Forthcoming events in this series
I will talk about recent work that uses recent ideas from stochastic analysis to develop robust and non-parametric statistical tests for stochastic processes.
In this talk, we will show how sharp bounds on the moments of the solutions to some stochastic heat equations can lead to various qualitative properties of the solutions. A major part of the method consists of approximating the solution by “independent quantities”. These quantities together with the moments bounds give us sharp almost sure properties of the solution.
The study of the Geometry of random nodal domains has attracted a lot of attention in the recent past, in particular due to their connection with famous conjectures such as Yau's conjecture on the nodal volume of eigenfunctions of the Laplacian on compact manifolds, and Berry's conjecture on the relation between the geometry of the nodal sets associated to these eigenfunctions and the geometry of the nodal sets associated to toric random waves.
At first, the randomness involved in the definition of random nodal domains is often chosen of Gaussian nature. This allows in particular the use of explicit techniques, such as Kac--Rice formula, to derive the asymptotics of many observables of interest (nodal volume, number of connected components, Leray's measure etc.). In this talk, we will raise the question of the universality of these asymptotics, which consists in deciding if the asymptotic properties of random nodal domains do or do not depend on the particular nature of the randomness involved. Among other results, we will establish the local and global universality of the asymptotic volume associated to the set of real zeros of random trigonometric polynomials with high degree.
(Joint work with Siva Athreya & Leonid Mytnik).
It is well known from the literature that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is “very bad” and has no solutions (or has multiple solutions), then the addition of a random noise leads almost surely to a “nice” ODE with a unique solution. The first part of the talk will be devoted to SDEs with distributional drift driven by alpha-stable noise. These equations are not well-posed in the classical sense. We define a natural notion of a solution to this equation and show its existence and uniqueness whenever the drift belongs to a certain negative Besov space. This generalizes results of E. Priola (2012) and extends to the context of stable processes the classical results of A. Zvonkin (1974) as well as the more recent results of R. Bass and Z.-Q. Chen (2001).
In the second part of the talk we investigate the same phenomenon for a 1D heat equation with an irregular drift. We prove existence and uniqueness of the flow of solutions and, as a byproduct of our proof, we also establish path-by-path uniqueness. This extends recent results of A. Davie (2007) to the context of stochastic partial differential equations.
[1] O. Butkovsky, L. Mytnik (2016). Regularization by noise and flows of solutions for a stochastic heat equation. arXiv 1610.02553. To appear in Annal. Probab.
[2] S. Athreya, O. Butkovsky, L. Mytnik (2018). Strong existence and uniqueness for stable stochastic differential equations with distributional drift. arXiv 1801.03473.
Numerical methods for SDEs typically use only the discretized increments of the driving Brownian motion. As one would expect, this approach is sensible and very well studied.
In addition to generating increments, it is also straightforward to generate time integrals of Brownian motion. These quantities give extra information about the Brownian path and are known to improve the strong convergence of methods for one-dimensional SDEs. Despite this, numerical methods that use time integrals alongside increments have received less attention in the literature.
In this talk, we will develop some underlying theory for these time integrals and introduce a new numerical approach to SDEs that does not require evaluating vector field derivatives. We shall also discuss the possible implications of this work for multi-dimensional SDEs.
Based on the notion of paracontrolled distributions, existence and uniqueness results are presented for rough convolution equations. In particular, this wide class of equations includes rough differential equations with possible delay, stochastic Volterra equations, and moving average equations driven by Lévy processes. The talk is based on a joint work with Mathias Trabs.
Let f be the planar Bargmann-Fock field, i.e. the analytic Gaussian field with covariance kernel exp(-|x-y|^2/2). We compute the critical point for the percolation model induced by the level sets of f. More precisely, we prove that there exists a.s. an unbounded component in {f>p} if and only if p<0. Such a percolation model has been studied recently by Beffara-Gayet and Beliaev-Muirhead. One important aspect of our work is a derivation of a (KKL-type) sharp threshold result for correlated Gaussian variables. The idea to use a KKL-type result to compute a critical point goes back to Bollobás-Riordan. This is joint work with Alejandro Rivera.
