# The 2017 Charles River Lectures

Jointly organized by Harvard University, Massachusetts Institute of Technology, and Microsoft Research New England, the Charles River Lectures on Probability and Related Topics is a one-day event for the benefit of the greater Boston area mathematics community.

The 2017 lectures will take place 9:15am – 5:30pm on Monday, October 2 at Harvard University in the Harvard Science Center.

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*UPDATED LOCATION*

### Harvard University

### 1 Oxford Street, Cambridge, MA 02138 (Map)

### Monday, October 2, 2017

### 9:15 AM – 5:30 PM

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Please note that registration has closed.

**Speakers:**

**Agenda:**

*In Harvard Science Center Hall C:*

8:45 am – 9:15 am**: ***Coffee/light breakfast*

**9:15 am – 10:15 am: Ofer Zeitouni**

*Title: *Noise stability of the spectrum of large matrices

*Abstract: *The spectrum of large non-normal matrices is notoriously sensitive to perturbations, as the example of nilpotent matrices shows. Remarkably, the spectrum of these matrices perturbed by polynomially (in the dimension) vanishing additive noise is remarkably stable. I will describe some results and the beginning of a theory.

The talk is based on joint work with Anirban Basak and Elliot Paquette, and earlier works with Feldheim, Guionnet, Paquette and Wood.

**10:20 am – 11:20 am:** **Andrea Montanari**

*Title: *Algorithms for estimating low-rank matrices

*Abstract: *Many interesting problems in statistics can be formulated as follows. The signal of interest is a large low-rank matrix with additional structure, and we are given a single noisy view of this matrix. We would like to estimate the low rank signal by taking into account optimally the signal structure. I will discuss two types of efficient estimation procedures based on message-passing algorithms and semidefinite programming relaxations, with an emphasis on asymptotically exact results.

11:20 am – 11:45 am**: ***B**reak*

**11:45 am – 12:45 pm:** **Paul Bourgade**

*Title: *Random matrices, the Riemann zeta function and trees

*Abstract: *Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.

1:00 pm – 2:30 pm: *Lunch*

*In Harvard Science Center Hall E:*

**2:45 pm – 3:45 pm: Roman Vershynin**

*Title: *Deviations of random matrices and applications

*Abstract: *Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This lecture will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.

3:45 pm – 4:15 pm: *Break*

**4:15 pm – 5:15 pm:** **Massimiliano Gubinelli**

*Title: *Weak universality and Singular SPDEs

*Abstract:* Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. This universality comes at a price: due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality and their relation with the Wilsonian renormalisation group framework of theoretical physics.

**Poster:**

**Organizers:**

**Alexei Borodin, Henry Cohn, Vadim Gorin, Elchanan Mossel, Philippe Rigollet, Scott Sheffield, and H.T. Yau**