We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Ito theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connected component is itself a continuous random tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with 3/2-stable branching mechanism. Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure M which is the analog of the Ito measure of Brownian excursions. This study is motivated in part by applications to the Brownian disks introduced by Bettinelli and Miermont, which also play an important role in recent work of Gwynne, Miller and Sheffield. We use our excursion measure M to develop a new approach to Brownian disks, which in particular makes it possible to prove that connected components of the complement of balls in the Brownian map are Brownian disks.

We will describe some recent and ongoing work (mostly joint with Wei Qian) on relations between gases non-interacting Brownian loops, their decompositions and how they are related to Conformal loop ensembles and the Gaussian free field.

Liouville Brownian motion is Brownian motion which is time-changed according to the exponential of the gaussian free field. In this talk I will discuss some conjectures and results concerning short time estimates for the LBM heat kernel.

The speaker will describe numerical and analytical results demonstrating universality in the fluctuations of the stopping times for various numerical algorithms with random data. This joint work with Tom Trogdon, and in part, with C.Pfrang, G.Menon and S.Olver.

We study a continuous-time random walk, X, on Z^d in an environment of dynamic random conductances taking values in (0,∞). We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle and local CLT for the Markov process X under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's and and De Giorgi's iteration scheme.

Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them in an equivariant way? "Fairly" means that each region has the same area. "Equivariant" means that if we rotate the sphere, the solution rotates along with the points. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). This is joint work with Nina Holden and Alex Zhai, based on earlier work of Chatterjee, Peled, Romik and the speaker (2010).

The TAP-equations (for Thouless-Anderson-Palmer) were first introduced and used for the SK-model. They can (heuristically) be derived from belief propagation (or message passing) equations which are widely used in coding theory and computer science. In [1], we introduced an iterative scheme which in contrast to earlier used ones converges in the full high temperature regime of the SK-model. The scheme can be used for the perceptron [2], and has recently found applications for instance in neural nets (see [3]). We review the structural properties of the scheme, and some of the recent applications.

References

[1] Bolthausen, E.: On iterative construction of solutions of the TAP equations for the Sherrington-Kirkpatrick model. Comm. Math. Phys. 325, 333-366 (2014).

[2] Bolthausen, E.: On the TAP equations for the perceptron. In preparation (2017).

[3] Mézard, M.: Mean-field message-passing equations in the Hopfield model and its generalizations. Phys. Rev. E 95, 022117 (2017).

I review some results on stochastic models of adaptive dynamics, emphasising the role of scaling limits and time scales. I present some ideas for an extension of the classical model that incorporates epigenetic switches of phenotyps together with mutations in genotype. This is motivated from problems arising in toumor evolution and treatment. I discuss a specific model for immunotherapy of melanoma. This talk is based on work with Martina Baar, Loren Coquille, Hannah Mayer, Rebecca Neukirch, and on a collaboration with Michael Hölzel and Thomas Tüting from the Bonn University clinics.

Branching branching random walk and Brownian motion have been the subject of intensive research recently. We consider branching random walk and investigate the effect of introducing a spatially random branching environment. We are primarily interested in the position of the maximum particle, for which we prove a CLT. Our result correspond, on an analytic level, to a CLT for the front of the solutions to a randomized Fisher-KPP equation, and also to a CLT for the parabolic Anderson model.

In this talk I will present some recent results obtained in collaboration with Maximilian Nitzschner concerning uniform estimates for the absorption by porous interfaces surrounding a compact set of Brownian motion starting in this compact set. I will also discuss some applications to coarse graining and certain large deviation upper bounds.

Consider independent continuous-time random walks on the integers to the right of a front R(t). Starting at R(0)=0, whenever a particle attempts to jump into the front, the latter instantaneously advances k steps to the right, absorbing all particles along its path. Sly (2016) resolves the question of Kesten and Sidoravicius (2008), by showing that for k=1 the front R(t) advances linearly once the particle density exceeds 1, but little is known about the large t asymptotic of R(t) at critical density 1. In a joint work with L-C Tsai, for the variant model with k taken as the minimal random integer such that exactly k particles are absorbed by the move of R(t), we obtain both scaling exponent and the random scaling limit for the front at the critical density 1. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the initial local fluctuations (with the scaling limit oscillating between instantaneous super and sub-critical phases).

