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. Andreas Seeger**University of Wisconsin-Madison,

**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.

**Please email to for Zoom link: Duvan.CardonaSanchez@UGent.be **

**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 , 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 , 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.