Related papers: Regularity structures and the dynamical $\Phi^4_3$…
A theory of structure is formulated for systems of many structureless classical particles with stable local interactions in Euclidean space. Such systems are shown to have their structure in thermodynamic equilibrium determined exactly by a…
The flow equations of the renormalization group allow to analyse the perturbative $n$-point functions of renormalizable quantum filed theories. Rigorous bounds implying renormalizability permit to control large momentum behaviour, infrared…
This set of five lectures provides an introduction to regularity structures and their use for the study of singular stochastic partial differential equations. Two appendices provide some additional informations that enter in the main text…
We derive a priori bounds for the $\Phi^4$ equation in the full sub-critical regime using Hairer's theory of regularity structures. The equation is formally given by \begin{equation} \label{e}(\partial_t-\Delta)\phi = -\phi^3 + \infty \phi…
We consider systems of stochastic evolutionary equations of the type $$du=\mathrm{div}\,S(\nabla u)\,dt+\Phi(u)dW_t$$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $$S(\xi)=(1+|\xi|)^{p-2}\xi,\quad \xi\in\mathbb…
Quantum fields are generally taken to be operator-valued distributions, linear functionals of test functions into an algebra of operators; here the effective dynamics of an interacting quantum field is taken to be nonlinearly modified by…
We introduce techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and blow ups) into more effective control. In the present paper, we focus on proving…
The aim of this article is to give a rather extensive, and yet nontechnical, account of the birth of the regularity theory for generalized minimal surfaces, of its various ramifications along the decades, of the most recent developments,…
This paper proposes a basic theory on physical reality, and a new foundation for quantum mechanics and classical mechanics. It does not only solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but…
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were…
Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems,…
This work is dedicated to the results were got in the model theory of the regular polygons. We give the characterization of the monoids with axiomatizable and model complete class of regular polygons. We describe the monoids with complete…
It is shown that Euclidean field theory with polynomial interaction, can be regularized using the wavelet representation of the fields. The connections between wavelet based regularization and stochastic quantization are considered.
The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some recent results on the structure of singular free boundary…
We construct the $\Phi^4_3$ measure on an arbitrary 3-dimensional compact Riemannian manifold without boundary as an invariant probability measure of a singular stochastic partial differential equation. Proving the nontriviality and the…
We establish codimension 4 regularity of noncollapsed sequences of metrics with bounds on natural generalizations of the Ricci tensor. We obtain a priori L2 curvature estimates on such spaces, with diffeomorphism finiteness results and…
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study the construction of the $\Phi^3_3$-measure and complete the program on the…
One of the interesting aspects in the study of atomic nuclei is the strikingly regular behaviour many display in spite of being complex quantum-mechanical systems, prompting the universal question of how regularity emerges out of…
We check the eigenvalue spectrum of the $\Phi^{4}_{1+1}$ Hamiltonian against Poisson or Wigner behavior predicted from random matrix theory. We discuss random matrix theory as a tool to discriminate the validity of a model Hamiltonian…
I give an overview over some work on rigorous renormalization theory based on the differential flow equations of the Wilson-Wegner renormalization group. I first consider massive Euclidean $\phi_4^4$-theory. The renormalization proofs are…