We shall explain, from variational point of view, why the Laplaciian operator introduced by Ebin-Marsden using deformations is suitable to describe the fluid motion in a milieu with viscosity.
Smooth Gaussian functions appear naturally in many areas of mathematics. Most of the talk will be about two special cases: the random plane model and the Bargmann-Fock ensemble. Random plane wave are conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator in a generic domain. The Bargmann-Fock ensemble appears in quantum mechanics and is the scaling limit of the Kostlan ensemble, which is a good model for a `typical' projective variety. It is believed that these models, despite very different origins have something in common: they have scaling limits that are described be the critical percolation model. This ties together ideas and methods from many different areas of mathematics: probability, analysis on manifolds, partial differential equation, projective geometry, number theory and mathematical physics. In the talk I will introduce all these models, explain the conjectures relating them, and will talk about recent progress in understanding these conjectures.
In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.
Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).
Abstract: In this talk we describe an invariance principle for a class of non-homogeneous martingale random walks in $\RR^d$ that can be recurrent or transient for any dimension $d$. The scaling limit, which we construct, is a martingale diffusions with law determined uniquely by an SDE with discontinuous coefficients at the origin whose pathwise uniqueness may fail. The radial component of the diffusion is a Bessel process of dimension greater than 1. We characterize the law of the diffusion, which must start at the origin, via its excursions built around the Bessel process: each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining a Riemannian metric $g$ on the sphere $S^{d−1}$, different from the one induced by the ambient Euclidean space, allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Browninan motion with drift on the Riemannian manifold $(S^{d−1}, g)$. In particular, this provides a multidimensional generalisation of the Pitman–Yor representation of the excursions of Bessel process with dimension between one and two. Furthermore, the density of the stationary law of the angular component with respect to the volume element of $g$ can be characterised by a linear PDE involving the Laplace–Beltrami operator and the divergence under the metric $g$. This is joint work with Nicholas Georgiou and Andrew Wade.
We propose a new splitting algorithm to solve a class of quasilinear PDEs with convex and quadratic growth gradients.
By splitting the original equation into a linear parabolic equation and a Hamilton-Jacobi equation, we are able to solve both equations explicitly.
In particular, we solve the associated Hamilton-Jacobi equation by the Hopf-Lax formula,
and interpret the splitting algorithm as a stochastic Hopf-Lax approximation of the quasilinear PDE.
We show that the numerical solution will converge to the viscosity solution of the equation.
The upper bound of the convergence rate is proved based on Krylov's shaking coefficients technique,
while the lower bound is proved based on Barles-Jakobsen's optimal switching approximation technique.
Based on joint work with Shuo Huang and Thaleia Zariphopoulou.
Recurrent major mood episodes and subsyndromal mood instability cause substantial disability in patients with bipolar disorder. Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. The signature method is derived from stochastic analysis (rough paths theory) and has the ability to capture important properties of complex ordered time series data. To explore whether the onset of episodes of mania and depression can be identified using self-reported mood data.
In this talk, I am going to report on some on-going research at the interface between Rough Paths Theory and Schramm-Loewner evolutions (SLE). In this project, we try to adapt techniques from Rough Differential Equations to the study of the Loewner Differential Equation. The main ideas concern the restart of the backward Loewner differential equation from the singularity in the upper half plane. I am going to describe some general tools that we developed in the last months that lead to a better understanding of the dynamics in the closed upper half plane under the backward Loewner flow.
Joint work with Prof. Dmitry Belyaev and Prof. Terry Lyons
Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties.
The shuffle Hopf algebra, which is fundamental in Lyons’s groundbreaking work on rough paths, is based on Lie algebras without additional properties. Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths.
