Talks

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of alge...

I will revisit some integrable difference equations arising in the study of the distance statistics of random planar maps (discrete surfaces built from polygons). In a paper from 2003 written jointly with P. Di Francesco and E. Guitter, we conjectured a general formula for the so-called ``two-point function'' characterizing these statistics. The first proof of this formula was given much later in a paper from 2012 joint with E. Guitter, where we used bijective arguments and the combinatorial theory of continued fractions. I will present a new elementary and purely analytic proof of the result, obtained by considering orthogonal polynomials with respect to a polynomial deformation of the Wigner semicircle distribution. This talk is based on a work in progress with Sofia Tarricone.
 

I will first present a generic argument to derive large deviations of a stochastic process when large deviations of certain functionals of that process are available. I will then apply such a general argument to the analysis of the lower tail of the height functions of the stochastic six vertex model starting with step initial conditions. One of the main novelties will be a proof of weak logarithmic concavity of the cumulative distribution function of the height function. This is a joint work with Sayan Das and Yuchen Liao.

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.

The quest for Anderson localization of light is at the center of many experimental and theoretical activities. Cold atoms have emerged as interesting quantum system to study coherent transport properties of light. Initial experiments have established that dilute samples with large optical thickness allow studying weak localization of light, which has been well described by a mesoscopic model. Recent experiments on light scattering with cold atoms have shown that Dicke super- or subradiance occurs in the same samples, a feature not captured by the traditional mesoscopic models. The use of a long range microscopic coupled dipole model allows to capture both the mesoscopic features of light scattering and Dicke super- and subradiance in the single photon limit. I will review experimental and theoretical state of the art on the possibility of Anderson localization of light by cold atoms.

The Cherenkov Telescope Array Observatory (CTAO) is the upcoming next-generation ground-based very-high-energy (VHE) gamma-ray observatory. The CTAO will significantly advance the study of VHE gamma-rays through a combination of wider field of view,  substantially increased detection area, and superior angular and spectral resolution over an energy range extending from tens of GeV to hundreds of TeV. Full-sky coverage will be achieved using two independent Imaging Air Cherenkov Telescope (IACT) arrays: one in the northern hemisphere (Canary Islands, Spain) and one in southern hemisphere (Paranal, Chile). The CTAO will explore a wide range of science topics in high-energy astrophysics, including the origin of higher-energy cosmic rays, mechanisms for particle acceleration in extreme environments, and astroparticle phenomena that may extend the Standard Model of particle physics.  In this talk, I will outline the broad science potential of the CTAO and provide the CTAO’s current status and timeline. I will also describe the contributions of the CTAO-US collaboration to CTAO, including the development of an ultra-high resolution Schwarzschild-Couder telescope for VHE astronomy and the emergence of UV-band optical astronomy at the sub-100 micro-arcsecond angular scale.

The generation of large scales white noise is a generic property of the dynamics of physical systems described by local non-linear partial differential equations. Non-linearities prevent the small scale dynamics to be erased by smoothing. Unresolved small scale dynamics act as an uncorrelated (white or Poissonian) noise (seemingly stochastic but actually deterministic) contribution to large scale dynamics. Such is the case for cosmic inhomogeneities. In the standard model of cosmology the primordial density power spectrum is taken to be sub-Poissonian and subsequent non-linear evolutions will inevitably produce white noise which will dominate on the largest scales. Non-observation of white noise on the Hubble scale precludes a power law extrapolation of the power spectrum below one comoving parsec and places severe constraints on a wide variety of phenomena in the early universe, including phase transitions, vorticity and gravitational radiation.

I will discuss the relation between non-compact spin chains and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. Guided by the desire to maintain the integrable structure, I will introduce boundary conditions for these models that are obtained from the boundary Yang-Baxter equation. This allows to define analogues of the open SSEP and ASEP with boundary reservoirs. Some recent exact results concerning these types of integrable non-equilibrium zero-range processes will be discussed.