In theoretical physics, the critical 2D Ising model typically serves as a toy example, in which many precursors of Conformal Field Theory objects and structures exist and can be studied directly in discrete, before passing to the small mesh size limit. Mathematically, a number of results on convergence and conformal invariance of such limits were established during the last decade, on the level of both correlation functions and interfaces (domain walls) arising in the model. The goal of this mini-course is to discuss a modern 'combinatorial' viewpoint on discrete holomorphic fermions shaping the nearest-neighbor Ising model in 2D and to show how it can be used
- to simplify some classical computations in the infinite-volume limit revealing the phase transition;
- to prove aforementioned convergence results for basic correlation functions (fermions, energy densities, spins) in bounded domains at criticality;
- to analyze the behavior of interfaces and more involved lattice fields in the small mesh size limit;
- time permitting, to discuss how, given an infinite planar graph, one can interpret a change of Ising interaction constants as a change of a 'discrete conformal embedding' of this graph naturally induced by the Ising model on it.
List of definitions, Exercise set 1, Hints
In this series of talks, I will present geometric description of topological recursion and quantum curves. We start with reviewing Mirzakhani's beautiful work on the Weil-Petersson volume of moduli spaces of bordered hyperbolic surfaces presented in her Harvard Thesis of 2004. This paper is the geometric origin of topological recursion. The surprise of this work is the polynomiality of the volume as a function in the boundary length of hyperbolic surfaces. The exact same polynomiality is found in algebraic geometry of Hurwitz numbers. The explanation comes from topological recursion and intersection theory of tautological classes on the moduli space of stable curves. The simplest example of such, the cotangent class intersections, will then lead us to the discussions of quantum curves. Classical examples of Airy and Hermite differential equations will be used as a motivation. We also mention Garoufalidis's AJ conjecture on hyperbolic knots of 2004, which is indeed the geometric origin of quantum curves. Quantum curves associated with Hitchin spectral curves of Higgs bundles will be explained in detail, in terms of opers and D-modules.
In the first half of my talk I will make a survey of the results describing asymptotic behaviour of spectra of Laplace operators on closed manifolds or Euclidean domains with boundaries. In particular, I will discuss the relationship between spectral asymptotics and the behaviour of the billiards trajectories. In the second half of the talk, I will introduce Steklov operators (also known as Dirichlet-to-Neumann maps) and discuss the asymptotic behaviour of their spectra. In the case of Steklov operators acting on domains with corners, the answer is quite surprising and depends on the arithmetic properties of the corners. This is a joint work (in progress) with M. Levitin, I. Polterovich and D. Sher.
The course will discuss: equivariant cohomology, characteristic classes, Atiyah- Bott-Berline-Vergne formula, Atiyah-Singer equivariant index formula, Mathai-Quillen formalism, and application to path integrals.