We will discuss some universal/ non universal results in Random Matrix Theory. Using ensembles for which explicit computations of eigenvalue statistics exist (determinantal random point fields), we discuss the universality question.

There are nice limiting spectral distributions for a host of patterned non-random matrices (Toeplitz operators and their generalizations, Szegö-type theorems). Looking at these with a probabalist's eyes leads to new theorems. There ARE connections with RMT but these connections remain to be developed. There are also nice applications outside RMT.

The KPZ equation is a model of interface growth which is supposed to be "universal" in the following sense. Take any one-parameter family of interface fluctuation models that are symmetric for some fixed value of the parameter (say a = 0). Then, for a small, one can find time and space scales such that the rescaled model converges to the solution to the KPZ equation as a -> 0. We show that this is indeed the case for a large family of models, including models that go beyond those with polynomial nonlinearities.

The cover time is the time needed for the simple random walk to visit all points on the lattice, and in the continuum, for the Wiener sausage of radius 1 to cover the torus of linear size n. As a maximum of correlated random variables (here, with logarithmic decay) it has interesting asymptotics. In higher dimension random interlacements, introduced by Sznitman to describe the local covering picture with a fixed intensity, still give a reliable account at large densities up to cover time. In dimension 2, it can be used as adescription of the neighborhood of an unvisited site, provided that the paths used in the interlacement are random walk and Brownian motion conditioned not to visit a ball. (Joint works with Serguei Popov and Marina Vachkovskaia.)

We describe the explicit transition probabilities for the scaling invariant Markov process at the centre of the Kardar-Parisi-Zhang universality class, and how they arise from the solution of the discrete model TASEP. This is joint work with Konstantin Matetski and Daniel Remenik.

The most obvious Hopf algebras came from considering the structure that appears when one considers the real valued functions on a group, and the enveloping algebra spanned by the left invariant (differential) operators on the group. The group injects into its enveloping algebra of operators.

Within the framework an important subclass are the combinatorial algebras – these are graded – and provide natural locally finite linear bases for the functions on the associated group. A particularly important example is the group of paths (or streams). In this case the Enveloping algebra is the tensor series, and the commutative algebra is the shuffle algebra. It provides an alternative way of formally describing a path segment or stream – via its evaluation on these features, instead of giving its value at the ends of the segment. This top down approach is fundamentally different to the evaluation at points approach of Kolmogorov and Doob and opens up the way to describing much rougher paths, and generalizing the notion of smooth path in a way that allows calculus.

From this starting point Rough path theory aims to build an effective calculus that can model the interactions between complex oscillatory (rough) evolving systems. At its mathematical foundations, it is a combination of analysis blended with algebra that goes back to LC Young, and to KT Chen. Key to the theory is the essential need to incorporate additional non-commutative structure into areas of mathematics we thought were stable.

The approach connects to modern data science in two ways. First it models multimodal data streams very naturally, and secondly, the principle of mapping to a curve into a linear space where the dual is the space of functions on the curve is a fundamental idea providing a “perfect” kernel approach (because it linearises polynomial functions). It allows a direct connection with standard machine learning thinking. The approach is fundamental, and descriptions via time series, and non-adaptive wavelet expansions are doomed to be relatively ineffective since they are linear features of the path. Classic results, by Clark, Cameron and Dickinson, demonstrate that a nonlinear approach to the data is often essential. Rough path theory lives up to this challenge and can be viewed as providing more efficient ways of approximately describing complex data; approaches that, after penetrating the basic ideas, are computationally tractable and lead to new scalable ways to regress, classify, and learn functional relationships from data.

One non-mathematical application that is already striking is the use of signatures on a daily basis (linked with deep learning) in the online recognition of Chinese Handwriting on mobile phones. Recognizing actions in video data is another highly competitive application.

I will argue that the Tutte embeddings (a.k.a. harmonic/barycentric embedding) of certain random planar maps converge to Liouville quantum gravity (LQG). Specifically, I will treat mated-CRT maps, which are discrete versions of the mating of two correlated continuum random trees, They can also be understood as random triangulations "decorated" by labeled space-filling paths of a particular kind. I will argue that the measure induced by the embedding converges to a Liouville quantum gravity measure, that the space-filling path scales to space-filling SLE in the annealed sense, and that simple random walk on the map scales to Brownian motion in the quenched sense. This is joint work with Ewain Gwynne and Jason Miller.

