Department of

Mathematics


Seminar Calendar
for Graduate Probability Seminar events the year of Thursday, July 2, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 30, 2020

2:00 pm in 347 Altgeld Hall,Thursday, January 30, 2020

The Semicircle Law for Wigner Matrices

Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. Talk 1 will sketch out the proof via the Stieltjes transform and via the energy entropy balance.

Thursday, February 6, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 6, 2020

The Semicircle Law for Wigner Matrices Part 2

Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. talk two will discuss the proof based on the method of moments and its relation to enumerative combinatorics.

Thursday, February 13, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 13, 2020

Distribution of eigenvalues of random matrices (part I)

Peixue Wu (UIUC Math)

Abstract: Last time we proved a famous semicircular law for the limit distribution of the empirical measure of the eigenvalues of Wigner's matrix (i.i.d. under the symmetry restriction). When we go over the proof in detail, we find two essential ingredients to the proof: 1. Stochastic independence of the entries. 2. Most matrix entries are centered and have the same variance. Using the similar idea (methods of moments) we will show that semicircular law holds for a much larger class of random matrices. We will also talk about the joint distribution for the eigenvalues of the Gaussian Orthogonal (Unitary) Ensembles (GOE or GUE).

Thursday, February 20, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 20, 2020

Distribution of eigenvalues of random matrices (part II)

Peixue Wu (UIUC Math)

Abstract: Last time we proved the classical Wigner's semicircular law for Wigner matrix. This time I will state a dynamical version of the semicircular law, which implies the classical Wigner's semicircular law. Our main tool will be stochastic analysis.

Thursday, February 27, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 27, 2020

Tracy Widom Distribution and Spherical Spin Glass (Part I)

Qiang Wu (UIUC Math)

Abstract: We studied the global behavior of eigenvalues of random matrices in previous talks. This time we are going to zoom into the bulk to study some local behavior of eigenvalues. In particular, the edge scaling limit of largest eigenvalue is given by the Tracy-widom (TW) distribution, which as a universal object also appears in some other areas, like growth process, spin system and many other interacting particle systems. Taking GUE as our example, we will try to derive the TW distribution represented as a Fredholm determinant with Airy Kernel. Time permits, we will briefly go through the integral representation of TW, and some universality results even extended to the underlying integrable system for general beta ensembles.

Thursday, March 12, 2020

2:00 pm in 347 Altgeld Hall,Thursday, March 12, 2020

Tracy-Widom distribution and spherical spin glass (Part II)

Qiang Wu (UIUC Math)

Abstract: I will talk about the connection between spherical spin glass(SSK) and random matrices, in particular, the fluctuation of free energy in SSK on low temperatures regime is given by GOE Tracy-Widom distribution.