I will explain an identity between scaling limits of the Ising correlations in arbitrary finitely connected planar domains, and correlations of a suitable version of a Gaussian free field, yielding explicit formulae for the former. The proof is based on (rigorously established) operator product expansions for the Ising and GFF correlation, and a limiting version of a classical identity between Szegö and Bergman kernel on Riemann surfaces. Joint work with Baran Bayraktaroglu, Tuomas Virtanen and Christian Webb.
I will explain what Gaussian multiplicative chaos (GMC) measures are, and prove new small deviations estimates for the total mass of GMC measures. Then I will show how to use GMC to construct the path integral formulation of the massless Sinh--Gordon model on the 2-dimensional torus. Finally, I will show how the small deviations bounds can be used to derive lower and upper bounds for the free energy of the massless Sinh--Gordon model on the infinite torus. Based on joint work with Nikolay Barashkov (MPI Leipzig) and Mo Dick Wong (Durham).
The conformal bootstrap is a powerful approach to two-dimensional conformal field theories (CFTs) developed forty years ago by Belavin, Polyakov and Zamolodchikov. The Virasoro fusion kernel appears naturally in this approach and carries non-trivial information about CFTs in general. In this talk, I will first review the conformal bootstrap approach to CFTs, and I will take the examples of Liouville theory for central charges c>25 and of its imaginary analog for c<1. I will finally introduce the Virasoro fusion kernel and describe its construction by Ponsot and Teschner for c>25. If time permits, I will discuss my recent construction of its imaginary analog for c<1.
In this talk, I will present a non-rigorous method known in the physics litterature as Coulomb Gas. I will focus on the O(N) loop models and, if time permits, I will discuss its higher rank counterpart, the A_2 web models. The Coulomb Gas method identifies scaling limit of partition and correlation functions of the lattice models with quantities obtained from the compactified imaginary Liouville, and A_2 Toda, conformal field theories. From a mathematician point of view, it can be seen as a tool to obtain conjectures on, for instance, conformal weights of (the scaling limit of) lattice observables.
The Yang--Baxter equation (YBE) plays a role in diverse and interrelated areas of mathematical physics, including statistical mechanics, quantum field theory, knot theory and quantum groups. Focusing on applications to statistical mechanics, I consider a generalisation of the YBE known as the multi-parametric YBE (mYBE), where each $R$-operator solution can be used to construct a family of commuting multi-parametric transfer operators. In an effort to identify algebraic structures admitting solutions to the mYBE I will introduce a construction called planar algebraisation, which can be thought of as embedding a particular algebraic structure (a tower of algebras) within a planar algebra. By considering the planar algebraisation of the tower of Temperley--Lieb algebras, I identify a tower of coupled Temperley--Lieb algebras and find that this tower admits a solution to the mYBE. If time permits, I will sketch how planar algebraisation may serve as a general tool for discovering further solutions.
The fermionic GFF (fGFF) a discrete version of the symplectic fermions CFT is known to be suitable to compute edge probabilities in the uniform spanning tree (UST). In turn, the UST has been linked to CFT at central charge -2 the same central charge as the symplectic fermions. In this talk, I will explain how to make a one-to-one correspondence between lattice local fields of the fGFF and local fields of the symplectic fermions in a way that the (suitably renormalized) correlation functions of the former converge, in the scaling limit, to the corresponding correlation functions of the latter. This result, applied to the UST, allows us to understand the scaling limit of correlation functions of local patterns that is, prescribed open collection of edges in a neighbourhood of a point as symplectic fermions correlation functions. Based on joint work with W. Ruszel (Uni. Utrecht).
When proving that a sequence of random variables converges, usually one first needs to establish precompactness, i.e. that every subsequence has a converging subsequence. If the random variables take values in some metric space, Prokhorov's theorem states that so called tightness of the sequence implies precompactness. I will provide another sufficient condition of precompactness which I call complete approximability. Every tight sequence is completely approximable, but the converse doesn't hold in general unless the metric space is separable and complete. The proof for sufficiency of complete approximability avoids some prerequisites that proofs for Prokhorov's theorem I'm aware of require. In the first half of the talk, I will briefly discuss notions of convergence for sequences of random variables, motivate the definition of complete approximability, and explore some ways in which it differs essentially from tightness. In the second half I will sketch the proof of sufficiency of complete approximability and, if time permits, demonstrate how to apply complete approximability for sequences of random curves.
The Loewner energy of a chord in a simply connected domain is defined as the Dirichlet energy of the driving function for the corresponding Loewner chain. In a pioneering work, Yilin Wang identified Loewner energy as the large deviation rate function for chordal Schramm--Loewner evolution as the parameter κ decreases to 0. With this interpretation, she deduced from the reversibility of chordal SLE that the Loewner energy or a chord does not depend on reversing its orientation. I will present a deterministic proof of this fact by reversing the chord in small increments, highlighting similarities and differences from Dapeng Zhan's proof of SLE reversibility.
The original Discrete Morse theory (Forman 1998) is a method for simplifying CW-complexes while preserving their homotopy equivalence. The combinatorial nature of this tool has proven its use in both theoretical mathematics and in applications. In this talk, we will take an algebraic view towards discrete Morse theory (Sköldberg 2005) while simultanously trying to keep the original geometric picture in mind. We will observe that that Sköldberg's formalisation generalizes from R-modules to additive categories. This allows for effective applications towards Khovanov homology, a homology theory for knots and links which categorifies the Jones polynomial. The results we present in Khovanov homology will be both of theoretical and computational in nature. Based on https://arxiv.org/abs/2306.11186 and recent work.
CAR algebras are fundamental operator algebras that appear in various fields of mathematics and mathematical physics. In this talk, we will focus on its relationship to random point processes, which are mathematical descriptions of random interacting particles. In particular, after discussing the relation between quasi-free states of CAR algebras and determinantal point processes, we will investigate how the operator algebraic framework provides stochastic processes on random point processes.
I will describe an extension of the integration measure for calculating scattering amplitudes in (super)string theory to a large (super)grassmannian space, known as the super Sato grassmannian. In comparison to bosonic string theory, superstring theory provides some simplifications in the properties of the integration measure, which I will highlight in my talk. On the other hand, superstring theory is intrinsically related to the supermoduli space of super Riemann surfaces, which poses challenges because of the supergeometric structure. I will explain why working with supergrassmannian space could be a good solution to these problems. This is based off of joint work with Alexander Voronov.
I will present a non-expert view on type theory taking formalization of natural numbers as an example. In particular, I will define natural numbers and the operation of addition, prove the unit laws, associativity and commutativity. If time permits, I will also prove Peano's 7th and 8th axioms. The demonstration will go in a proof assistant Agda. (This presentation is not about a conventional topic from mathematical physics.)
After a review of the geometric axiomatisation scheme to two-dimensional conformal field theories (CFTs) in the spirit of Kontsevich and Segal, I will explicitly show how to construct the chiral CFT for the free, massless boson on arbitrary compact Riemann surfaces. The construction provides a rich playground for the interplay of distant areas of mathematics (Kähler spaces, C*-algebras, geometric function theory, etc.). Time permitting, I will sketch connections to Teichmüller theory, global analysis, and the Atiyah-Singer index theory.
The reflection equation was introduced in relation to quantum integrable systems with boundary condition and is closely related to the Yang-Baxter equation. The reflection equation algebra was in turn introduced to allow the algebraic study of solutions of the reflection equation, similar to the connection between quantum groups and the Yang-Baxter equation. We will describe the basic properties of the reflection equation algebra and explain the classification of its bounded *-representations. This is based on joint work with Kenny De Commer.
In von Neumann's formulation of Quantum Mechanics, physical observables are represented by self-adjoint operators in some Hilbert space. The spectrum of the operator is interpreted as the set of possible outcomes of a measurement of the observable, which is perhaps the most important physical prediction of this mathematical model. In Physics literature, especially on the introductory level, a somewhat heuristic approach to studying the spectrum is sometimes employed: "generalized eigenvectors" outside the original Hilbert space are acceptable, unless they are too wild to be "physical". The aim of the talk is to provide a rigorous justification of such heuristics under certain conditions, but also to show that these heuristics may miserably fail in some situations.
I will give an overview of functorial field theories, a mathematical formalisation of quantum field theories as functors out of some (higher) category of bordisms between manifolds. It is particularly well suited for describing topological quantum field theories. In the first half of this talk I will explain the Cobordism Hypothesis: a classification result for topological quantum field theories. In the second half, I will explain how one thinks about defects (interfaces between different quantum field theories) in this formalism, and how this relates to the bulk-boundary correspondence between three-dimensional topological quantum field theories and two-dimensional conformal field theories. I will aim to make this talk accessible to a broad audience.
In this talk we provide some basic facts about Loewner chains driven by both deterministic functions and stochastic processes. In the second half of the talk we provide some insights about the current development on Loewner chains driven by deterministic complex-valued functions and complex Brownian motion.
A fundamental goal in random matrix theory is to understand the eigenvalues of a given random matrix model. Schramm-Loewner evolution SLE(κ) is a one-parameter family of random curves arising from two-dimensional conformal geometry. In this talk I will show how the evolution of eigenvalues of certain Itô-diffused random matrices coincide with the driving functions of multiple interacting SLE(κ) curves of some special values of κ. If time permits, in the end I will present a large deviation principle which quantifies the exponential rate of convergence of SLE(κ)-curves to deterministic SLE(0) in the limit κ->0.
Toda conformal field theories form a family of two-dimensional quantum field theories initially introduced in the physics literature. They are natural generalizations of Liouville theory that enjoy, in addition to conformal invariance, an enhanced level of symmetry encoded by W-algebras. In this presentation we will explain how one can study these theories from a mathematically rigorous perspective. For this purpose we will describe a probabilistic framework designed to make sense of these models and provide some insight on how the introduction of this framework can help to understand the model. To be more specific, we will prove ---we will not enter into much details but rather try to convey the main ideas--- that one can compute some basic correlation functions of the theory based on probabilistic tools. Along the proof of this statement we will shed light on some unexpected interplays between probability theory and conformal field theory such as a generalized Brownian path decomposition.
In this talk, we start with an overview of space-filling SLE, the quantum sphere, and the mating of trees theorem. We then define permutons and illustrate their natural construction from SLE-decorated LQG. Focusing on two special cases, Baxter permutons and meandric permutons, we reveal how they arise as limits of Baxter permutations and meanders, respectively. Through this exploration, permutons offer insights into the connection between discrete models and random geometry.
In this two-part talk we will present recent results on W3 conformal blocks. These are solutions of a system of PDEs arising from W3 algebra (an extension of the Virasoro algebra) null-vectors and Ward identities. At c=2, we give an explicit subspace of solutions having the dimension predicted by CFT. The solutions are expressed in terms of Specht polynomials in a simple way. After giving heuristic physical motivations, we will state a conjecture relating these functions to connection probabilities in the triple dimer model, recently computed by Kenyon and Shi. All along the talk, we will present the analogous known results for the Virasoro case.
In Teichmuller theory, a theorem due to H. Masur and W. Veech states that for generic Abelian differentials, the leaves of the horizontal foliation are uniquely ergodic. Said differently, the orbits of the horizontal straight-line flow on a generic translation surface are uniquely ergodic. Subsequently, G. Forni proved an effective form of this statement, establishing in particular precise power-laws for the deviations of ergodic averages of smooth functions from the power law (as predicted by the ergodic theorem). The main goal of this talk is to introduce and motivate translation surfaces, and to explain the main ideas behind an analytic approach (via anisotropic Banach spaces) to effective unique ergodicity for straight-line flows on a generic translation surface. This is a joint work in progress with D. Galli (University of Zurich).
