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.

**Methusalem Colloquium Lectures:**

**Prof. Sonja Hohloch**** **University of Antwerp, Belgium

**Topic:*** *Integrable Hamiltonian systems with S^1-symmetry**Time: ***15:00 – 16:00 CET,* 13* December 2022***Where:** *Leslokaal 3.1 (Krijgslaan 281, Building S8, Ghent University)*

We consider examples of completely integrable Hamiltonian systems with -symmetry, study their geometry and dynamics, and then explain the advances towards a global symplectic classification of such systems on compact, symplectic, 4-dimensional manifolds.

**Prof. E. N. Belitser **** **VU Amsterdam, The Netherlands

**Topic:*** *Statistical inference, general framework for projection structures**Time: ***15:00 – 16:00 CET,* 15* December 2022***Where:** *Leslokaal 2.1 (Krijgslaan 281, Building S8, Ghent University)*

In the first part of the talk, we provide a short (and hopefully gentle) overview of general statistical inference problems. We shortly discuss the problems of estimation, testing, uncertainty quantification (which used to be called construction of confidence sets) and structure recovery, also intending to make some connections with the statistical (machine) learning theory. We will touch upon the Bayesian methodology (and perhaps its empirical Bayes version), and explain a frequentist perspective at the quality assessment of Bayesian procedures.

In the second part, we present some recent original results concerning the so called general framework for projection structures. We propose a class of procedures based on data dependent measures (DDM) and make connections with empirical Bayes and penalization methods. The main inference problem is the uncertainty quantification (UQ), but along the way we solve the estimation, DDM-contraction problems, and a weak version of the structure recovery problem. If time permits we discuss the so called deceptiveness phenomenon in the UQ problem. The proposed general framework unifies a very broad class of high-dimensional models and structures, interesting and important on their own right. We apply the developed theory and demonstrate how the general results deliver a whole avenue of local and global minimax results (some new ones, some known results from the literature are improved) for specific models and structures as consequences.

## Abstracts of talks

**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.

** **

**Prof. Julio Delgado**** **Universidad del Valle,

**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 Ni****stor **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**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.

**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.

**Methusalem Workshop on Global Analysis**

**Prof. Todor D. Todorov**California Polytechnic State University

**California, USA**

**Topic:*** Non-Standard Version of Egorov’s Algebra of Generalized Functions***Time: ***11:00 – 12:00 CET, 6 October 2022***Where:** *Auditorium 1 Valére Billiet (Krijgslaan 281, Building S8, Ghent University)*

We consider a non-standard version of Egorov’s algebra of generalized functions, which improves the properties of the generalized scalars and the embedding of Schwartz distributions in the original standard version. The embedding of distributions is similar to, but different from author’s works in the past and independently done by Hans Vernaeve. The emphases is on the simplicity and potential for axiomatization. We also define a regular subalgebra of generalized functions in the presence of Vernaeve’s counterexample.

**Prof. Joachim Toft**Linnaeus University

**Sweden**

**Topic:*** Fractional Fourier transform, harmonic oscillator propagators and Strichartz estimates*

* ***Time: ***14:00 – 15:00 CET, 6 October 2022***Where:** Auditorium 1 Valére Billiet (Krijgslaan 281, Building S8, *Ghent University*)

We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces, and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces.

Especially we extend some results in [1,2,3,4]. We also show that general forms of fractional harmonic oscillator

propagators are continuous on suitable Pilipović spaces. Especially we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are continuous on any Pilipović space and corresponding distribution space, which are *not* Gelfand-Shilov spaces.

*The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.*

References

[1] D. G. Bhimani The nonlinear Schrödinger equations with harmonic dinger equations with harmonic potential modulation spaces, Discrete Contin. Dyn. Syst. 39 (2019), 5923–5944.

[2] D. Bhimani, R. Balhara, S. Thangavelu Hermite multipliers on modulation spaces, in: Analysis and partial differential equations: perspectives from developing countries, Springer Proc. Math. Stat., 275, Springer, Cham, 2019, pp. 42–64.

