__October 8__

Jorge Garza-Vargas (UC Berkeley)

Title: Spectral stability under random perturbations

Abstract: Consider an \(n\times n\) deterministic matrix \(A\) and a random matrix \(M\) with independent standard Gaussian entries. In this talk I will discuss recent results that state that, if \(||A||\leq 1\), for any \(\delta>0\), with high probability \(A+\delta M\) has eigenvector condition number of order poly\((n/\delta)\) and eigenvalue gaps of order poly\((\delta/n)\), which implies that the randomly perturbed matrix has a stable spectrum. This has useful applications to numerical analysis and was used to obtain the fastest known provable algorithm for diagonalizing an arbitrary matrix.

This is joint work with Jess Banks, Archit Kulkarni and Nikhil Srivastava.

__October 29 (Special time: 10 AM to 11 AM PST)__

Guillaume Dubach (IST Austria)

Title: Overlaps between eigenvectors of non-Hermitian random matrices

Abstract: Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps.
The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics.
Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998).
For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade).
Similar results have been obtained in other integrable settings, namely quaternionic Gaussian matrices,
as well as matrices from the spherical and truncated unitary ensembles.

__November 19__

Ben Landon (MIT)

Title: Fluctuations of the 2-spin spherical Sherrington Kirkpatrick model

Abstract: The 2-spin spherical Sherrington Kirkpatrick (SSK) model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual Sherrington-Kirkpatrick model with Ising spins. The SSK model admits tractable formulas for many of its observables, allowing for a detailed analysis of its fluctuations using techniques from random matrix theory. We discuss recent results on the fluctuations of the SSK model with a magnetic field as well as at critical temperature. Based on joint work with P. Sosoe.

__December 3__

Amber Puha (CSU San Marcos and UC San Diego)

Title: Scaling Limits for Shortest Remaining Processing Time Queues

Abstract: In an SRPT queue, the job with the shortest remaining processing time is served first, with preemption. The SRPT scheduling rule is of interest due to its optimality properties; it minimizes queue length (number of jobs in system). However, even with Markovian distributional assumptions on the processing times, an exact analysis is not possible. Hence, approximations in the form of a fluid (functional law of large numbers) limit or a diffusion (functional central limit theorem) limit can provide insights into system performance. It was shown by Gromoll, Kruk and Puha (2011) that, if the processing time distribution has unbounded support, then, under standard heavy traffic conditions, the diffusion limit of the queue length process is identically equal to zero. This exhibits the queue length minimization property of SRPT in sharpest relief. It also demonstrates that the SRPT queue length process is orders of magnitude smaller than the workload process in the diffusion limit. (The workload process tracks the time it will take the server to process the work associated with each job in system.) In this talk, we report on progress in characterizing this order of magnitude difference. We find that distribution dependent scaling must be used to obtain a nontrivial limit for the queue length and the associated measure valued state descriptor. The scaling captures the order of magnitude difference, and the nature of the limit is dependent on the tail decay of the processing time distribution.

This work is joint with Sayan Banerjee (UNC) and Amarjit Budhiraja (UNC).