These lectures will explain how multiscale methods developed for analysis of Feynman diagrams can be extended to prove stochastic estimates on renormalised products that are a crucial input for the theory of regularity structures.
I will not assume any knowledge of Feynman diagrams or renormalisation in these lectures and aim to clearly describe the power-counting that appears in the analysis of these diagrams.
Time permitting, I will discuss how these diagrammatic methods also allow the renormalization of the Sine-Gordon model in the subcritical regime.

These lectures will explain how multiscale methods developed for analysis of Feynman diagrams can be extended to prove stochastic estimates on renormalised products that are a crucial input for the theory of regularity structures.
I will not assume any knowledge of Feynman diagrams or renormalisation in these lectures and aim to clearly describe the power-counting that appears in the analysis of these diagrams.
Time permitting, I will discuss how these diagrammatic methods also allow the renormalization of the Sine-Gordon model in the subcritical regime.

These lectures will explain how multiscale methods developed for analysis of Feynman diagrams can be extended to prove stochastic estimates on renormalised products that are a crucial input for the theory of regularity structures.
I will not assume any knowledge of Feynman diagrams or renormalisation in these lectures and aim to clearly describe the power-counting that appears in the analysis of these diagrams.
Time permitting, I will discuss how these diagrammatic methods also allow the renormalization of the Sine-Gordon model in the subcritical regime.

These lectures will explain how multiscale methods developed for analysis of Feynman diagrams can be extended to prove stochastic estimates on renormalised products that are a crucial input for the theory of regularity structures.
I will not assume any knowledge of Feynman diagrams or renormalisation in these lectures and aim to clearly describe the power-counting that appears in the analysis of these diagrams.
Time permitting, I will discuss how these diagrammatic methods also allow the renormalization of the Sine-Gordon model in the subcritical regime.

These lectures will explain how multiscale methods developed for analysis of Feynman diagrams can be extended to prove stochastic estimates on renormalised products that are a crucial input for the theory of regularity structures.
I will not assume any knowledge of Feynman diagrams or renormalisation in these lectures and aim to clearly describe the power-counting that appears in the analysis of these diagrams.
Time permitting, I will discuss how these diagrammatic methods also allow the renormalization of the Sine-Gordon model in the subcritical regime.

We will introduce the notion of decorated trees and their associated multi-pre-Lie/Hopf algebraic structures. We will see how this structure can be used in numerous applications ranging from singular SPDEs up to a numerical scheme for dispersive PDEs. This structure is also well adapted for handling renormalisation and symmetries.
The key references are:
[1] I. Bailleul, Y. Bruned, "Renormalised singular stochastic PDEs", 16 pages, arxiv:2101.11949 .
[2] Y. Bruned, F. Gabriel, M. Hairer, L. Zambotti, "Geometric stochastic heat equations", Journal of the American Mathematical Society, 2022, Volume 35, Issue 1, pp 1-80. arxiv:1902.02884 . doi:10.1090/jams/977 .
[3] Y. Bruned, D. Manchon, "Algebraic deformation for (S)PDEs", 43 pages, submitted, arxiv:2011.05907 .
[4] Y. Bruned, K. Schratz, "Resonance based schemes for dispersive equations via decorated trees", Forum of Mathematics, Pi, 2022, Volume 10, e2, pp 1 76. arxiv:2005.01649 . doi:10.1017/fmp.2021.13.