[3] E. Cordero, K. H. Gröchenig, F. Nicola, L. Rodino Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys. 55 (2014), 081506.

[4] E. Cordero, F. Nicola Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Func. Anal. 254 (2008), 506–534.

[5] J. Toft Images of function and distribution spaces under the Bargmann transform, J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.

[6] J. Toft, D. Bhimani, R. Manna Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipović and modulation spaces, (preprint), arXiv:2111.09575.

**Prof. Alexander Cardona**Universidad de los Andes

**Colombia**

**Topic:*** Kirillov’s construction, dequantization and pseudo-differential calculus***Time: ***15:00 – 16:00 CET, 6 October 2022* **Where:** *Auditorium 1 Valére Billiet (Krijgslaan 281, Building S8, Ghent University)*

In this talk, we review the aspects of the Kirillov theory, and from the point of view of dequantization we analyze its interplay with the pseudo-differential calculus.

**Colloquia**

**Prof. Jean VAN SCHAFTINGEN**Catholic University of Louvain

**Belgium**

**Topic:*** Endpoint Sobolev inequalities for vector fields and cancelling operators***Time: ***15:00 – 16:00 CET, 26 October 2022***Where:** *Auditorium Leslokaal 3.1 (Krijgslaan 281, Building S8, Ghent University)*

Sobolev embeddings control the integrability of some power of a function by an integral of the derivative of the function at a lower power. The limiting case where the latter power is taken to be 1 due to Gagliardo and Nirenberg, is inaccessible to classical methods of harmonic analysis and turns out to be a functional version of the isoperimetric inequality. If one considers vector fields instead of functions, one can hope that some redundancy in the derivative would allow to obtain estimates with an integrand that does not involve all the components of the derivative. Such sparse estimates have been obtained for the deformation operator (M.J. Strauss) and for the Hodge complex (Bourgain and Brezis). I have characterized the homogeneous autonomous linear differential operators for which they hold as elliptic and canceling differential operator. I will also present various further questions that have been solved or remain as open problems.

**Methusalem Workshop on Functional Analysis**

**P**hD **Medea Tsaava**** **The University of Georgia, Georgia

**Topic:*** Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain***Time: ***11:00 – 11:30 CET, * 1*6 November 2022***Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*

We have investigated the model mixed boundary value problem for the Helmholtz equation in a planar angular domain of magnitude . The BVP is considered in a non-classical setting when a solution is sought in the Bessel potential spaces , , . The problems are investigated using the potential method by reducing them to an equivalent boundary integral equation (BIE) in the Sobolev-Slobodečkii space on a semi-infinite axes , which is of Mellin convolution type. By applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko & R. Duduchava, explicit conditions of the unique solvability of this BIE in the Sobolev-Slobodečkii and Bessel potential spaces for arbitrary are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the above mentioned non-classical setting.

**PhD student:** **Margarita Tutberidze**** **The University of Georgia, Georgia

**Topic:*** Mixed and transmission type boundary value problems for the Helmholtz equation on a surface with Lipschitz boundary***Time: ***11:30 – 12:00 CET, * 1*6 November 2022***Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*

We have studied the solution of a transmission type boundary value problem for the anisotropic Helmholtz equation in the non-classical setting on a surface with Lipschitz boundary. At first the problem is reduced by the localization method (cf. [1]) to several model problems at flat angles consisting of two and three beams emerging from point 0. Mixed type (Dirichlet-Neumann) boundary conditions on the boundary beams and transmission condition on the interface beam are given. We apply the potential method and reduce the boundary problem to the system of boundary integral equations, which represent a system of Mellin-type convolution equations in the Bessel potential space on the half-axis. By using the resent results on such equations, obtained in (cf. [2]), we write symbol of equations and formulate criteria for solvability (Fredholmness) of such systems of equations in the Bessel potential spaces.

