Strong Convergence

Starting 15.10.25, I plan to give a series of informal lectures about the strong convergence phenomenon, mostly following the excellent survey of Ramon van Handel. I will try to make an effort to make the talks accessible to a wide audience. The lectures will be broadcasted on Zoom.

• First meeting:

Details: Wednesday Oct 15, Room 155 Ziskind Building, Weizmann Institute, Zoom link.

--- Morning session,10:15 - 12:00: Introduction (Section 1 and Section 5.3 of [van Handel]), lecture notes, recording

Synopsis: We will introduce the main topic of strong convergence, and mention some of its remarkable applications to almost optimal spectral gaps of graphs and manifolds, C* and von Neumann algebras and random matrix theory. 

--- Afternoon session,14:15 - 16:00: Basic properties of strong convergence (Section 2 of [van Handel]), lecture notes (no recording)

Synopsis: We will give a brief introduction to C*-probability spaces, and present basic general properties of strong convergence and weak convergence of random matrices. this includes a relation to finite dimensional approximations of operator algebras. All necessary background from operator algebras will be provided.

• Second meeting: 

    Details : Tuesday Oct 21 at 10:15 - 12:00, in Room 155 Ziskind Building, Weizmann Institute, Zoom link. 

Strong convergence of random permutation matrices I: (Sections 3.1 and 3.2 in  [van Handel]), lecture notes, recording

synopsis: We will introduce the general proof strategy of [Chen, Garza-Vargas, Tropp, van Handel] to prove the Bordenave-Collins theorem on strong asymptotic freeness of random permutations, known as the polynomial method. We will then dive into the first step of the proof, involving moments of word measures.

• Third meeting:

    Details : Wednesday Nov 12at 10:15 - 12:00, in Moross Lab meeting room 34 (Ziskind ground floor), Zoom link. 

   Strong convergence of random permutation matrices I: (Sections 3.1 and 3.2 in  [van Handel])

   synopsis: We complete the proof of [Chen, Garza-Vargas, Tropp, van Handel] to prove the Bordenave-Collins theorem. This will be done via classical tools from the analytic theory of polynomials (such as Markov brother's inequality and Chebyshev polynomials), together with Haagerup's inequality and the positivisation trick (due to de la Salle).