"Precise large deviations for S(P)DEs: theory, numerics, applications"

Oberseminar Analysis-Probability at MPI, G3 10 (Lecture hall), 15:15

Abstract: In this talk, I will first review the classical theory of Laplace asymptotics for small-noise diffusions in R^n and extensions to stochastic partial differential equations (SPDEs), with particular emphasis on sharp asymptotics. Going beyond the usual log-asymptotics of large deviation theory, we focus on prefactors and thus Hessian contributions that are essential for accurate probability estimation.

I will then describe how these theoretical results can be turned into a practical computational framework for estimating rare event probabilities without sampling. The approach is designed to scale to high-dimensional state spaces, as required for SPDEs, and involves solving ODE/PDE-constrained optimization problems to identify optimal paths, as well as computing Hessian determinants around minimizers. Modern tools, such as automatic differentiation, make these steps tractable.

As an application, I will present results for intermittency in the stochastic 1D viscous Burgers equation, as a canonical toy model of fluid turbulence. Time permitting, I will conclude with an outlook on future improvements based on ideas from the functional renormalization group.

Based on joint work with Tobias Grafke, Rainer Grauer, Georg Stadler, Shanyin Tong, in Stat. Comput. 33(6), 137 (2023), arXiv:2502.20114, and arXiv:2512.03841.

"Post-Lie deformations of pre-Lie algebras and their applications in Regularity Structures"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: We prove that the post-Lie algebraic structure on the decorated trees which appears spontaneously in Regularity Structures is a post-Lie deformation of a pre-Lie algebra. This post-Lie structure is the conerstone of the algebraic foundations of the theory of Regularity Structures, a theory that solves a large class of subcritical singular stochastic partial differential equations which were out of reach without a proper algebraic structure. This deformation allows to reduce the origin of these structures to a simpler object. We use the formalism of deformation theory in order to understand the nature of this deformation. This leads us to construct the differential graded Lie algebra that governs post-Lie deformations of a pre-Lie algebra, to develop the post-Lie cohomology theory for a pre-Lie algebra, by which we classify infinitesimal post-Lie deformations of a pre-Lie algebra using the second cohomology group. The rigidity of such kind of deformations is also characterized using the second cohomology group. This is a joint work with Yvain Bruned and Yunhe Sheng.

TBA

Oberseminar Analysis-Probability at MPI, E1 05 (Leibniz-Saal), 15:15

Abstract: TBA

TBA

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30

Abstract: TBA

TBA

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: TBA

TBA

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30

Abstract: TBA

TBA

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: TBA

TBA

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30

Abstract: TBA

"Generalised graphical representations"

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30

Abstract: Numerous dynamic spatial models feature interactions between particles that are based on waiting times; the most prominent examples being first passage percolation, where a fluid spreads from a source, and the contact process, where an infection spreads on a graph with the possibility of recovery. However, waiting times are unnatural for certain models. One can argue that people have a daily routine and meet each other on a relatively fixed schedule, rather than a random waiting time after having met someone else. With the help of graphical representations, we develop a framework that moves away from waiting times towards so called contact times. As an application, we present results on a first contact percolation model, where people follow a random daily meeting routine, as well as contact processes with periodic recoveries.

"Quasilinear parabolic equations in the Bessel scale"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: In this talk, I will discuss and analyze scalar quasilinear parabolic equations posed on nonsmooth domains with generally irregular input data. The lat ter refers to merely measurable and bounded coefficients, mixed boundary conditions, possibly inhomogeneous Robin- or Neumann boundary data, and also to nonlinear forcing functions which may act on the gradient of the solution. The driving force for the analysis will be maximal parabolic regu larity of the differential operators in the scale of (duals of) Bessel potential spaces incorporating mixed boundary conditions, with the scale parameter depending on the smoothness of the solution within the quasilinear operator. These depend on permanence principles for the elliptic counterparts which are leveraged from bilinear interpolation. A careful analysis allows to close the loop for the quasilinear equation and thus obtain local-in-time solutions under minimal assumptions. I further mention some sufficient criteria for solutions to exist globally, and how these can be leveraged in the framework of optimal control problems to achieve a satisfying theory for nonsmooth problems with strong nonlinearities. 

"On the global well-posedness of the stochastic Yang-Mills-Higgs equations in two dimensions"

Oberseminar Analysis-Probability at MPI, E2 10 (Leon-Lichtenstein), 15:15

Abstract: The Yang-Mills-Higgs (YMH) model plays a central role in several areas of mathematics, including analysis, geometry, and probability theory. In this talk, we implement the stochastic quantization procedure for the YMH model in two dimensions. That is, we construct the two-dimensional YMH measure by proving global well-posedness and uniform-in-time bounds for the two-dimensional stochastic YMH equations. As part of our proof, we first discuss covariant stochastic objects and a covariant para-controlled Ansatz. Then, we discuss a decay mechanism for the stochastic YMH equations near unstable Yang-Mills connections. This is joint work with S. Cao, M. Hairer, and W. Zhao.

"Oriented matroids of planar point configurations"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Oriented matroids are purely combinatorial objects that were devised, independently, by Bland to study the simplex algorithm, by Folkman and Lawrence to study face lattices of polytopes, and by Las Vergnas to study questions in graphs and combinatorics. In this talk, I will discuss the oriented matroids that arise from finite, planar point configurations. We will consider them from the point of views of computational complexity and discrete geometry, with a particular emphasis on their symmetries and typical properties.