Consider the dimer model sampled on a general Riemann surface. In this setup, the dimer height function becomes additively multivalued with a random monodromy. Given a sequence of graphs approximating the conformal structure of the surface in a suitable way, the underlying sequence of height functions is expected to converge to the compactified free field on the surface. Recently, this problem was addressed by Berestycki, Laslier and Ray in the case of Temperley graphs. Using various probabilistic methods, they obtained the following universal result: given that a sequence of graphs satisfies certain set of probabilistic conditions (which link it with the conformal structure of the surface), the limit of height functions exists, is conformally invariant and does not depend on a particular sequence of graphs. However, the identification of the limit with the compactified free field was missing in this result. In my recent work I fill this gap by studying the same problem from the perspective of discrete complex analysis. For this purpose, I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain local geometric conditions. In this setup I obtain an analytic description of the limit which allows to identify it with a suitable version of the compactified free field; I also prove the convergence in some non-Temperlian cases when the surface is generic. A core part of this approach is the regularity theory on t-embeddings recently developed by Chelkak, Laslier and Russkikh. In this talk we discuss the aforementioned results, in particular, how the methods of discrete complex analysis are generalized to the case of a Riemann surface, and how the geometry of the surface affects the limit.
For gamma in (0,2), the gamma-mated-CRT map is a random triangulation in the plane encoded by a pair of Brownian motions with correlation depending on gamma. It can be obtained as a discretized version of the infinite-volume peanosphere construction that glues together two continuum random trees to get a sphere-homeomorphic surface decorated by a space-filling path, or alternatively as the adjacency graph of cells filled in by a space-filling Schramm--Loewner evolution parametrized by Liouville quantum gravity volume. Many other models of random planar maps (RPMs) can be encoded by 2D random walks, which can be approximated by Brownian motion using Skorokhod-type embeddings to allow statements about these RPMs to be reduced to statements about mated-CRT maps. For instance, one can understand graph distances in uniform infinite planar triangulations by studying the embedding of the mated-CRT map into the plane given by the SLE construction. In this talk I will give an introduction to mated-CRT maps and suggest directions for future work.
Abstract: Consider the Lorentz mirror model on the 2d lattice: at each lattice site, independently place a mirror at 45 degrees to the lattice with some probability p. The orientation of the mirror is chosen independently, say north-west with probability q in (0,1). Loops can then be formed which bounce off the mirrors, or pass straight through lattice sites with no mirror. What is the probability that the loop through some given edge is infinite? For p=1 it is known to be zero, but for p in (0,1) the problem is open. We study this model where we re-weigh the measure by n^#loops. We discuss a form of breaking of translation invariance, where for n large, almost all the loops are trivial loops surrounding black faces, or trivial loops surrounding the white faces. We can see that the method applied also works for a model of loops coming from O(n)-invariant quantum spin chains, where the breaking of translation invariance is known as dimerisation. Joint work with Jakob Björnberg.
In the first half of this talk I will give an introduction to the Landau-Ginzburg (LG) / Conformal Field Theory (CFT) correspondence, which predicts a relationship between certain categories of matrix factorisations (for the ``LG potential'') and modular tensor categories (on the CFT side). This prediction has its origin in physics, and comes from observations about 2-dimensional N=2 supersymmetric quantum field theory. I will explain how this prediction is to be interpreted mathematically and what difficulties one encounters in doing this. In the second half of the talk I will discuss joint work with Ana Ros Camacho in which we realise the LG/CFT correspondence for the potentials x^d. The main ingredient in this is an enriched category theoretical versions of the classical Temperley-Lieb/Jones-Wenzl construction of the representation category of quantum su(2).
The Shapiro Conjecture (Theorem of Mukhin, Tarasov and Varchenko as of 2009) is a statement about subspaces of univariate complex polynomials. It gives a sufficient condition for the existence of a real basis in terms of the Wronski polynomial of the subspace. In the case of 2-dimensional subspaces, the conjecture becomes a statement about rational functions and their critical points. This presentation outlines an "elementary" proof for this special case presented by A. Eremenko and A. Gabrielov in 2005. The main tools of the proof are combinatorial invariants called nets for rational functions. Eremenko and Gabrielov then show that the statement holds for particular rational functions, and use the nets to argue that analytic continuation can be used to obtain a complete proof.
