Summer Graduate School

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Consider two independent and identically distributed continuous-time Markov chains $X$ and $Y$ on a graph with vertex set $V$. The collision measure $\Pi$, which describes the times and positions of collisions of $X$ and $Y$, is given as a random measure on $\mathbb{R}_{\geq 0} \times V$ by  $\Pi([0,t] \times E) = \int_{0}^{t} 1_{\{X(s) = Y(s)\}} ds.$. We have shown that if graphs converge to a metric space in an appropriate Gromov-Hausdorff-type topology with respect to the effective resistance metric, and if there is a suitable upper bound on the heat kernels of the Markov chains on the graphs, then the two independent Markov chains on the graph and its collision measure converge to stochastic processes on the limiting space and its collision measure. The collision measure of the limiting stochastic processes is constructed using positive continuous additive functionals (PCAFs) in Dirichlet form theory. This applies to many low-dimensional spaces, such as the Sierpi\'{n}ski gasket and the Brownian random tree.

Stein’s Method is a powerful tool for studying distances between distributions of random variables. It has been used widely in the context of statistical mechanics to obtain approximations for various thermodynamic quantities. We will discuss an application of Stein’s Method to the $O(N)$ model, a model of magnetism, and show how recent advances in Stein’s theory allow us to study the limiting behaviour of the magnetisation at the phase transition.

The principal Dirichlet Laplacian eigenfunction $\varphi_U$ describes the invariant measure of a Brownian motion conditioned to remain inside a bounded domain $U\subseteq \mathbb{R}^n$. In this talk, I will present results about how domain perturbations affect $\varphi_U$. These results lead to explicit expressions for $\varphi_U$ in certain Euclidean domains, thus implying improved Dirichlet heat kernel estimates. This is joint work with Laurent Saloff-Coste.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Planar maps are graphs embedded in the sphere such that no two edges cross, where we view two planar maps as equivalent if we can get one from the other via a continuous deformation of the sphere. Planar maps are studied in several different branches of mathematics and physics. In particular, in probability theory and theoretical physics random planar maps are used as natural models for discrete random surfaces. In this mini-course we will present scaling limit results for random planar maps and we will focus in particular on a notion of convergence known as convergence under conformal embedding. The limiting surface is a highly fractal surface called a Liouville quantum gravity (LQG) surfaces, which has its origin in string theory and conformal field theory.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Understanding the (near-)critical behaviour of Bernoulli percolation is a very challenging task. One case where things are slightly simpler is the high-dimensional case. It is expected that ,for $d>6$, the model should exhibit what is known as mean-field behaviour. This is something that is now well understood using a technique called lace expansion.   In this talk, I will present a new, more probabilistic approach to studying high-dimensional percolation. While this method currently gives weaker results than the lace expansion, it offers new insights and a different perspective on the problem. Based on joint work with Hugo Duminil-Copin, Romain Panis and Gordon Slade

Consider percolation, the Ising model, and the self-avoiding walk on $\mathbb Z^d$. Above their upper critical dimensions, the lace expansion gives a convolution equation for the two-point (correlation) function. We present two simple ways to analyse the convolution equation, to derive the asymptotic behaviour of the critical two-point function. For short-range models, we prove a deconvolution theorem, using only Hölder’s inequality, weak derivatives, and basic Fourier theory, which yields $|x|^{-(d-2)}$ asymptotics for the critical two-point functions. For (spread-out) long-range models, we prove an exact random walk representation of the two-point function, which allows us to derive the asymptotic behaviour via random walk computations. Our methods significantly simplify previous analyses based on the lace expansion. This is based on joint work with Gordon Slade.

We study the model of spread-out percolation, originally introduced in \mathbb{Z}^d by Hara and Slade. This model depends on a distance parameter, but the percolated graph retains the same expected degree at each vertex as the parameter grows. We present a natural generalization to all vertex-transitive graphs. In the case of transitive graphs with superlinear polynomial growth, we prove that the critical value for the expected degree of each vertex converges to 1 as the distance parameter tends to infinity. This extends a well-known result in  \mathbb{Z}^d for $d≥2$, established by Penrose. Based on joint work with Matthew Tointon.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

In this talk, I will discuss a particular system of partially exchangeable Markov chains. I will begin with a motivating example, the averaging process, which we will reinterpret within a framework involving a countable collection of Markov chains. After reviewing the definition of exchangeability and key known results, I will present a characterization of partially exchangeable Markov chains whose finite subcollections remain Markovian. Finally, I will explore concrete examples and highlight their connection to the averaging process. This talk is based on current work with Adrián González Casanova, Noemi Kurt and José-Luis Pérez.

The Poisson boundary of a random walk is an abstract measure space that captures all the nontrivial asymptotic events of the Markov chain. In many examples, one can find a concrete topological “realization” of this space, and it is a well-studied question when one can find a universal such realization for all random walks on the same group. We show that, in some sense, this is never possible. This is joint work with Josh Frisch.

In genetics, certain statistical quantities are governed by a tree structure, called a coalescent, whose law is determined by a scaling limit of random walks on random graphs. Classical coalescent theory describes a limiting law for these coalescents by, implicitly, averaging over these graphs. Recent work has examined what random limiting laws one may describe when one instead conditions on these random graphs. The limiting random laws and the averaged-over laws are known to coincide when there is a suitable "propagation of chaos" for the random walks.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Starting from an initial root, build a random tree of size $n$ by successively attaching each new vertex to an existing node chosen uniformly at random. If you were given only the final tree, how would you find the root? How easy would it be? In this talk, I will present the best root recovery algorithm and the notion on which it is based: rumor centrality. Then, I will describe the structure of the set of most central vertices and explain how to use it to characterize sharply the algorithm's accuracy. Based on joint work with Louigi Addario-Berry, Catherine Fontaine, Louis-Roy Langevin, and Simone Têtu.

We study properties of the connected components in rank-1 inhomogeneous random graphs in the critical regime. In this model, the graph has vertex set $[n]$, and the vertices are assigned weights according to a vector $w^{(n)}$. For each pair of vertices, an edge is added with probability proportional to their weights, independently of other edges. For a connected component C of the resulting graph, we consider both the “size” of C (its number of vertices), and the “weight” of C (the combined weight of its vertices). A natural question emerges: are the largest connected components also the heaviest components, with high probability as n goes to infinity? We answer this question in the affirmative, when the vectors $(w^{(n)},n \ge 1)$ satisfy certain regularity conditions (namely, convergence of the third moment and a maximal weight that is $o(n^{⅓})$). We also provide the limiting distributions of the sizes and weights of the connected components, and show that these convergences holds in the $l_2$-topology. This is joint work with Louigi Addario-Berry, Prabhanka Deka, Serte Donderwinkel, Sourish Maniyar, Minmin Wang and Anita Winter.

Consider two configurations, C_1 and C_2, in Bernoulli bond percolation on a general graph G. Take the set S of edges where one endpoint belongs to the cluster of a given vertex “a” in C_1. Now, construct a new configuration that matches C_1 on S and C_2 outside of S; this configuration will retain the same Bernoulli distribution. We identify a broader class of sets S with this property and leverage them to derive new bounds on the three-point function.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

The planar FK percolation on Z^2 has been extensively studied, with many critical exponents computed for the q=1(Bernoulli) and q=2(FK-Ising) cases. However, for a long time, most exponents beyond these two cases were not computable. In this talk, I will present the recent progress of computing certain critical exponents. This result builds on recent progress in the convergence of the six-vertex model.

Nilpotent groups are the closest class of noncommutative groups to abelian groups. Many results on Euclidean spaces can be considered there. The celebrated Gromov polynomial growth theorem asserts that a finitely generated discrete group has polynomial growth if and only if it is virtually nilpotent. More generally, for compactly generated locally compact groups of polynomial growth, structure theorems are given in a series of papers by Losert. In this minicourse, we will explore random walk models on groups of polynomial growth, starting from simple random walks on discrete groups, to more general random walks on locally compact ones, walks of unbounded range, etc. We will explain techniques to prove various estimates, limit theorems, and some applications beyond polynomial growth.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

When can a sample of an invariant random process on a directed tree G can be transformed to a sample of an invariant random process on a different tree H, with the further requirement that two samples are the same up to a graph automorphism of G if and only if the same is true for the resulting samples on H? When such processes exists, we say the trees G,H are measure equivalent. In this talk, we survey measure equivalence and give new results in the classification of trees.

This talk will review some recent results concerning a strongly correlated percolation model called the metric graph Gaussian free field. In particular, we will focus on the probability that the boundary of a large box is connected to the origin at criticality. We will demonstrate how one can derive this probability by working with the underlying random walk.

Last passage percolation (LPP) is a model of random geometry where the main observable is a path evolving in a random environment. When the environment distribution has light tails and a fast decay of correlation, the random fluctuations of LPP are predicted to be explained by the Kardar–Parisi–Zhang (KPZ) universality theory. However, in strongly correlated environments — such as those arising from branching random walks, Gaussian free field and other log-correlated fields— KPZ predictions are believed to break down, and much less is known. In this talk we will present recent progress towards developing an understanding of LPP in such environments. In particular, we will report results on LPP in fractal percolation, a random fractal set introduced by Mandelbrot, a hierarchical approximation of the “thick points” of log-correlated fields. We show that in this setting, the LPP energy grows sub-linearly — in contrast to the linear growth under KPZ universality. I will also discuss results in high dimension which provides a rich setting to investigate the structural distinction between directed and undirected models of random geometry. This is based on joint work with Shirshendu Ganguly and Victor Ginsburg.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

The Ising model is one of the most known model of statistical mechanics. It exhibits a phase transition and, in the planar case it possesses at its critical point a very strong property: it is conformal invariant (and it is one of the very few models for which this has been derived rigorously). These reasons justify (among many others) the interest mathematicians put and are putting into it. However, even in dimension two, there are still things that remain mysterious. One of the aim of my joint work with Garban and Sepúlveda is to investigate the relationships between two different viewpoints that were used to analyze conformal invariance: interfaces between clusters of spins and the magnetization field.

I will discuss Liouville Brownian motion, the canonical diffusion in the random geometry defined by Liouville quantum gravity (LQG). In particular I will present some recent results on the spectral geometry of LQG, showing that the eigenvalues satisfy a Weyl law. We will also discuss a number of striking conjectures which aim to relate LQG to a phenomenon known as "quantum chaos", which will also be explained.