In 2002, Fontes-Isopi-Newman introduced a diffusion (which is now called a FIN diffusion) as a scaling limit of the 1-dimensional Bouchaud trap model. It is a time change of 1-dimensional Brownian motion. In this talk, we will consider more general setting and discuss convergence of random walks for trap models on disordered media. We first provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. We then apply the general theory to trap models on recurrent fractals and on random media such as the Erdos-Renyi random graph in the critical window. If time permits, we also discuss heat kernel estimates for the relevant time-changed processes. This is a joint work with D. Croydon (Warwick) and B.M. Hambly (Oxford).

We study a general class of birth-and-death processes that describe the size of populations going to extinction with probability one. The scale of the population is measured in terms of a carrying capacity K. When K is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. We quantify the time for the process to reach the quasi-stationary regime and the mean time to extinction in the quasi-stationary distribution as a function of K, for large K. In dimension one, we also give a quantitative description of this quasi-stationary distribution.

We will discuss the effect that biasing the random walk away from the root of a Galton-Watson tree with no leaves can have on its asymptotic distance from the root and asymptotic entropy. One of the fundamental questions in the study of the former is the so-called ``monotonicity conjecture’’ of Lyons-Peres-Pemantle (1996), which was confirmed for a large enough bias by Ben Arous, Fribergh and Sidoravicius in 2011, followed by an improved bound by Aidekon. We will discuss how the study of the latter is related to the mixing time of random walk on random graphs, in light of which a similar monotonicity conjecture seems plausible for the entropy, and present some recent progress on this in joint work with A. Ben-Hamou and Y. Peres.

Around 2004 it was conjectured that once-reinforced random walk in dimensions d > 2 undergoes a transition: if reinforcement is strong enough the walk is recurrent, and if reinforcement is weak, the walk is transient. Until very recently it was not clear if such type of statements are true. In two joint works with D. Kious and A. Colleveccio and D. Kious we show that this is true on variety of tree type graphs, depending on its branching-ruin number. If time permits, I will discuss more general cases of Markov chains.

After introducing Poissonian ensembles of Markov loops on graphs and Brownian loops on manifolds, we present a few results about their distribution in different topological types.

I discuss some macroscopic equations obtained in kinetic or hydrodynamic limits, of systems of interacting particles in contact with heat bath modeled by Langevin stochastic dynamics.

We show how, the discovery/design of robust scalable numerical solvers for arbitrary bounded linear operators, can, to some degree, be addressed/automated as a Game/Decision Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space is a Banach space $\mathcal{B}$ endowed with a quadratic norm $\|\cdot\|$, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of $u\in \mathcal{B}$, given partial measurements $([\phi_i, u])_{i\in \{1,\ldots,m\}}$ with $\phi_i\in \mathcal{B}^*$, using relative error in $\|\cdot\|$-norm as a loss) is a centered Gaussian field $\xi$ solely determined by the norm $\|\cdot\|$, whose conditioning (on measurements) produces optimal bets. When measurements are hierarchical, the process of conditioning this Gaussian field produces a hierarchy of elementary gambles/bets (gamblets). These gamblets generalize the notion of Wavelets and Wannier functions in the sense that they are adapted to the norm $\|\cdot\|$ and induce a multi-resolution decomposition of $\mathcal{B}$ that is adapted to the eigensubspaces of the operator defining the norm $\|\cdot\|$. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of continuous linear operator (including arbitrary continuous linear bijections from $H^s_0(\Omega)$ to $H^{-s}(\Omega)$ or to $L^2(\Omega)$) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to $\mathcal{O}(N \operatorname{polylog}(N))$ solvers and eigenspace adapted Multiresolution Analysis. This is a joint work with Clint Scovel.

In joint work with Federico Camia and Jianping Jiang we study the problem of proving exponential decay of correlations (and hence existence of a mass gap) for the Euclidean field theory that is the scaling limit of the near-critical (small magnetic field) planar Ising model.

Our understanding of trapping for random walks in random environments has greatly improved in the last decade. The goal of this talk is to give an overview of the progress made by describing and explaining the possible scaling limit behaviours in two different contexts: biased random walks on trees or in $\mathbb{Z}^d$, and ants in labyrinths.