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries.
In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.
Joint work with: Charles Curry and Dominique Manchon
Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties.
The shuffle Hopf algebra, which is fundamental in Lyons’s groundbreaking work on rough paths, is based on Lie algebras without additional properties. Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths.
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries.
In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.
Joint work with: Charles Curry and Dominique Manchon
We consider L^2-approximations of white noise within the framework of regularity structures. Possible applications include support theorems for SPDEs driven by degenerate noises and numerics. Joint work with Ilya Chevyrev, Peter Friz and Tom Klose.
This talk is based on a joint work with Dmitry Beliaev.
We study the volume distribution of nodal domains of families of naturally arising Gaussian random field on generic manifolds, namely random band-limited functions. It is found that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.
Having made sense of differential equations driven by rough paths, we now have a new set of models available but when it comes to calibrating them to data, the tools are still underdeveloped. I will present some results and discuss some challenges related to building these tools.
Consider the Loewner equation associated to the upper-half plane. This is an equation originated from an extremal problem in complex analysis. Nowadays, it attracts a lot of attention due to its connection to probability. Normally this equation is driven by a real-valued function. In this talk, we will show that the equation still makes sense when being driven by a complex-valued function. We will relate this situation to the classical situation and also to complex dynamics.
In the talk, we discuss how to combine the recurrent neural network with the signature feature set to tackle the supervised learning problem where the input is a data stream. We will apply this method to different datasets, including the synthetic datasets( learning the solution to SDEs ) and empirical datasets(action recognition) and demonstrate the effectiveness of this method.
In this talk, we will revisit the proof of the large deviations principle of Wiener chaoses partially given by Borell, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Z_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the so-called Wigner matrices without Gaussian tails of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large deviations result to the last-passage time which yields a large deviations principle when the weight matrix has law $\mu_{\alpha}^{n^2}$, where $\mu_{\alpha}$ is the probability measure on $\mathbb{R}^+$ with density $2Z_{\alpha}^{-1}e^{-x^{\alpha}}$ when $\alpha \in (0,1)$.
The signature of a path has many properties that make it an excellent feature to be used in machine learning. We exploit this properties to analyse a stream of data that arises from a psychiatric study whose objective is to analyse bipolar and borderline personality disorders. We build a machine learning model based on signatures that tries to answer two clinically relevant questions, based on observations of their reported state over a short period of time: is it possible to predict if a person is healthy, has bipolar disorder or has borderline personality disorder? And given a person or borderline personality disorder, it is possible to predict his or her future mood? Signatures proved to be very effective to tackle these two problems.
Abstract. As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).
This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.
Inverting the signature of a path with ideas from linear algebra with implementations.
We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.
The (kinetic) Langevin equation is an SDE with degenerate noise that describes the motion of a particle in a force field subject to damping and random collisions. It is also closely related to Hamiltonian Monte Carlo methods. An important open question is, why in certain cases kinetic Langevin diffusions seem to approach equilibrium faster than overdamped Langevin diffusions. So far, convergence to equilibrium for kinetic Langevin diffusions has almost exclusively been studied by analytic techniques. In this talk, I present a new probabilistic approach that is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. The approach yields rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime, and it may help to shed some light on the open question mentioned above.
There is a routine for obtaining formulae for derivatives of smooth heat semigroups,and for certain heat semigroups acting on differential forms etc, established some time ago by myself, LeJan, & XueMei Li. Following a description of this in its general form, I will discuss its applicability in some sub-Riemannian situations and to higher order derivatives.
