matrices. Statistical mechanics interpretation. Selberg integral.


Notes 
Lec 15  16 Mar 
The special case β=2. Determinantal form of the density. Marginal densities, counting number of points in a subset.



Lec 16  21 Mar 
Circular unitary ensemble. Mean and variance of linear statistics. Asymptotic normality for N(I).



Lec 17  23 Mar 
Fredholm determinants and hole probability in determinantal processes. Asymptotics of gap probability in CUE.



Lec 18  25 Mar 
Hermite polynomials and their basic properties. Integral representations. ChristoffelDarboux formula.


Notes 
Lec 19  28 Mar 
Laplace's method and the saddle point method. Semicircle law, bulk and edge scaling for GUE. 


Lec 20  30 Mar 
Elements of free probability theory. Cumulants in classical probability theory. 

Notes 
Lec 21  06 Apr 
Noncommutative probability spaces.



Lec 22  11 Apr 
Free independence. Examples. Free cumulants.



Lec 23  13 Apr 
Free cumulants and fee independence. Free CLT.



Lec 24  18 Apr 
Relationship to random matrix theory. Voiculescu's theorem on the sum of wo Hermitian random matrices. 


 
Subhamay Saha Symmetric band matrices and Wigner's semicircle law 

The paper 
 
Rajesh Sundaresan Random matrices in communication theory 

Slides 
 
Tulasi Ram Reddy Smallest singular value of an i.i.d matrix


The paper 
 


