# Ghent Methusalem Colloquium

The Ghent Methusalem Colloquium is intended for a broad audience of PhD students, postdocs and professors at the Ghent Analysis & PDE Center and beyond. The series includes colloquia from visiting and invited guests.

## Upcoming Colloquia

Prof. Durvudkhan Suragan
Navarbayev University,
Kazakhstan

Topic: On pointwise estimates for polyharmonic Green’s function

Time: 10:00 AM – 10:45 AM (CET) – 15 July 2022
Where: The venue is Leslokaal 3.1

Abstract: We talk about Maz’ya’s article “Seventy five (thousand) unsolved problems in analysis and partial differential equations”. Particularly, we discuss Problem 32 and give a simple proof for pointwise estimates for (Dirichlet) polyharmonic Green’s functions in balls with an arbitrary radius.

Colombia

Topic: S(m,g) calculus and some applications

Time: 11:00 AM – 11:45 AM (CET) – 15 July 2022
Where: The venue is Leslokaal 3.1

Abstract: We will  discuss some recent applications to the analysis of  partial differential operators and PDEs in the setting of the  S(m,g) calculus introduced by Lars Hörmander.

Prof. Johannes Sjostrand
The Institut de Mathématiques de Bourgogne,
France

Topic: TBA

Time: 11:00 AM (CET) – 20 June 2022
Where: ZOOM

Abstract: TBA

Prof. Victor Nistor
The Institut Élie Cartan de Lorraine,
France

Topic: Analysis on non-compact manifolds and index theory

Time: 11:00 AM (CET) – 13 June 2022
Where: ZOOM

Abstract: I will review a general approach to analysis on singular and non-compact manifolds using Lie algebras of vector fields. I will begin by presenting, as a motivation, the case of the Laplace operator in cylindrical and then in spherical coordinates. This will justify the introduction of Lie manifolds, which is a class of manifolds that have a compactification to a manifold with corners and whose geometry is modeled by projective Lie algebras of vector fields. Lie manifolds can be used to study (in particular) the analysis on (asymptotically) euclidean, conical, cylindrical, and hyperbolic spaces. In the same vein, the desingularization of a polyhedral domain leads to a Lie manifold with boundary. As one of the main results, will recall the construction of a class of adapted pseudodifferential operators on a Lie manifold following Ammann, Lauter, and Nistor, a construction that extends earlier constructions of Connes and Melrose. Then I will discuss the Fredholm property of these pseudodifferential operators on Lie manifolds and the resulting index problem. Throughout the paper, we will distinguish three types of examples based on:
1. compact manifolds (the classical case which we want to generalize);
2. manifolds with cylindrical ends (the “almost classical case”, which again we want to generalize)
3. the rest of Lie manifold.
The second case plays a crucial role since it is easier to understand, and will thus be treated in more detail. I will conclude by formulating the problem of combining these definitions with the approach of Ruzhansky, a problem that arises in the analysis of G-invariant operators. The results presented here are joint works with: Ammann, Carvalho, Lauter, and Yu Qiao.

Prof. Vitaly Volpert
Institut Camille Jordan, University of Lyon,
France

Topic: Mathematical Modeling of respiratory viral infections

Time: 3 PM (CET) – 1 June 2022
Where: Auditorium 1, S8, Ghent University

Abstract: Mathematical modelling of respiratory viral infections (including COVID-19) will be discusses, with a combination of modelling and analysis.

Prof. Andreas Seeger
US

Topic: Lp improving bounds for maximal and variational operators associated with spherical means

Time: 2 PM (CET) – 18 April 2022
Where: ZOOM online-Ghent University

Abstract: This talk will be about spherical means with various recent results on Lp improving estimates for associated variational operators and maximal operators with restricted dilation sets. I will discuss how different notions of dimension of a dilation set enter in this problem.

Prof.  Eugene Shargorodsky
King’s College

London, UK

Topic: Estimates for pseudodifferential operators in terms of the norms of Fourier multipliers

Time: 20 April 2022
Where: Ghent University

Abstract: I will discuss methods of reducing the problem of boundedness of pseudodifferential operators (operators “with variable coefficients”) to the theory of Fourier multipliers (operators “with constant coefficients”). They allow one to obtain boundedness results for pseudodifferential operators under minimal restrictions on their symbols with respect to the Fourier variable. This is achieved by placing appropriate restrictions on the behaviour of the symbols with respect to the variable x. Optimality of the latter will also be discussed.

Prof.  Roland Duduchava
University of Georgia

Tbilisi, Georgia

Topic: Convolution Equations on Lie Groups and Their Applications

Time: 20 April 2022
Where: Ghent University

Abstract: Lie group G is a manifold, equipped with the group structure-an associative multiplication of elements and all elements have inverses. On such Lie groups exists the unique Haar measure and is defined integration, as well as Fourier transformation F_G, which is an isomorphism of the Lebesgue-Hilbert spaces L2(G) to L2(G^), where G^ is the associated group of unitary representations.
Most interesting in applications are convolution equations on different Lie groups. The most known examples are Wiener-Hopf convolution equations, Mellin convolution equations and the discrete (Toeplitz) convolution equations, which play outstanding role in many applications.
The purpose of the presentation is to present a new type of convolution equations on the Lie group (-1,1), where the group operation for the pair 1<x,y<1 is defined as follows (x+y)(1+xy)-1. As a result emerge new type of convolution equations, which are solved precisely.
To the new class of convolution equations belong celebrated airfoil (Prandtl) equation,
singular Tricomi equation and Lavrentjev-Bitsadze equation, which encounter in many applications. Moreover, some partial differential equations fall in this class, including Laplace-Beltrami equation on the unit sphere.

## Previous Colloquia

Prof.  Philippe Souplet
Université Sorbonne

Paris Nord, France

Topic: Singularities of solutions of the diffusive Hamilton-Jacobi equation

Time: postponed
Where: Ghent University

We consider the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$, with homogeneous Dirichlet conditions, which plays an important role in stochastic control theory, as well as in certain models of surface growth (KPZ). Despite its simplicity, it displays, in the surquadratic case $p>2$, a variety of interesting behaviors and we will discuss two classes of phenomena:
– Gradient blowup (GBU): boundary localisation of singularities, single point point blowup, blowup rates, space and space-time profiles, Liouville type theorems and applications;
– Continuation after GBU as a global viscosity solution: GBU with or without loss of boundary conditions (LBC), recovery of of boundary conditions (RBC) with or without regularisation, multiple time GBU and LBC
In particular, in one space dimension, we will present the, recently obtained, complete classification of GBU and RBC rates and profiles.
Based on a series of joint works with A. Attouchi, R. Filippucci, Y. Li, N. Mizoguchi, A. Porretta, P. Pucci, Q. Zhang.