# London-Ghent Microlocal Analysis Workshop: 23-24 March 2022

The next ‘London-Ghent Microlocal Analysis Workshop’ will take place on

23-24 March 2021, online on ZOOM

Link to zoom Here! [Meeting ID: 880 0883 0881 – Passcode: 671966]

This Microlocal Analysis Workshop will be organised by Ph.D. students of the Analysis and PDE group of Prof. Dr. Michael Ruzhansky (Ghent University and Queen Mary University of London) and Ph.D. students of the research group of Prof. Dr. Boguslaw Zegarlinski (Imperial College London).

The aim of this workshop is to present some modern trends in the microlocal analysis of PDE and its applications.

Speakers

Schedule

Opening Ceremony

Workshop Talks:
Plenary Talk: Prof. Dr. Roland Duduchava
Institute of Mathematics, the University of Georgia & A.Razmadze Mathematical Institute, Tbilisi, Georgia

Shell equation derived by the Г-convergence

Invited Talk: Karina Navarro Gonzalez

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

In this talk I will present sharp estimates for the covering numbers of the embedding of the Reproducing Kernel Hilbert Space (RKHS) associated with the Weierstrass fractal kernel into the space of continuous functions. The method is based on the characterization of the infinite dimensional RKHS generated by the Weierstrass fractal kernel and it requires estimates for the norm operator of orthogonal projections on the RKHS.

Invited Talk: Dr Serena Federico
Università di Bologna, Italy

On some variable coefficient Schrödinger operators on ${{ \mathbb{R}\times \mathbb{T}^2}}$

In this talk we will investigate the validity of sharp Strichartz estimates for some variable coefficient Schrödinger operators on $\mathbb{R}\times \mathbb{T}^2$. In the first part of the talk we shall consider the problem for a class of time-degenerate Schrödinger operators on $\mathbb{R}\times \mathbb{T}^2$.  Next, still in the same setting, we will focus on a class of nondegenerate space-variable coefficient Schrödinger operators. Finally, local well-posedness results for the seminilinear initial value problem for the aforementioned classes will be given.

Invited Talk: Dr Bolys Sabitbek
Queen Mary University of London, UK

Global existence and nonexistence of semilinear wave equation with a new condition

In in this talk, we consider the initial-boundary problem for semilinear wave equation with a new condition

\begin{aligned} \alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma,\end{aligned}

for some positive constants $\alpha, \beta, \sigma$, where $\beta < \frac{\lambda_1(\alpha -2)}{2}$ with $\lambda= \lambda_1$ being a first eigenvalue of Laplacian. By introducing a family of potential wells, we establish the invariant sets, vacuum isolation of solutions, global existence and blow-up solutions of semilinear wave equation for initial conditions $E(0) and $E(0)=d$.

Imperial College London, UK

Sobolev, Hardy and CLR inequalities in two dimensions

We present a new result for eigenvalue bounds in two dimensions and present links to other well-known inequalities.

Invited Talk: Mengchun Zhang
Imperial College London, UK

Logarithmic Schrödinger equation in infinite dimensions

We introduce logarithmic Schrodinger equations on the infinite dimensional space $\mathbb{R}^{\mathbb{Z}^d}$ and prove the existence of weak solution by finite-dimensional approximations.

Invited Talk: Shreya Mehta
Imperial College London, UK

Noncommutative extension of diffusions

In quantum mechanics, the noncommutative framework is necessary for understanding of the events at quantum scale. We want to find the noncommutative analogue of the Hörmander theorem involving the Hörmander type diffusion operators. We use the Quantum Dirichlet forms to generate the corresponding Markov semigroups for these diffusions.

Invited Talk: Yaozhong Qiu
Imperial College London, UK

Applications of heat kernels

We discuss some applications of positivity preserving semi groups to spectral theory and give some examples in the form of Markov semi groups. We discuss the equivalence between the semi group and the generator as well and how properties in one domain can be transferred to another.

Invited Talk: Niyaz Tokmagambetov
Ghent University, Belgium

Non-harmonic Analysis of Boundary Value Problems

In this talk we present the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We present the applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.

Invited Talk: Brian Grajales

Geodesics on adjoint orbits of $SL(n,\mathbb{R})/P_{\emptyset}$

The aim of this talk is to present some properties of geodesics on adjoint orbits of $SL(n,\mathbb{R})/P_{\emptyset}$, where $P_{\emptyset}$ is a particular parabolic subgroup of $SL(n,\mathbb{R})$. The homogeneous space $SL(n,\mathbb{R})/P_{\emptyset}$ can be identified with the tangent bundle of certain $SO(n)$-flag manifold, so it is possible to use results from invariant geometry of this flag manifold in order to obtain some explicit description of families of geodesics on $SL(n,\mathbb{R})/P_{\emptyset}$.

Closing Ceremony

For any inquiries about the workshop, you can also Email Mrs Kim Verbeeck.

Organisers:
Duvan Cardona Sanchez, Ghent University
Marianna Chatzakou, Ghent University