The foundations of modern conformal field theory (CFT) were introduced in a 1984 seminal paper by Belavin, Polyakov and Zamolodchikov (BPZ). Though the CFT formalism is widespread in the physics literature, it remains a challenge for mathematicians to make sense out of it. Liouville CFT (or quantum field theory), introduced by Polyakov in his…

We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to…

We consider interacting electrons in a one dimensional lattice with an incommensurate Aubry-Andre' potential in the regime when the single-particle eigenstates are localized. We rigorously establish persistence of ground state localization in presence of weak many-body interaction.The proof uses a quantum many body extension of methods adopted…

Geometric inequalities, such as entropy power inequality or the isoperimetric inequality, relate geometric quantities, such as volumes and surface areas. Classically, these inequalities have useful applications for obtaining bounds on channel capacities, and deriving log-Sobolev inequalities. In my talk I provide quantum analogues of certain well…

In this talk I will review some selected aspects of the theory of interacting electrons on the honeycomb lattice, with special emphasis on the Haldane-Hubbard model: this is a model for interacting electrons on the hexagonal lattice, in the presence of nearest and next-to-nearest neighbor hopping, as well as of a transverse dipolar magnetic field…

In this talk I will discuss the dynamics of interacting fermionic systems in the mean field regime. Compared to the bosonic case, fermionic mean field scaling is naturally coupled with a semiclassical scaling, making the analysis more involved. As the number of particles grows, the quantum evolution of the system is expected to be effectively…

For integers N let H_N(x) be an isotropic Gaussian field on the N-dimensional unit sphere, meaning that Cov(H_N(x),H_N(y)) is a function, f_N, of the inner product of . The spherical spin glass models of statistical mechanics are such random fields, with f_N = N f with the function independent of the dimension N. The intricate landscape…

We consider a non-relativistic quantum particle surrounded by a detecting surface and ask how to compute, from the particle's initial wave function, the probability distribution of the time and place at which the particle gets detected. In principle, quantum mechanics makes a prediction for this distribution by solving the Schrodinger equation of…

Starting with the work of Lebowitz-Rose-Speer in 1988, there has been great interest and significant progress in establishing invariance of certain Gibbs-type measures with respect to dispersive equations in 1 and 2 dimensions. The invariant measures in question are absolutely continuous with respect to the free field, and thus concentrated…

Starting with an analysis of chiral edge currents in 2D electron gases exhibiting the quantum Hall effect I will discuss the role of anomalous chiral edge currents and of anomaly inflow in 2D insulators with explicitly broken parity and time-reversal and in time-reversal-invariant 2D topological insulators exhibiting edge spin-currents. I will…

The lace expansion provides a formula for the two point function which has been useful for critical percolation, self-avoiding walk and related problems in high dimensions. Recently Akira Sakai has shown that the Ising model and the one component $\phi^4$ model admit similar formulas. I will review the basic features of the lace expansion…

Einstein's classic treatment of Bose-Einstein condensation for non-interacting bosons predicts at what temperature the phase transition to BEC occurs, but how this critical temperature changes when the bosons start to interact weakly is an old problem, first posed by Feynman in 1953. In this talk, I will address this question in a variational…

I will describe a potential mechanism for a singular solution of the Euler equation. The mechanism involves the interaction of vortex filaments, but occurs sufficiently quickly and at a small enough scales that could have plausibly evaded experimental and computational detection. Joint work with Sahand Hormoz and Alain Pumir.

Using the coherent state transform I willI establish the asymptotical behaviour of the Riesz mean for functional-difference operators associated to mirror curves of special del Pezzo Calabi–Yau threefolds. Furthermore, I will prove the Weyl law for the eigenvalue counting function of these operators, therefore implying that their inverses are…

Understanding the various aspects, and even the qualitative structure of phase diagrams of interacting many-body systems in the presence of static disorder still poses a big challenge. In this context, and in view of the woefully short list of rigorous results on disordered systems with interaction, limiting or integrable model systems present a…