The aim of this lectures is to introduce the stochastic calculus for the flows of interacting Brownian particles on the real line. Such families of Brownian motions lose some Gaussian features and obtain new. So, the establishing of such results as Girsanov type theorem, Large Deviations Principle, chaotic expansion, Clark representation, requires some new techniques and arguments. In the lectures we will present such new tools. In particular the time of free motion in the continuum particle system and quadratic entropy for such systems will be introduced. Using these new notions we obtain the above mentioned Gaussian style results and some essentially infinite-dimensional properties of the Brownian stochastic flows on the real line as well.
There are 4 lectures per 50 min.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton--Jacobi- Bellman PDE. We show how this technique (the reduction to a Hamilton--Jacobi--Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of $A_2$ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation.As an illustration we also compute the numerical value of the norm in $L^p$ of the real and imaginary parts of the Ahlfors--Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors--Beurling operator itself and explain the connection with Morrey's problem.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton--Jacobi- Bellman PDE. We show how this technique (the reduction to a Hamilton--Jacobi--Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of $A_2$ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation.As an illustration we also compute the numerical value of the norm in $L^p$ of the real and imaginary parts of the Ahlfors--Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors--Beurling operator itself and explain the connection with Morrey's problem.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton--Jacobi- Bellman PDE. We show how this technique (the reduction to a Hamilton--Jacobi--Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of $A_2$ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation. As an illustration we also compute the numerical value of the norm in $L^p$ of the real and imaginary parts of the Ahlfors--Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors--Beurling operator itself and explain the connection with Morrey's problem.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton--Jacobi- Bellman PDE. We show how this technique (the reduction to a Hamilton--Jacobi--Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of $A_2$ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation. As an illustration we also compute the numerical value of the norm in $L^p$ of the real and imaginary parts of the Ahlfors--Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors--Beurling operator itself and explain the connection with Morrey's problem.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton--Jacobi--Bellman PDE. We show how this technique (the reduction to a Hamilton--Jacobi--Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of $A_2$ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation.As an illustration we also compute the numerical value of the norm in $L^p$ of the real and imaginary parts of the Ahlfors--Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors--Beurling operator itself and explain the connection with Morrey's problem.

Flows of ideal incompressible fluid can be regarded as geodesics on the group of volume preserving diffeomorphisms of the flow domain. It is natural to look for the shortest geodesics. However, the minimizers of the length functional deliver merely weak solutions of the Lagrange equations, which are continuous analogues of braids. The main part of the work is the proof of their regularity which means that there exists a velocity field corresponding to the length minimizing braid. This field is a weak solution of the Euler equation.
Then we introduce an extension of the braids named "folded braids". Their minimization results in a weak solution of the Euler equation which tends to zero in a weak sense as $t\to 0$. This can be interpreted as a weak solution which is zero for all $t

The feature of infinite dimensional energy landscapes that is best understood, when it exists, is the absolute energy minimizer. Many recent results in applied analysis can be viewed as works that probe additional details of the energy landscape. We will study a sample of such problems that focus on features such as the height of energy barriers, the crossover from one minimizer to another as a function of a critical parameter value, applications of the Lojasiewicz-Simon condition for long-time convergence, and sufficient conditions for dynamic metastability.

The feature of infinite dimensional energy landscapes that is best understood, when it exists, is the absolute energy minimizer. Many recent results in applied analysis can be viewed as works that probe additional details of the energy landscape. We will study a sample of such problems that focus on features such as the height of energy barriers, the crossover from one minimizer to another as a function of a critical parameter value, applications of the Lojasiewicz-Simon condition for long-time convergence, and sufficient conditions for dynamic metastability.

The feature of infinite dimensional energy landscapes that is best understood, when it exists, is the absolute energy minimizer. Many recent results in applied analysis can be viewed as works that probe additional details of the energy landscape. We will study a sample of such problems that focus on features such as the height of energy barriers, the crossover from one minimizer to another as a function of a critical parameter value, applications of the Lojasiewicz-Simon condition for long-time convergence, and sufficient conditions for dynamic metastability.

The feature of infinite dimensional energy landscapes that is best understood, when it exists, is the absolute energy minimizer. Many recent results in applied analysis can be viewed as works that probe additional details of the energy landscape. We will study a sample of such problems that focus on features such as the height of energy barriers, the crossover from one minimizer to another as a function of a critical parameter value, applications of the Lojasiewicz-Simon condition for long-time convergence, and sufficient conditions for dynamic metastability.