For the model angle the mixed type BVP were investigated in [3]. We obtain necessary and sufficient conditions for the solvability of the model BVP for two angles with the interface along a beam. The final result is the solvability criteria for the mixed type BVP for the anisotropic Helmholtz equation in the non-classical setting on a surface with interface and with Lipschitz boundary.

References

[1] T. Buchukuri, R. Duduchava, D. Kapanadze and M. Tsaava, Localization of a Helmholtz boundary value problem in a domain with piecewise-smooth boundary, Proc. A. Razmadze Math. Inst, 162, 37-44, (2013).

[2] V. Didenko and R. Duduchava, Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels. Journal of Analysis and Applications, 443, 2016, 707-731.

[3] R. Duduchava, M. Tsaava, Mixed boundary value problems for the Laplace-Beltrami equation. Complex Variables and Elliptic Equations, 63, 10, 2018, 1468-1496

**Prof. Roland Duduchava**** **The University of Georgia, Georgia

**Topic:*** BVPs on Lipschitz domains in the Bessel potential spaces***Time: ***14:00 – 14:30 CET, * 1*6 November 2022***Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*

Studying Boundary Value Problem on Lipschitz domain (or a Lipschith surface ) we encounter problem with spaces. If the BVP is considered in the Bessel potential space Lac-Milgram Lemma guarantees the existence of a weak solution, but we can not conclude even continuity of a classical solution. To establish continuity of a solution, we will study BVPs in the Bessel potential space , , .

We apply two different Bessel potential spaces, defined by A) Fourier transforms (the classical case); B) By Mellin transform .

For the investigation of BVPs in the Bessel potential spaces of both types we apply localization, obtaining model problems in angles. The obtained model problems is reduced further to convolutions on the half axes and study this problem by lifting them to the space setting .

**Prof. George Tephnadze**** **The University of Georgia, Georgia

**Topic:*** Almost everywhere convergence of partial sums and certain summability methods of trigonometric and Vilenkin systems***Time: ***14:30 – 15:00 CET, * 1*6 November 2022***Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*

The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, combined with martingale theory.

This talk is devoted to investigating tools which are used to study almost everywhere convergence of the partial sums of trigonometric and Vilenkin systems. In particular, these methods combined with martingale theory helps to give a simpler proof of an analogy of the famous Carleson-Hunt theorem for Fourier series with respect to the Vilenkin system. We also define an analogy of Lebesgue points for integrable functions and we will describe which certain summability methods are convergent in these points.

**Methusalem Mini-Course on Spectral Asymptotics**

**Prof**. **Dmitri Vassiliev**** **University College London, UK

**Topic:*** Spectral theory of differential operators: what’s it all about and what is its use***First Session: ***14:00 – 15:30 CET,* *22 November 2022***Where:** *Leslokaal 3.1 (Krijgslaan 281, Building S8, Ghent University)*

**Second Session: ***10:30 – 12:00 CET, * *24 November 2022***Where:** *Leslokaal 2.2 (Krijgslaan 281, Building S8, Ghent University)*

I will give an overview of the spectral theory of partial differential operators, charting its development from the non-rigorous works of physicists to modern rigorous mathematical results.

**Methusalem Mini-Course on Principles of Machine Learning**

**Dr. Sven Nõmm**** **Tallinn University of Technology, Estonia

**Topic:*** Principles of Machine Learning***First Session: ***10:30 – 12:00 CET,* *6 December 2022***Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*

**Second Session: ***14:00 – 15:30 CET, * * 8 December 2022***Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*

The mini course is devoted to the introduction and main practical principles of statistical machine learning. The first session will introduce two main types of machine learning problem, supervised- and unsupervised-learning. Also, we will discuss in detail typical workflow of machine learning. The second session will present more advanced approaches to improving model quality and introduce simple artificial neural networks.

**Methusalem Colloquium Lecture: **Machine learning supported diagnostics of the Parkinson’s disease

**Time:** *14:00 – 15:00 CET, 6 December 2022*

**Where:** *Leslokaal 1.2 (Krijgslaan 281, Building S8, Ghent University)*