Strong Convergence

Here you can find information on the informal lectures about the strong convergence phenomenon, mostly following the excellent surveys of R. van Handel and of M. Magee . I will try to make an effort to make the talks accessible to a wide audience. We meet on Mondays 11:15-13:00 in Ziskind 155 at the Weizmann Institute. The lectures will be broadcasted on this Zoom link.

• First Meeting: Monday Jan 26 at 11:15 - 12:00, in Ziskind 155, Zoom link. 

   Introduction and origins of strong convergence (Sections 3.1 and 3.2 in  [van Handel]), lecture notes, recording (note first hour is from an older lecture).

   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. We will prove Pisier's Linearization trick and de la Salle's positivization trick. All necessary background from operator algebras will be provided. 

• Second Meeting: Monday Feb 9 at 11:15 - 12:00, in Ziskind 155, Zoom link. 

   Strong convergence for surfaces groups and resolution of Buser's conjecture I ([Magee]) .

   Synopsis: We will discuss strong convergence for general group representations and actions. we will clarify the situation for amenable groups, then give a survey of which "rank one" groups are known to have strongly convergent actions. As a spectacular application, we present the resolution of Buser's conjecture, by constructing closed, arithmetic hyperbolic surfaces of growing genera whose spectral gap approaches 1/4, following [Louder-Magee], [Hide-Magee]. 

• Second meeting Monday  Feb 9 11:15-13:00

• Third meeting: Monday  Feb 16 11:15-13:00

• Fourth meeting: Monday  Feb 23 11:15-13:00

• Fifth meeting: Monday  March 2 11:15-13:00

Past Meetings (2025):

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

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

   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).

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

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.

• Wednesday Oct 15, Room 155 Ziskind Building, Weizmann Institute.

--- Morning session,10:15 - 12:00: Introduction and motivation (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.