Tag - Differential equations

Nicolas Perkowski: Energy solutions and generators for singular SPDEs

13th June, 2023

I will discuss how to use tools from Gaussian analysis and operator semigroups together with some commutator estimates to construct Markov semigroups for some singular SPDEs. This yields in particular uniqueness for Goncalves-Jara-Gubinelli type energy solutions. The method applies to some critical equations and, in finite dimensions, even for some supercritical equations. In infinite dimensions we get Markov semigroups for supercritical equations but we lack a uniqueness result for supercritical energy solutions in infinite dimensions. The main SPDE examples where this works are of Burgers type: quadratic, divergence-free nonlinearity and Gaussian quasi-invariant measure.

Hao Shen: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, I

13th June, 2023

In an earlier work with Chandra, Chevyrev and Hairer, we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge covariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev, where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic. A novel step in the argument is a geometric way to identify the limit using Wilson loops. This universality of the dynamics is crucial for obtaining a sequence of important results for 2D Yang-Mills, including for instance the invariance of the 2D Yang-Mills measure for its Langevin dynamic, which will be discussed by Ilya Chevyrev.

Martin Hairer: The role of symmetry in renormalisation

13th June, 2023

There are several interesting situations where the solutions to singular SPDEs exhibit a symmetry at a formal level that could in principle be broken by the renormalization procedure required to define them. We’ll discuss a relatively simple argument showing that, in many cases, the renormalization can be chosen in such a way that the symmetry does indeed hold and we’ll apply it to the stochastic quantization of the 3D Yang-Mills theory.

Nikos Zygouras: SPDEs at the critical dimension

13th June, 2023

I will make an overview of the progress on treating SPDEs at the critical dimension, the current status and further challenges. Examples will include stochastic heat equations and a more recent Allen-Cahn.

Luca Fresta: The forward-backward SDE for subcritical Euclidean fermionic field theories

12th June, 2023

In this talk, I will describe a synergy between the renormalization group (RG) in the form of Polchinski's equation and the stochastic quantisation in the form of a forward-backward stochastic differential equation (FBSDE). This approach can be used for constructing subcritical Grassmann Gibbsian measures and is based on controlling the solution of the FBSDE by means of a flow equation with respect to a scale parameter. However, unlike the standard RG approach, we only need to solve Polchinski’s equation in an approximate way, resulting in a great simplification of the analysis.

Jonas Sauer: Time-Periodic Weighted Lp-Estimates

20th February, 2023

In between elliptic PDEs, which do not depend on time (think of the steady-state Stokes equations), and honest parabolic PDEs, which do depend on time and are started at a given initial value (think of the instationary Stokes equations), there are time-periodic parabolic PDEs: On the one hand, time-independent solutions to the elliptic PDE are also trivially time-periodic, which gives periodic problems an elliptic touch, on the other hand solutions to the initial value problem which are not constant in time might very well be periodic.

I want to advocate for time-periodic problems not being the little sister of either elliptic or parabolic problems, but being a connector between the two and a class of its own right. This is highlighted by a direct method for showing a priori Lp estimates for time-periodic, linear, partial differential equations. The method is generic and can be applied to a wide range of problems, for example the Stokes equations and boundary value problems of Agmon-Douglas-Nirenberg type. In the talk, I will present these ideas and show how they can be extended to the setting of weighted Lp estimates, which is advantageous for extrapolation techniques and rougher boundary data.

Theodore Vo: Canards, Cardiac Cycles, and Chimeras

8th March, 2022

Canards are solutions of singularly perturbed ODEs that organise the dynamics in phase and parameter space. In this talk, we explore two aspects of canard theory: their applications in the life sciences and their ability to generate new phenomena.

More specifically, we will use canard theory to analyse a canonical model of the electrical activity in a heart muscle cell. We demonstrate that pathological heart rhythms, called early afterdepolarisations, are canard-induced phenomena. We use this knowledge to explain the rich set of model behaviours, some of which have also been observed in experiments. Then, we explore a new class of canard-induced patterns in reaction-diffusion PDEs which exhibit coexisting domains of mutually synchronised oscillators and complementary domains of decoherent (asynchronous) oscillators.

Ivan Guo: Stochastic Optimal Transport in Financial Mathematics

22nd February, 2022

In recent years, the field of optimal transport has attracted the attention of many high-profile mathematicians with a wide range of applications. In this talk we will discuss some of its recent applications in financial mathematics, particularly on the problems of model calibration, robust finance and portfolio optimisation. Classical topological duality results are extended to probabilistic settings, connecting stochastic control problems with non-linear partial differential equations and providing interesting practical interpretations in finance. We will also look at how numerical methods, including machine learning algorithms, can be implemented to solve these problems.

Svitlana Mayboroda: PDEs vs. Geometry: analytic characterizations of geometric properties of sets

7th February, 2022

In this talk we will discuss connections between the geometric and analytic/PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including domains and equations with singularities and structural complexity. The main result establishes that in all dimensions d < n, a d-dimensional set in ℝn is regular (rectifiable) if and only if the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes).