# Seminars & Events for Mathematical Physics Seminar

##### Critical points and the Gibbs measure of a spherical spin glass model

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 <x,y>. 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 of the graph of H_N(x) may be studies through its critical points and the corresponding values. Focusing on the pure p-spin models, I will review recent developments concerning the distribution of the number of critical values at a given height and the associated extremal point process.

##### Mean field evolution of fermonic systems

**Please note different day (Wednesday), time and location (Jadwin 303). **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 described by Hartree-Fock theory. The next degree of approximation is provided by a classical effective dynamics, corresponding to the Vlasov equation. I will consider initial data which are close to quasi-free states, at zero (pure states) or at positive temperature (mixed states), with an appropriate semiclassical structure.

##### Universality of transport coefficients in the Haldane-Hubbard model

**Please note special day (Wednesday), but usual room and time. **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. I will discuss the key properties of its phase diagram, most notably the phase transition from a standard insulating phase to a Chern insulator, across a critical line, where the system exhibits semi-metallic behavior.

##### Quantum analogues of geometric inequalities for Information Theory

** Please note special day (Monday), time (2:30) and location (Jadwin 303). **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-known inequalities from classical Information Theory, with the most notable being the isoperimetric inequality for entropies. The latter inequality is useful for the study of convergence of certain semigroups to fixed points.

##### Localization of interacting fermions with quasi-random disorder

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 for the stability of tori of nearly integrable hamiltonian systems, and relies on number-theoretic properties (Diophantine conditions) of the potential incommensurate frequency and phase.

##### Can one hear the shape of a random walk?

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 the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes.

Joint work with Marek Biskup

##### Ward and Belavin-Polyakov-Zamolodchikov (BPZ) identities for Liouville quantum field theory on the Riemann sphere

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 1981 theory of Liouville quantum gravity, is a class of CFTs which can be seen as a random version of the theory of Riemann surfaces. In a recent work, we constructed the correlation functions (and the random measures) of Liouville CFT in the Feynman path formalism using probabilistic techniques. In this talk, I will present a rigorous derivation of the so-called Ward and BPZ identities for Liouville CFT. These identities are the building blocks of the CFT formalism. Based on joint works with F.