Titles and abstracts will be added here later.
Talks on Monday 17th August (afternoon session)
Konstantin Izyurov (Helsinki): Laurent expansions of discrete holomorphic functions
Abstract: We will discuss a simple lemma concerning Laurent expansions of discrete holomorphic functions.
Kalle Kytölä (Aalto): Should we formalize mathematical physics?
Eero Saksman (Helsinki): Close to quasiconformal but not quite - analysis methods for random geometry
Alex Karrila (Åbo): Virasoro representations on boundary correlations in critical models
Abstract: I will try to explain a longer, still ongoing program that aims to relate certain representations of Virasoro algebra to boundary correlations in critical lattice models. Related articles have been co-authored by the speaker and K. Kytölä, A. Lafay, E. Peltola, J. Roussillon and L. Schoug.
Talks on Tuesday 18th August/morning
Antti Haimi (Åbo): Gaussian Weyl-Heisenberg functions
Abstract: I will present some basic results on the zeros of Gaussian Weyl–Heisenberg functions. These are random functions on the plane which are invariant under the action of the Weyl–Heisenberg group. The best-known example is the Gaussian Entire Function (GEF), but many other examples arise naturally in time-frequency analysis. Crucially, the GEF is the only analytic function in this family, so many of the techniques that have been used to study the zeros of the GEF do not extend to the other members of the family.
István Prause (Åbo): Quasiconformal methods for limit shape variational problems
Abstract: I will review some quasiconformal methods for variational problems as they came up in my research. The main motivation here is the variational principle for limit shape phenomena. The latter says that, in many models of statistical mechanics, random configurations are deterministic at a macroscopic scale albeit very sensitive to boundary conditions. I'll concentrate on structural questions in the relevant equations.
Mikhail Basok (Helsinki): Kenyon's identities for the dimer height function and compactified free field.
Abstract: In his seminal work published in 2000 Kenyon developed a method for analyzing the height function in the 2d dimer model by means of discrete complex analysis. The core of this method is formed by combinatorial identities expressing the correlations of heights at different points through the inverse of the so-called Kasteleyn matrix. Scaling limits of these identities, if exist, become identities between the correlations of the limiting field and the entries of the Dirac Green's kernel with some (potentially very complicated) boundary conditions depending on the combinatorics of the dimer model. A natural question is what the structure of these identities tell about the limiting field. In the case when the dimer model is sampled in a simply connected domain, a definitive answer was given in a recent work of Chelkak, Laslier and Russkikh: as soon as the correlations (but not the Dirac operator itself!) satisfy Dirichlet boundary conditions, the continuous Kenyon's identities imply that the limiting field is the Gaussian free field. In this talk we will discuss a generalization of this identification result to arbitrary Riemann surfaces.
Talks on Tuesday 18th August/afternoon
Liam Hughes (Aalto): Large deviations for SLE loops via conformal welding of quantum discs
Abstract: I will talk about ongoing work to prove a large deviations principle for the gamma-LQG boundary length on the unit circle, i.e. the Gaussian multiplicative chaos measure associated to gamma/2 times a Gaussian free field on the circle, as gamma goes to 0. When one conformally welds two such GMC measures, the interface is an SLE_kappa loop with kappa = gamma^2. Using the analytic study of this welding by Kupiainen, McAuley and Saksman, we show that the LDP for the measures implies an LDP for the SLE loops equipped with their natural parametrization.
Benoît Laslier (Paris Cité): TBA
Abstract: TBA
Talks on Wednesday 19th August/morning
Janne Junnila (KTH): On singularity lines of projective structures with real holonomy
Abstract: 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.
S.C. Park (Michigan): Bosonization of Planar Near-Critical Ising model
Abstract: We prove that the continuum primary field correlations of the doubled massive Ising model in 2D coincide with correlations of a Sine-Gordon model. This is a manifestation of the celebrated correspondence theorized by Sidney Coleman in 1975, at the free fermion point. We independently construct and match correlations on both sides in the case of compactly supported mass, then show that the correlations converge to well-defined quantities under a constant mass limit, which in particular corresponds to the regime where Painlevé transcendent appears in the full plane. Based on a joint work with Christian Webb and Tuomas Virtanen.
Tuomas Virtanen (Åbo): On the sine-Gordon Model: OPEs and Bosonization
Abstract: The sine-Gordon model is a fundamental example of an interacting quantum field theory. In this talk I will give a brief overview of the model and present the main results of my PhD thesis.
First, we explore the local structure of the model below the first threshold of collapse (β < 4π) by analyzing the singular terms in the OPEs of the derivative fields. This is based on joint work with A. Karrila and C. Webb.
Second, we look at the global duality of the model at the free fermion point (β = 4π) through bosonization - a remarkable correspondence linking a bosonic field theory to a fermionic one. In joint work with S. C. Park and C. Webb we present a rigorous "dictionary" that maps doubled Ising primary fields (spin, disorder, fermion, and energy) exactly to specific sine-Gordon fields inside correlation functions.
Talks on Thursday 20th August/morning
Alexander Bobenko (TU Berlin): Dimer Models and M-curves. Solutions via Inverse Problem.
Abstract: We develop a general approach to solution of the inverse problem for dimer models, in particular explicit description of limit shapes for the Aztec diamond and the Hexagon boundary conditions. This leads to dimer models on doubly periodic bipartite graphs with quasiperiodic positive weights. Dimer models with periodic weights are recovered as a special case. This generalization provides a qualitatively natural description of the models' phase states. Using variational descriptions, explicit representations for limit shapes are obtained in terms of Abelian integrals. Based on Schottky uniformization of Riemann surfaces, we compute the weights and dimer configurations. The computational results are in complete agreement with the theoretical predictions.
Mattias Eriksson (Åbo): Limit shapes of random lozenge tilings with holes
Abstract: A lozenge is a rhombus with a 60 degree angle. When placed edge-to-edge, lozenges create tilings of 2D regions. There are usually many distinct ways to tile a given region using lozenges. If we fix a region, let the lozenge size tend to zero and pick a tiling uniformly at random, the tiling will with overwhelming probability be close to a typical arrangement which has an associated shape. This shape is called a limit shape and it is deterministic. In this presentation, we outline the tangent plane method which can be used to find such limit shapes. Of particular interest are limit shapes of regions which contain a hole.
Shinji Koshida (Aalto): A homological model of a quantum group and conformal blocks
Abstract: The Kazhdan-Lusztig duality refers to a tensor equivalence of module categories for a vertex operator algebra and a quantum group. In a relatively recent instance, such a duality has been given between the Virasoro algebra and the quantum sl2 at generic parameter. There are in fact different ways to prove it, but one way is by directly constructing Virasoro conformal blocks out of the quantum sl2 passing through its homological model. I will report on the ongoing joint work with Alexis Langlois-Rémillard, where we are trying to push this picture to the case when the quantum parameter is a root of unity.