The affine and periodic Temperley-Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In the first part of the talk, we will present the diagrammatic calculus related to these algebras and define finite-dimensional quotients of these algebras, which we name uncoiled algebras in reference to the diagrammatic interpretation. Afterwards, we construct a family of Jones-Wenzl idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. The second part of the talk will go in depth on the construction of the Jones-Wenzl idempotents and present some of their applications, mainly looking at their trace.
The talk will consider modular forms and crossing probabilities. In particular, I will review how Cardy's formula can be expressed in terms of the modular Eta function, and further, that Cardys function is the unique function that satisfies f(r)+f(1/r)=1 and has an expansion in form e−2παr times a power series in e−2πr for some α∈R. The first property is implied by a symmetry of the problem, but there is no physical argument for the latter.
I will review how one can obtain integrable local transfer matrices for the O(N) loop model from the relevant evaluation representation of the appropriate quantum affine algebra. After motivating the definition of a recently introduced rank 2 counterpart, the G_2 web models, I will show how to use similar ideas to obtain integrable transfer matrices in this context.
The HOMFLY-PT homology of links in the three-sphere was defined by Khovanov-Rozansky, following various earlier constructions and conjectures. Many of these were motivated by string and gauge theories, in addition to questions in low-dimensional topology. HOMFLY-PT homology is a link invariant valued in triply graded vector spaces (or modules over certain polynomial rings) which categorifies the HOMFLY-PT polynomial. In the first part of the talk, I will introduce the mathematical construction of HOMFLY-PT homology. In the second part, I will try to explain some of our current understanding of its physical underpinnings, which continues to develop in parallel with the mathematical theory.
I will review some ongoing work on constructing the conformal field theory (d'après G. Segal) of the massless, free boson.
Conformal invariance of general critical planar lattice models was conjectured by Belavin, Polyakov and Zamolodchikov in the early 80s. Using the (non-rigorous) renormalisation group flow, they deduced that any scaling limit of such a model must be rotationally invariant. Since any such limit must also be translation and scale invariant, the argument goes that it will also be invariant under the action of the group of transformations which are locally a composition of these three types of transformations. This is exactly the conformal group, which, in the planar case, is infinite-dimensional and therefore, particularly rich as a symmetry group. Another conjecture of theoretical physics going back to Griffiths and Kadanoff is that of universality: That the scaling limits of various models with different microscopic details, e.g. the graph on which it is defined, turn out to be the same across so-called universality classes. During the last 25 years, the first conjecture has received immense attention from the probabilistic community after Schramm's introduction of the SLE. In this talk, however, we shall turn our attention to the second question. Building on work by Duminil-Copin, Kozlowski, Krachun, Manolescu and Oulamara, we prove that the critical random-cluster models each satisfy a universality property across a large class of planar graphs including the hexagonal and triangular lattices. A consequence thereof will be that any scaling limit in one of these universality classes is rotationally invariant and thus, this may also be thought of as a stepping stone towards proving conformal invariance for all critical random-cluster models. Based on joint work with Ioan Manolescu.
The powerful knot invariant Jones polynomial is defined by local skein relations and normalisation. On the other hand, the original categorification of the Jones polynomial was defined on a global scale - Khovanov homology takes whole knots and links as an input. It was later realized, by Bar-Natan, that Khovanov's construction can be defined locally and that these local pieces can be composed by a planar algebra structure. We take a look at Bar-Natan's framework and how it can be used to obtain new results about the original Khovanov homology.
V.F.R. Jones initially introduced planar algebras to describe the standard invariant of a subfactor. Since then, planar algebras have found many applications in mathematics and physics. Informally, a planar algebra describes the interaction of elements of a graded vector space in the plane. The 'two-dimensional' structure of planar algebras makes them natural objects to describe planar statistical-mechanical models. In this talk, I will focus on the role played by planar algebras in relating statistical mechanics and knot theory. In particular, I will introduce a notion of YangBaxter integrability and show that statistical-mechanical models with this property give rise to polynomial link invariants. If time permits, I will report on recent results (joint with J. Rasmussen) classifying all singly generated planar algebras admitting a YangBaxter integrable model.
