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.
I will discuss recent approaches to characterising the Gaussian free field in the plane, and in higher dimensions. The talk will be based on joint work with Juhan Aru, Nathanael Berestycki, and Gourab Ray.
The forward evolution of chaotic systems notoriously washes out inexact information about their state. When advected by a chaotic system, physically relevant measures therefore often converge to some reference measure, usually the SRB measures. This property implies various important statistical behaviours of chaotic systems. In this talk we discuss the behaviour of slices of these physical measures along smooth submanifolds that are reasonably generic (e.g. not stable or unstable manifolds). We give evidence that such conditional measures also have exponential convergence back to the full SRB measures, even though they lack the regularity usually required for this to occur (for example, they may be Cantor measures). Using Fourier dimension results, we will prove that CDoC holds in a class of generalised baker's maps, and we will give rigorous numerical evidence in its favour for some non-Markovian piecewise hyperbolic maps. CDoC naturally encodes the idea of long-term forecasting of systems using perfect partial observations, and appears key to a rigorous understanding of the emergence of linear response in high-dimensional systems.
We consider a random height function associated with the dimer model on a graph embedded into a Riemann surface. Given a sequence of such graphs approximating the surface in a certain sense we prove that the corresponding sequence of height functions converges to the compactified free field on the surface. To establish this result we follow approach developed by Dubédat: we introduce a family of observables of the model which can be expressed as determinants of discrete perturbed Cauchy-Riemann operators, we analyze the latter using Quillen curvature formula.
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