Lyons’ theory of rough paths allows us to solve stochastic differential equations driven by a Gaussian processes X of finite p-variation. The rough integral of the solutions against X again exists. We show that the solution also belong to the domain of the divergence operator of the Malliavin derivative, so that the 'Skorohod integral' of the solution with respect to X can also be defined. The latter operation has some properties in common with the Ito integral, and a natural question is to find a closed-form conversion formula between this rough integral and its Malliavin divergence. This is particularly useful in applications, where often one wants to compute the (conditional) expectation of the rough integral. In the case of Brownian motion our formula reduces to the classical Stratonovich-to-Ito conversion formula. There is an interesting difference between the formulae obtained in the cases 2<=p<3 and 3<=p<4, and we consider the reasons for this difference. We elaborate on the connection with previous work in which the integrand is generally assumed to be the gradient of a smooth function of X_{t}; we show that our formula can recover these results as special cases. This is joint work with Nengli Lim.
Let Q be a uniformly random quadrangulation with simple boundary decorated by a critical (p=3/4) face percolation configuration. We prove that the chordal percolation exploration path on Q between two marked boundary edges converges in the scaling limit to SLE(6) on the Brownian disk (equivalently, a Liouville quantum gravity surface). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. Our method of proof is robust and, up to certain technical steps, extends to any percolation model on a random planar map which can be explored via peeling. Joint work with E. Gwynne.
In certain cases of (linear) partial differential equations random perturbations have been observed to cause regularizing effects, in some cases even producing the uniqueness of solutions. In view of the long-standing open problems of uniqueness of solutions for certain PDE arising in fluid dynamics such results are of particular interest. In this talk we will extend some known results concerning the well-posedness by noise for linear transport equations to the nonlinear case.
Recent work in regularity structures has provided a robust solution theory for a wide class of singular SPDEs. While much progress has been made on understanding the analytic and algebraic aspects of renormalisation of the driving signal, the action of the renormalisation group on the equation still needed to be performed by hand. In this talk, we aim to give a systematic description of the renormalisation procedure directly on the level of the PDE, which allows for explicit computation of the form of the renormalised equation. Joint work with Yvain Bruned, Ajay Chandra, and Martin Hairer.
In this work we study a stochastic three-dimensional Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $\mathbb{S}^3$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Utpal Manna (Triva
This talk will address a new link from stochastic differential equations (SDEs) to nonlinear parabolic PDEs. Starting from the necessary and sufficient condition of the path-independence of the density of Girsanov transform for SDEs, we derive characterisation by nonlinear parabolic equations of Burgers-KPZ type. Extensions to the case of SDEs on differential manifolds and the case od SDEs with jumps as well as to that of (infinite dimensional) SDEs on separable Hilbert spaces will be discussed. A perspective to stochastically deformed dynamical systems will be briefly considered.
I will described how ideas from constructive quantum field theory can be adapted to produce a systematic approach for analytic renormalization in the theory of regularity structures.
The Hastings-Levitov models describe the growth of random sets (or clusters) in the complex plane as the result of iterated composition of random conformal maps. The correlations between these maps are determined by the harmonic measure density profile on the boundary of the clusters. In this talk I will focus on the simplest case, that of i.i.d. conformal maps, and obtain a description of the local fluctuations of the harmonic measure density around its deterministic limit, showing that these are Gaussian. This is joint work with James Norris.
If we fix a rectangle in the affine real space and if we choose at random a real polynomial with given degree d, the probability P(d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P(d) uniformly bounded from below when d increases? I will explain a positive answer to a very close question involving real analytic functions. This is a joint work with Vincent Beffara.
The Ising model is one of the most classical statistical mechanics model, which has seen spectacular mathematical and physical developments for almost a century. The description of its scaling limit at the phase transition is at the center of a fascinating (conjectured) connection between statistical mechanics and field theories. I will discuss how recent mathematical progress allows one to make the connection between the two-dimensional Ising model and Conformal Field Theory rigorous. If time allows, I will discuss the insight this gives one into related models and field theories.