Given a random loop ensemble in some domain on the plane, we can define the corresponding nesting field at a point by computing the number of loops surrounding this point and subtracting its mean. If the number of loops in all samples of the loop ensemble is bounded by some constant, then we get a well-defined random variable pointwise. Miller, Watson and Wilson have shown that, applying a suitable regularization procedure, one can extend this definition to the conformal loop ensemble with parameter k (CLE(k)) for all 8/3
I will discuss planar algebras of Jones' style associated with the Young graph and harmonic functions on it. In particular, I will explain that, for the harmonic function originating from the Plancherel measure, the associated planar algebra recovers the local relations for the Khovanov Heisenberg category as algebraic relations. We will also see that various functions of Young diagrams including moments, Boolean cumulants, and normalized characters are identified with elements in the planar algebra. These results provide an alternative proof to the Rattan--Sniady conjecture that I proved in the past.
It is an understatement to say that oscillatory integrals play a major role in analysis. In usual settings, we are able to understand the behavior of such integrals by using results like the (non)-stationary phase lemma or the Van Der Corput lemma. But for such results to hold, the phase has to be regular enough. What can we say when the phase is only Hölder regular? This talk aims to explore some results, ideas, and methods for understanding oscillatory integrals in such cases. If the general case is not reachable for now, a particular setting seems to yield some interesting results: when the phase exhibits self-similarity, a recent method introduced by Bourgain and Dyatlov, and generalized by Li, Naud, Pan, Sahlsten, Steven, Jordan, and Baker allows us to prove power decay of the associated oscillatory integral. We will recall some results in the smooth case, discuss some ideas in a probabilistic setting, and finally we will discuss the main tool behind the aforementionned method: the sum-product phenomenon.
Recently (past 25 years), there has been interest in the (double-)dimer model because of the conformally invariant properties of its scaling limit. Physicists, on the other hand, have proposed this model to be described by a logarithmic Conformal Field Theory (logCFT) of central charge c=−2. In the first half of this talk, I will introduce a discretization of the symplectic fermions a logCFT at c=−2 as observables on the double dimer model. From these observables, I will explain how to build a space of local fields which carries a Virasoro representation of central charge −2. On the second half, I will dive into the logarithmic structure of the representation. In particular, I will show that it contains an L_0 Jordan block of primary fields with conformal weight 0.
Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic, such as hyperbolic surfaces. More recently, several authors have investigated quantum ergodicity for sequences of spaces which ``converge'' in the sense of Benjamini-Schramm to their common universal cover, such as a sequence of hyperbolic surfaces whose injectivity radii go to infinity, and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs (Anantharaman-Le Masson '15, Brooks-Le Masson-Lindenstrauss '16), rank one symmetric spaces (Le Masson-Sahlsten '17, Abert-Bergeron-Le Masson '18), and some higher rank symmetric spaces (Brumley-Matz '21). We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to $PGL(3, F)$ where $F$ is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.
Many dynamical systems are in a state of transition between two regimes. Examples are firing patterns of neurons, disordered quantum systems or pseudo-integrable systems. A common feature which is often observed for critical states of such systems is a multifractal self-similarity in a certain scaling regime which cannot be captured by a single fractal exponent but only by a spectrum of fractal exponents. I will discuss a proof of multifractality of solutions for certain stationary Schrödinger equations with a singular potential on the square torus (joint with Jon Keating). Towards the end of the talk, I will allude to some new work on multifractal scaling and solutions to nonlinear PDE in fluid dynamics on cubic tori.
This is a 5-lecture mini-course (Mon-Fri 14-16). Abstract: Liouville quantum gravity (LQG) is a universal one-parameter family of random fractal surfaces. These surfaces have connections to string theory, conformal field theory, and statistical mechanics, and are expected to describe the scaling limits of various types of random planar maps. Recent works have shown that one can endow an LQG surface with a metric (distance function). This metric has many interesting geometric properties. For example, it induces the same topology as the Euclidean metric, but its Hausdorff dimension is strictly greater than two and its geodesics merge into each other to form a tree-like structure. I will discuss the definition of and motivation for LQG, the construction and properties of the metric, and some of the techniques for proving things about it.
