Schedule

On singularity lines of complex projective structures with real holonomy

 I will discuss certain curve systems on the punctured Riemann sphere that are induced by complex projective structures with real holonomy. Such curves appear e.g. in semiclassical limits of the Schramm-Loewner evolution. A central question is whether every isotopy class of a curve system can be realized as the singularity lines of some projective structure, and whether it is unique. With Bonk, Rohde and Wang we answered this affirmatively in the special case when the projective structure is induced by a differentiable Jordan curve passing through the punctures and satisfying the following geodesic property: Every arc on the curve is a hyperbolic geodesic in the domain bounded by all the other arcs. These curves turn out to be Loewner energy minimizing in their isotopy classes, and we also proved that the accessory parameters of the associated Schwarzian derivative satisfy an identity analogous to the famous formula conjectured by Polyakov and proven by Takhtajan and Zograf in the Fuchsian case. In addition to explaining these results, I will also showcase a few examples of other such curve systems whose projective structures can be explicitly described thanks to certain symmetries and holonomical constraints.

Integrable topological solitons in the Gross-Neveu model

It is believed that the Gross-Neveu QFT features factorized scattering and the S-matrix has been proposed. We start from the scattering amplitudes of solitonic particles carrying the spinor representation of the global symmetry group O(2N) and characterize the ground state and excitations in a chemical potential. In the large-N limit, we find that the dispersion is described by a Dirac operator with an elliptic potential, which relates the system to the Lamé equation and mKdV.

Loewner Chains and their driving measures

Charles Loewner’s seminal paper in 1923 Loewner chains have been a powerful tool in Mathematics. Originally, applied to distortion estimates for univalent functions such as Bieberbach’s conjecture/de Branges theorem, Loewner chains have been rediscovered as a method to describe random curves, e.g. Schramm-Loewner Evolutions, and random growth models such as Diffusion Limited Aggregation (DLA) or Hasting-Levitov type growth models. This works as by the Riemann Mapping Theorem there is a bijection between Loewner chains and (continuously) growing simply connected sets in the plane.

Crucially, there is also a bijection between Loewner chains and finite Borel measures which constitutes an analytic representation theorem. One direction of this bijection is given by the famous Loewner-Kufarev Equation. We were interested in the converse question: How can the driving measure be obtained from its Loewner chain. In particular, we obtain explicit formulations of the driving measure and an expression for the density of the driving measure.

Non-isotropic fractal percolation and random cascade measures

In recent decades, considerable progress has been made in understanding the fine geometric and measure-theoretic properties of fractal percolation and cascade measures, which found applications for explicit proofs of the KPZ relation in one dimension (Benjamini–Schramm 2009, Barral–Kupiainen–Nikula–Saksman–Webb 2014).

A recent direction is to move beyond stochastically self-similar models towards non-isotropic, or stochastically self-affine, variants. In this talk, I will survey some of these developments, focusing on self-affine percolation models, including directed percolation, and discuss connections with random cascade measures and related applications.

(Based on joint work with István Kolossváry.)

Superdiffusivity of the stochastic Burger’s equation and of a critical diffusion

In this talk I discuss two “driven diffusive systems”, that is models that combine diffusive effects with a forcing mechanism. The first is a diffusion given by the SDE

dX_t = F (X_t)dt + dB_t

where F is a random drift field and the second is an SPDE given by

∂_tu = ∆u + (w · ∇u²) + ∇ · ξ,

where w is a fixed vector, and ξ is space time white noice.

The large scale behaviour of these models depends highly on the dimension. In dimension 3 and higher, the diffusive effects prevails, while in dimension 1 the nonlinearity dominates. In the scaling critical dimension 2 the behaviour is more subtle and a logarithmic correction to the diffusivity occurs. 

In the talk I will show how to use Fock space analysis to show this behaviour and give an overview of other results in this area. In particular I will contrast the two different universality classes of these models.

Based on joined work with Giuseppe Cannizzaro, Damiano De Gaspari and Fabio Toninelli

W_3 BPZ equations in critical lattice models from sl_3 webs

Webs are combinatorial objects (some graphs and reduction rules) that are intimately connected with the representation theory of quantum groups. The simplest example is the sl_2 case and Temperley-Lieb link patterns. These link patterns describe connection probabilities of sets of curves in various lattice models, for instance Ising/percolation interfaces, double-dimers paths. In the scaling limit these probabilities can be described in terms of multiple Schramm Loewner evolution partition functions, that are in particular solutions of a set of so-called Beliavin-Polyakov-Zamolodchikov equations originating from conformal field theory. In this talk, I will discuss recent work on a higher rank analogue of this story. I will present rigorous results on the triple-dimer model as well as some numerical conjectures on the 3-states Potts model.

Noise Sensitivity in Last Passage Percolation

In this talk, we will discuss the effect of small perturbation in the initial weights on the travel time in last passage percolation between two distant points. In the last passage percolation model we assign non-negative weights to nodes of a square lattice and consider paths between two fixed points that are allowed to move only right or up. The travel time is defined as the maximal total weight collected along such paths. We will show that after applying a certain type of perturbation, the resulting noisy travel time becomes asymptotically uncorrelated with the original travel time.

Ising local fields and their Virasoro counterparts

We construct a correspondence between local field in the critical Ising model on a lattice, i.e., formal polynomials in lattice spins, and vectors in a vector space carrying two commuting representations of the c=1/2 Virasoro algebra. As expected in the physics literature, the module in question is a direct sum of the highest weight unitary modules of weights 0, 1/16 and 1/2, and the highest weight vectors correspond to the constant field, the spin at a single lattice vertex, and the energy at a single lattice edge.

We show that a local field is a null field, i.e., vanishes in all correlations with any other sufficiently distant fields, if and only if it corresponds to the zero vector. Otherwise, when rescaled by the lattice mesh taken to a suitable power, it converges, inside correlations, to a non-trivial scaling limit. All these limits can be expressed in terms of correlations of primary fields in the continuum, which are well understood. This is a joint work with Dmitry Chelkak and Kalle Kytölä.