Based off joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, R. Gheissari, K. Izyurov, F. Johansson-Viklund, K. Kytölä, S. Park and S. Smirnov
If a dynamical system has a conservation law, i.e. a constant along the trajectory of the motion, the study of its evolution along the trajectories of a perturbed system becomes interesting. Conservation laws can be seen everywhere, especially at the level of probability distributions of a reduced dynamic. We explain this with a number of models, in which we see a singular perturbation problem and identify a conservation law, the latter is used to seek out the correct scale to work with and to reduce the complexity of the system. The reduced dynamic consists of a family of ODEs with rapidly oscillating right hands side from which in the limit we obtain a Markov process. For stochastic completely integrable system, the limit describes the evolution of the level sets of the family of Hamiltonian functions over a very large time scale.
The Yang-Mills heat equation is the gradient flow corresponding to the Yang-Mills functional. It was initially introduced by S. K. Donaldson to study the existence of irreducible Yang-Mills connections on the projective plane. In this talk, we will consider this equation over compact three-manifolds with boundary. It is a nonlinear weakly parabolic equation, but we will see how one can prove long-time existence and uniqueness of solutions by gauge symmetry breaking. We will also demonstrate some strong regularization results for the solution and see how they lead to detailed short-time asymptotic estimates, as well as the long-time convergence of the Wilson loop functions.
Motivated by a problem in quasiconformal mapping, we introduce a new type of problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity.The problem will be referred to as \emph{geometric zero packing}, and is somewhat analogous to studying Fekete point configurations.The associated quantity is a density, denoted $\rho_\C$ in the planar case, and $\rho_{\mathbb{H}}$ in the case of the hyperbolic plane.We refer to these densities as \emph{discrepancy densities for planar and hyperbolic zero packing}, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures.The universal asymptoticvariance $\Sigma^2$ associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: $\Sigma^2= 1-\rho_{\mathbb{H}}$.We obtain the estimates$2.3\times 10^{-8}<\rho_{\mathbb{H}}\le0.12087$, where the upper estimate is derived from the estimate from below on $\Sigma^2$ obtained by Astala, Ivrii, Per\"al\"a, and Prause, and the estimate from below is much more delicate.In particular, it follows that $\Sigma^2<1$, which in combination with the work of Ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached.Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics$\mathrm{B}(k,t)\sim\frac14\Sigma^2 k^2|t|^2$ for small $t$ and $k$, the conjectured formula $\mathrm{B}(k,t)=\frac14k^2|t|^2$ is not true.As for the actual numerical values of the discrepancy density $\rho_\C$, we obtain the estimate from above $\rho_\C\le0.061203\ldots$ by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The values of $\rho_{\mathbb{H}}$ is expected to be somewhat close to the value of $\rho_\C$.
One of the challenges of 21st-century science is to model the evolution of complex systems. One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations. Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.
We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations. Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions. We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.
Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation.
Identifying correlations within multiple streams of high-volume time series is a general but challenging problem. A simple exact solution has cost that is linear in the dimensionality of the data, and quadratic in the number of streams. In this work, we use dimensionality reduction techniques (sketches), along with ideas derived from coding theory and fast matrix multiplication to allow fast (subquadratic) recovery of those pairs that display high correlation.
Joint work with Jacques Dark
I will give a light introduction to the theory of regularity structures and then discuss recent developments with regards to renormalization within the theory - in particular I will describe joint work with Martin Hairer where multiscale techniques from constructive field theory are adapted to provide a systematic method of obtaining needed stochastic estimates for the theory.
Abstract: Equations with small scales abound in physics and applied science. When the coefficients vary on microscopic scales, the local fluctuations average out under certain assumptions and we have the so-called homogenization phenomenon. In this talk, I will try to explain some probabilistic approaches we use to obtain the first order random fluctuations in stochastic homogenization. If homogenization is to be viewed as a law of large number type result, here we are looking for a central limit theorem. The tools we use include the Kipnis-Varadhan's method, a quantitative martingale central limit theorem and the Stein's method. Based on joint work with Jean-Christophe Mourrat.