The hypergraph stochastic block model is a statistical model for sampling inhomogeneous random hypergraphs associated with a partition of the vertex set. In this talk I will discuss fundamental concepts and recent developments on the statistical analysis of hypergraph stochastic block models. The focus is on universal information-theoretic bounds and phase transitions that help to understand requirements on data sparsity and model dimensions under which consistent learning of the underlying vertex partition is possible.
I will discuss the basic tools of discrete complex analysis on Z 2 that we use to construct a Virasoro representation on the space of local fields of the discrete Gaussian Free Field. I will also go over some recent results we obtained with Delara Behzad and Kalle Kytölä.
In this expository talk, I will give an outline of the developments in the field of critical and massive scaling limits in the Ising model in two dimensions starting from the breakthrough work of Smirnov, which defined the notion of s-holomorphicity. Emphasis will be placed on explaining the steps and techniques used to relate this discrete complex analytic notion to analysis in continuum, yielding conformal invariance of the model in the critical case.
The correlation functions in conformal field theories, in particular, in minimal models, enjoy a number of properties. Among them are, in particular, Belavin-Polyakov-Zamolodchikov equations, which are second order partial differential equations, and the Operator product expansion (OPE) hypothesis concerning the asymptotic expansions of correlations to all orders. The correlations in scaling limit of the critical the Ising model have been recently computed rigorously. However, the program of relating them on mathematical level to correlations in a minimal CFT is not completed, and deriving the BPZ equations and the OPE from the explicit expressions is not straightforward. In this talk, I will discuss our recent results in this direction. This is a joint work with Christian Webb.
The free fermion was one of Segal's original examples of a theory satisfying his Conformal Field Theory (CFT) axioms. This was made fully rigorous relatively recently by James Tener. I will review Segal's definition of a CFT, and then describe the free fermion. Then, I will report on some of my ongoing work attempting to extend the free fermion, following Stolz and Teichner.
Introduced by Polyakov in the 1980s, Liouville quantum gravity (LQG) is in some sense the canonical model of a random fractal Riemannian surface, constructed using the Gaussian free field. Sheffield showed that when a certain type of LQG surface, called a quantum wedge, is decorated by an appropriate independent SLE curve, the wedge is cut into two independent surfaces which are themselves quantum wedges, and that these resulting wedges uniquely determine the original surface as well as the SLE interface. We prove that the original surface can in fact be obtained as a metric space quotient of the LQG metrics on the two wedges. This was proven by Gwynne and Miller in the special case $\gamma = \sqrt{8/3}$, for which $\gamma$-LQG surfaces are equivalent to Brownian surfaces, allowing an explicit description of the metric in terms of Brownian motion that is not available in general. I will explain how our work uses GFF techniques to extend their results to the whole subcritical regime $\gamma \in (0,2)$, while establishing new estimates describing the boundary behaviour of LQG. Joint work with Jason Miller.
The possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk we will be interested in the properties of these measures - in particular, whether they are absolutely continuous. First, I will quickly describe the case of Bernoulli convolutions, which can be seen as generalisations of the Cantor middle third set, and then the case of random iterations of matrices in SL(2, R) acting on the real projective line, where the stationary measure is unique under certain conditions, and is called the Furstenberg measure. It had been conjectured that the Furstenberg measure is always singular when the random walk has a finite support. There have been several counter-examples, and the aim of the talk will be to describe that of Bourgain, where the measure even has a very regular density. I will explain why the construction works for any simple Lie group, using the work of Boutonnet, Ioana, and Salehi Golsefidy on local spectral gaps in simple Lie groups.
