In a seminal paper in 2012, Fyodorov & Keating proposed a series of conjectures describing the statistics of large values of zeta in short intervals of the critical line. In particular, they relate these statistics to the ones of log-correlated Gaussian fields. The objective of the two lectures is to present a general strategy for proving these conjectures relying on previous works on branching random walks and log-correlated fields. In the first lecture, we will focus on a probabilistic model of zeta introduced by Harper where one can get a precise description of the maximum and of the moments. In particular, we will see how the Fyodorov-Keating conjectures can be extended to intervals of diverging length. In the second lecture, we will adapt these arguments to obtain the leading order of the maximum and of the moments of the zeta function in short intervals unconditionally.

We recall some basic properties of Gaussian multiplicative chaos and point out several model where it plays a role.

I will discuss how certain random fractal measures, known as multiplicative chaos measures, can be used to study extreme values of logarithmically correlated fields, that is stochastic processes whose covariance has a logarithmic singularity on the diagonal. Moreover, I will briefly discuss the role of such an approach in the setting of random matrix theory. Time permitting, I will also discuss some open questions such as the role of complex multiplicative chaos.

We shall present the links between Gál sums and the resonance method and describe applications of recent bounds for these quantities obtained with La Bretèche. These concern : lower bounds for localised maxima of the Riemann zeta function, large values of Dirichlet L function at 1/2, and large values of character sums.

In a seminal paper in 2012, Fyodorov & Keating proposed a series of conjectures describing the statistics of large values of zeta in short intervals of the critical line. In particular, they relate these statistics to the ones of log-correlated Gaussian fields. The objective of the two lectures is to present a general strategy for proving these conjectures relying on previous works on branching random walks and log-correlated fields. In the first lecture, we will focus on a probabilistic model of zeta introduced by Harper where one can get a precise description of the maximum and of the moments. In particular, we will see how the Fyodorov-Keating conjectures can be extended to intervals of diverging length. In the second lecture, we will adapt these arguments to obtain the leading order of the maximum and of the moments of the zeta function in short intervals unconditionally.

In the first part of this talk, we shall provide details for the proof of estimates for Gál sums obtained jointly with Gérald Tenenbaum. In the second part, we present joint results with Marc Munsch related to the minimal values of certain sums of Gál type, with applications.

I will discuss some recent work with de la Bretèche on the variance of divisor functions in progressions as well as some of my past work on Hooley's conjecture on primes in progressions. I will show how these questions are linked with random matrix theory as well as large deviations of random variables.

One can form a "partition function" from the Riemann zeta function by looking at integrals of powers of zeta over short intervals. A particularly interesting case, which turns out to be closely connected to critical multiplicative chaos, is that of short integrals of the mean square of zeta. I will describe how one can obtain sharp upper bounds for the low moments of this partition function, and the consequences for the maximum of zeta in short intervals.

For different random fields whose correlation is logarithmic with respect to the distance between the points, we observe similar behavior for their extreme values, either proven or conjectured, depending on the model. In this talk, we present two different examples of such fields: the logarithm of the characteristic polynomial of the Circular Beta Ensemble (an ensemble of random unitary matrices generalizing the Circular Unitary Ensemble), and the logarithm of the Riemann zeta function, on a random interval of the critical line.

In this talk, I would like to advertise an equality between two objects from rather different areas of mathematical physics. This bridges the Gaussian Multiplicative Chaos, which plays an important role in certain conformal field theories, and a reference model in random matrices, the CBE (Circular Beta Ensemble). The main tool is an explicit description of canonical moments aka Verblunsky coefficients. On the one hand, in 1985, J.P Kahane introduced a random measure called the Gaussian Multiplicative Chaos (GMC). Morally, this is the measure whose Radon-Nikodym derivative w.r.t to Lebesgue is the exponential of a log correlated Gaussian field. In the cases of interest, this Gaussian field is a Schwartz distribution but not a function. As such, the construction of GMC needs to be done with care. In particular, in 2D, the GFF (Gaussian Free Field) is a random Schwartz distribution because of the logarithmic singularity of the Green kernel in 2D. Here we are interested in the 1D case on the circle. On the other hand, it is known since Verblunsky (1930s) that a probability measure on the circle is entirely determined by the coefficients appearing in the recurrence of orthogonal polynomials. Furthermore, Killip and Nenciu (2000s) have given a realization of the CBE, an important model in random matrices, thanks to random orthogonal polynomials of the circle. I will give the precise statement whose loose form is CBE = GMC modulo a relation between the coupling constant \beta in the CBE and the coupling constant of GMC.

I will motivate and present results concerning moments of moments of characteristic polynomials of random unitary matrices. The study of moments of moments has been furthered through connections with statistical mechanics, Gaussian multiplicative chaos, representation theory, and additionally can be seen to model certain number theoretic functions. This is joint work with Jon Keating.

The Alternative Hypothesis concerns a hypothetical (and unlikely) arrangement of the zeros of the Riemann zeta function in which the distances between zeros are very nearly half-integer multiples of the average spacing. Ruling out the alternative hypothesis would have a number of interesting consequences in number theory. In this talk I will discuss joint work with J. Lagarias and independent work of T. Tao demonstrating that what is presently known about the local distribution of zeros is not sufficient to rule out the Alternative Hypothesis. This is done via the construction of an explicit counterexample point process. I will also discuss some broader problems related to mimicking one point process by another at a "blurry" band-limited resolution.