Unit of Mathematics, Pure and Applied

We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus. We set temperature, chemical potential and coupling constant in a regime where classical field theory gives leading order asymptotics. In the same limit, the repulsive interaction potential is set to be short-range: it converges to a Dirac delta function with a rate depending polynomially on the other scaling parameters. We prove that the free-energy of the interacting Bose gas (counted relatively to the non-interacting one) converges to the free energy of the $Φ^4_2$ non-linear Schr{ö}dinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin-Nam-Rougerie to connect, with controled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the $Φ^4_2$ free energy then follows from arguments of Fr{ö}hlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.

We study the Gibbs equilibrium of a classical 2D Coulomb gas in the determinantal case = 2. The external potential is the sum of a quadratic term and the potential generated by individual charges pinned in several extended groups. This leads to an equilibrium measure (droplet) with flat density and macroscopic holes. We consider ''correlation energy'' (free energy minus its mean-field approximation) expansions, for large particle number . Under the assumptions that the holes are sufficiently small, separated, and far from the droplet's outer boundary, we prove that (i) the correlation energy up to order 1 is independent of the holes' locations and orientations, and (ii) the difference between the correlation energies of systems differing by their number of holes essentially consists of ``topological'' $O(\log N)$ and $O(1)$ terms.

We show that the recent work by G{\'e}rard-Kappeler-Topalov can be used in order to construct new non degenerate invariant measures for the Benjamin-Ono equation on the Sobolev spaces H s , s > --1/2.

We compute explicitly the Goncharov regulator integral associated to $K_4$ classes on modular curves in terms of $L$-values of modular forms. We use this expression to connect it with the Beilinson regulator integral.

Inspired by the Mordell–Tornheim double zeta function, in this paper, we introduce the Eisenstein series of Mordell–Tornheim type. Based on the theory of Cohen series and partial fraction decomposition, we explicitly compute these series. We end by discussing several identities about products of Eisenstein series and their Fourier expansions.

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.

Abstract The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.