The double dimer model (DDM) on a planar graph is a model of random loops, and the Gaussian free field (GFF) is a model of a height function. The two models are linked by a conjecture that the DDM loops converge in the scaling limit to loops in the continuum, which are level lines of the GFF. I will introduce these two models and outline two results. First, certain 2n-point correlation functions in the DDM are known to be determinants in the 2-point functions; we give a new proof of this, which in particular we show can be extended to the GFF. Second, it is known that the GFF exhibits a "height gap": if one sets the boundary height to be +\lambda on one half of the boundary, and -\lambda on the other, then if \lambda = \sqrt(pi/8), the zero level line actually exhibits a sharp "jump" of 2\lambda. We give a simple derivation of this special value of the height gap, using the determinental result. Joint work with Marcin Lis (TU Vienna)
In the late 80's, the physicists Dotsenko and Fateev used the Coulomb-gas formalism to solve for the structure constants of Belavin--Polyakov--Zamolodchikov's minimal models of 2D CFT. To this day, their analysis has not been made mathematically rigorous. In this talk, we will discuss progress towards a rigorous Coulomb-gas formalism, along with its application to the construction of the minimal models.
In the late 80's, the physicists Dotsenko and Fateev used the Coulomb-gas formalism to solve for the structure constants of Belavin--Polyakov--Zamolodchikov's minimal models of 2D CFT. To this day, their analysis has not been made mathematically rigorous. In this talk, we will discuss progress towards a rigorous Coulomb-gas formalism, along with its application to the construction of the minimal models.
In recent decades, there have been interesting connections made between a number of areas of mathematics, including statistical mechanics, knot theory, quantum groups, and subfactors. The Temperley-Lieb algebras are a family of finite dimensional algebras that were the original source of these connections. In this talk we review the representation theory of the finite Temperley-Lieb algebras. We then discuss extremal traces, their classification, and applications to the representation theory of an infinite dimensional generalization of the Temperley-Lieb algebra.
Conformal nets provide a rigorous mathematical framework for conformal field theory, assigning an algebra of observables to each region of the underlying spacetime manifold. Here, we consider so-called discrete conformal nets whereby the spacetime is not a smooth manifold but instead has an 'atomic' structure. It turns out that planar algebras can be used to construct 'almost' examples of discrete conformal nets. In this talk, I will review this business and detail recent efforts to construct fully-fledged examples of discrete conformal nets. Inspired by the statistical mechanics literature, I will also introduce some integrable operators that act on the spacetime and detail some of their algebraic structure.
Log-correlated Gaussian fields such as the Gaussian free field are random Schwartz distributions whose covariances have a logarithmic singularity on their diagonal. They appear for instance in Liouville quantum gravity, characteristic polynomials of random matrices, the dimer model etc. In this talk I will present ways to decompose general log-correlated fields into a sum of a canonical log-correlated field with a particularly nice covariance structure and a Hölder-continuous error term. I will also discuss applications to the study of the Gaussian multiplicative chaos of the field. The talk is based on joint works with Eero Saksman and Christian Webb as well as Juhan Aru and Antoine Jego.
I discuss the recent preprint [arXiv:2211.16111] of Nikolay Barashkov and I, where we show the almost sure well-posedness of a deterministic nonlinear wave equation (cubic Klein-Gordon equation) on the plane. Here "almost sur" is in respect to the \\\\phi^4 quantum field theory. I briefly introduce the invariant measure argument and outline the solution on 2D torus due to Oh and Thomann. I then explain our main contributions: extension of periodic solutions to infinite volume, and a weaker result for nonlinear Schrödinger equation. The viewpoint is functional-analytic with a dash of probability.
One method of introducing external randomness to a Gibbs state, as opposed to the internal randomness of the Gibbs state itself, is to perturb the Hamiltonian with a term corresponding to the coupling of a random external field to the system. For the mean-field spherical model, the corresponding perturbed model can be exactly solved, in some sense, in the infinite volume limit. In this talk, we will introduce, motivate, and present some constructions and results concerning the so-called infinite volume metastases of the mean-field spherical model in a random external field. The aim of this talk is to present the general theory of disordered systems as it pertains to this particular model, and highlight the particular aspects of this model which lead to its curious behaviour as a disordered system. This talk is based on work in a recently accepted paper to appear in the Journal of Statistical Physics.
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