Schedule
| Wed. 8 | Thu. 9 | Fri. 10 | |
|---|---|---|---|
| 09:00–09:45 | Speicher Room 734 | Gabriel Room 734 | Cébron Room 734 |
| 10:00–10:45 | Dabrowski Room 734 | Manchon Room 734 | Malham Room 734 |
| 11:00–11:45 | Schott Room 734 | Curry Room 734 | Diehl Room 734 |
| 12:00–14:00 | Lunch | ||
| 14:15–15:00 | Tapia Room 734 | Dahlqvist Room 734 | Bruned Room 734 |
| 15:00–18:45 | Discussions Rooms 734 and 822 | ||
Talks
Quasi-generalised KPZ equation
(Slides)Yvain Bruned
The generalised KPZ equation has been fully understood in a geometric context where one constructs a solution satisfying the invariance under the action of diffeomorphisms and Itô's isometry. But for the quasilinear variant of the equation, writing the renormalised equation with only local counterterms is an open problem. Indeed, some non-commutative structure appears and one may expect this structure to vanish at the level of the renormalised equation. This is a work in progress with Mate Gerencser.
A quantitative fourth moment theorem in free probability theory
Guillaume Cébron
Kemp, Nourdin, Peccati, and Speicher showed that, for certain non-linear functionals of the free Brownian motion, convergence of the fourth moment implies convergence in distribution towards the semicircular law. It is the free analogue of the fourth moment theorem. Using the combination of Stein’s method and the free Malliavin calculus, I will explain how to obtain a quantitative version of the free fourth moment theorem.
Geometrization of elliptic operators and numerical methods for SDEs
(Slides)Charles Curry
Following work of Cruzeiro, Malliavin and Thalmeier, we show how the Eells-Elworthy-Malliavin construction of Riemannian Brownian motion using orthonormal frame bundles finds a surprising application in numerical integration of SDEs obeying an ellipticity condition. We discuss this in the context of stochastic Lie group integration, and highlight the challenges faced on attempting to generalize to the hypoelliptic/sub-Riemannian setting.
Free differential analysis in planar algebras
Yoann Dabrowski
First, I will give some background on Vaughan Jones's planar algebras, a particularly interesting class of diagrammatic algebras. This is a natural algebraic object produced from (finite index) inclusions of von Neumann algebras with trivial centers (so-called subfactors). The simplest example comes from the fixed point algebra of an (outer finite) group action on a factor, but planar algebras also generalize (finite dimensional) Hopf algebras. They are also related to knot invariants and are naturally related to free probability. Conversely, starting from natural non-commutative laws on planar algebras, Guionnet-Jones-Shlyakhtenko were able to reconstruct such inclusions of von Neumann algebras. In the second part of the talk, I will explain some results in a joint work with Stephen Curran and Dima Shlyakhtenko explaining how to control algebraic properties (typically computation of commutants) of algebras reconstructed in this way from very general non-commutative laws on planar algebras. The key idea is to use derivations defined (diagrammatically) on a larger algebra and to exploit its behavior on a better understood subalgebra in order to control commutants of the small algebra. This idea has been used again recently in type III settings by Brent Nelson and is probably of general interest, beyond the planar algebra case.
Yang-Mills measure and Free Unitary Brownian Loops
Antoine Dahlqvist
In this talk I shall present results about different diffusion on the unitary groups, in the regime where the size of the matrix goes to infinity. I will explain how it relates to the Yang-Mills measure and the notion of master fields on surfaces. This leads to a first definition of a Free Unitary Brownian Loop.
Discrete integration for extracting information of time series
(Slides)Joscha Diehl
A proven tool for extracting information of a discrete time-series is to interpolate it linearly and calculate the iterated-integrals signature. It turns out though that for a one-dimensional time-series the extracted information consists only in the increment over the whole time interval. Using discrete (Ito- and Backward-Ito-type) integration a lot more data is gathered, while at the same time preserving some invariant properties that the classical signature enjoys. I discuss the algebraic structures involved and applications. This is work in progress with Kurusch Ebrahimi-Fard (NTNU), Max Pfeffer (MPI Leipzig) and Nikolas Tapia (NTNU).
Permutation-Invariant Matrix-Valued Lévy Processes and Partition Algebra
(Slides)Franck Gabriel
Convergence of large-size permutation-invariant matrix-valued Lévy processes toward a deterministic limit can be understood using a new triangular transformation and two co-products on the partition algebra. Schur-Weyl-Jones duality allows one to define cumulants for (permutation invariant) random matrices which generalize usual and n.c. cumulants. This allows one to define an additive and multiplicative convolutions on the partition algebra which mimic addition and multiplication of --large independent permutation invariant-- random matrices. From the study of additive and multiplicative infinitesimal characters on partitions, we deduce a general theorem for the convergence in probability of matrix-valued Lévy processes. These general results will be illustrated using two Lévy processes: the unitary Brownian motion and random walks on the symmetrical group.
Fredholm Grassmannian flows and their applications to nonlinear PDEs and SPDEs
(Slides)Simon J.A. Malham
We present a programme for generating the solutions of large classes of nonlinear partial differential equations, by pulling the equations back to a linear system of equations. The idea underlying this programme is to lift the standard relation between Riccati equations and linear systems to the infinite dimensional setting. This generalisation is well-known in linear-quadratic optimal control theory where the off-line Riccati solution mediates the optimal current state feedback. The solution procedure can be presented at an elementary level and many examples will be included. Such example applications are partial differential equations with nonlocal nonlinearities, for example the nonlocal FKPP equation, Smoluchowski's coagulation equation and, by association, the standard inviscid Burgers equation. The latter arises in nonlinear optimal control theory. We also formally include the viscous stochastic Burgers equation via random characteristics and show how the procedure extends to analogous classes of SPDEs.
Monomial bases
Dominique Manchon
I will present an original construction of monomial bases for free pre-Lie, Lie and post-Lie algebras, using two free magmatic products on the linear span of planar rooted trees. Joint work with Mahdi Al Kaabi and Frédéric Patras.
Regularity of non-commutative distributions and the free skew field
Roland Speicher
I will report on recent joint work with Tobias Mai and Sheng Yin, when the division closure (within affiliated unbounded operators) of concrete operators in finite von Neumann algebras is isomorphic to an abstract algebraic object, the free skew field (aka the field of non-commutative rational functions).
Integration with respect to the non-commutative fractional Brownian motion
(Slides)René Schott
We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian motion in a non-commutative probability setting. When the Hurst index H of the process is stricly larger than ½, integration can be handled through the so-called Young procedure. The situation where H = ½ corresponds to the specific free case, for which an Itô-type approach is known to be possible. When H < ½, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined Lévy area process. We show that such an object can indeed be “canonically” constructed for any H ∈ (¼, ½). Finally, when H ≤ ¼, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) Lévy area above the standard fractional Brownian motion. (joint work with Aurélien DEYA, Institut Elie Cartan, University of Lorraine)
Noncommutative Wick polynomials
(Slides)Nikolas Tapia
In a series of works, Ebrahimi-Fard and Patras have developed a Hopf-algebraic approach to different moments-cumulants relations in noncommutative probability. We present here a similar approach to noncommutative Wick polynomials, introduced by Anshelevich in the early 2000's. This approach unifies three classes of such polynomials, namely free, bolean and conditionally free Wick polynomials. Finally, we discuss a relation between this combinatorial description and a certain substitution operation on non-commutative power series.