Talks

In this talk, I'll describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make these connections explicit. Recently, a new tableau formula was found for the modified Macdonald polynomial $\widetilde{H}_{\lambda}$ in terms of a queue inversion statistic that is naturally related to the dynamics of the TAZRP. We give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ using the same queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle, and the queue inversion statistic encodes to the dynamics of the ASEP.  Our tableaux are in bijection with Martin's multiline queues, from which we obtain an alternative multiline queue formula for $P_{\la...

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...

In this talk, the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice is revisited. A central role is played by a quantum oscillator-weighted five vertex model, which features an unusual weight conservation distinct from the conventional one. This approach clarifies the interrelations among several known results and refines their derivations, including the multiline queue construction and matrix product formulas. (Joint work with Masato Okado and Travis Scrimshaw)

The quantum K-theory of the flag variety is a ring defined by introducing a quantum product to the K-theory of the flag variety. Under appropriate localization, it is known that the following three rings (i), (ii), and (iii) are isomorphic, and this property allows for a detailed investigation of each ring: (i)the coordinate ring of the phase space of the relativistic Toda lattice, (ii) the quantum equivariant K-theory of the flag variety, and (iii) the K-equivariant homology ring of the affine Grassmannian.

The isomorphism between (i) and (ii) is derived from the Lax formalism of the relativistic Toda lattice [Ikeda-Iwao-Maeno]. The isomorphism between (ii) and (iii) is referred to as the K-Peterson isomorphism [Lam-Li-Mihalcea-Shimozono, Kato, Chow-Leung, Ikeda-Iwao-Maeno]. In this talk, I will outline how techniques from classical integrable systems, such as the construction of algebraic solutions and Bäcklund transformations, are applied to the study of geometry. This talk is ba...

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.

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 present an overview of the work I have done over the last few years on the phase space structure of gauge theories in the presence of boundaries. Starting with primers on the covariant phase space and symplectic reduction, I then explain how their generalization when boundaries are present fits into the reduction-by-stages framework. This leads me to introduce the concept of (classical) superselection sectors, whose physical meaning is clarified by a gluing theorem. Applying the framework developed this far to a null hypersurface, I then discuss how the extension of the Ashtekar-Streubel symplectic structure by soft modes emerges naturally, and how electric memory ties to superselection. If time allows, and depending on the audience’s interests, I will finally compare reduction-by-stages with the edge-mode formalism or discuss its relation to dressings and “gauge reference frames”. An overarching theme will be the nonlocal nature of gauge theories. This seminar is based on work done with Gomes and Schiavina.   References: The general framework: 2207.00568 Null Yang-Mills: 2303.03531 Gluing: 1910.04222 A pedagogical introduction: 2104.10182 Dressings and reference frames: 1808.02074, 2010.15894, 1608.08226

Accretion flows aroud black-holes, neutron stars or white dwarfs are studied since almost 60 years. Although they are ubiquitous and somewhat similar over scales reaching billions in mass and size, their study has been limited because they remain unresolved point like sources in the optical/ultraviolet and X-rays, where they emit. Two main modes of accretion have been identified in Active Galactic Nuclei. In most sources the accretion rate is low and a high pressure, low density, low collision rate, optically thin, radiatively inefficient, two temperature plasma can form (Shapiro 1976; Narayan & Yi 1994,1995). This solution is stable only for low luminosities (<1% LEDD). The Event Horizon Telescope has recently resolved such flows in Sgr A and M87, confirmed several aspects of the model and could detect particles accelerated close to the horizon of Sgr A (Wielgus, 2022) a likely signature of the Blandford-Znajek (1977) process. When the accretion rate is higher, momentum can be dissipated by viscosity and the flow proceeds via geometrically thin disk-shaped structures. These accretion disks provide feedback to their environment by accelerating winds and launching jets in their central regions. The apparent size of accretion disks are of the order of 1-40µarcsec in nearby quasars, Seyfert galaxies and galactic cataclysmic variables and of 0.1-1µarcsec in of low mass X-ray binaries in our Galaxy. Hanbury-Brown & Twiss (1954) invented intensity interferometry and measured the size of some bright stars by correlating the arrival times of photons detected by two optical telescopes. The physics has been explained as a quantum effect in the early 60s (Fano 1961) and has triggered the development of quantum optics (Glauber 1963). Its root is found in the quantum theory of statistical fluctuations in an ideal gas (Einstein 1925). The achievable signal-to-noise depends on the telescope size, the detector time resolution, and the number of spectral channels observed simultaneously. Extremely large telescope and 10ps resolution single photon detectors bring the key improvements to reach in the optical angular resolutions better than these achieved in the radio by the Event Horizon Telescope and to obtain the first images of accretion disks around galactic and extragalactic compact objects, a breakthrough. I will present the goals and the status of the QUASAR project, which started one year ago, aiming at bringing a 10ps resolution optical spectrometer on very large telescope.