The Microlocal Day is an event featuring a brief and intensive series of lectures on various aspects of microlocal analysis and related topics. The program consists of both research presentations and survey lectures intended for researchers and PhD students interested in the field. Everyone is welcome to attend.

The next “Ghent Methusalem Microlocal Day #14” will take place on Friday, 22 August 2025

Venue: Leslokaal 3.1, Campus Sterre, S8, Ghent University, Belgium

Zoom link to join: https://us02web.zoom.us/j/83095584409?pwd=c3huaUhZcWFrZ2NNMU05Ujg4aUlWQT09

Meeting ID: 830 9558 4409

Passcode: 135803

Schedule: all times CEST

Opening 14:00-14:05

14:05-15:00.  S. Thangavelu (Indian Institute of Science, Bangalore, India)

Title: What are the operator analogues Hermite functions?

Abstract: With the standard notation, the normalised Hermite functions  defined by the equation $$ \Phi_\alpha(x) e^{-\frac{1}{2}|x|^2} = c_\alpha  \partial^\alpha e^{-|\x|^2},\,\, \alpha \in \mathbb N^n$$ form an orthornormal basis for $ L^2(\R^n) .$ A search for operator analogues of Hermite functions have led us to the operators defined by  $$ S_\mu e^{-\frac{1}{2}H} = c_\mu\, \mathcal{D}^\mu e^{-H},\,\, \mu \in \mathbb N^{2n}$$ where $ \mathcal{D}_j $ are certain non-commutative derivations on  the space of bounded linear operators action on $ L^2(\R^n)$  and $ H = -\Delta+|x|^2 $ is the Hermite operator. We show that the family $ S_mu, \mu \in \mathbb N^{2n} $ forms an othonormal basis for $ \mathcal{S}_2,$ the space of Hilbert-Schmidt operators on $ L^2(\R^n).$ Moreover, they share many properties of the Hermite functions.

This talk is based on the ongoing work with Rahul Garg.

15:05 – 16:00.  Julio Delgado (Universidad del Valle, Cali-Colombia)

Title: A brief Introduction to some basic tools of Analysis on the phase-space and some Applications to PDES 

Abstract: We present a short introduction to some basic elements of the theory of metrics on the phase-space and some examples of applications to the analysis of PDEs, in particular spectral properties and well-posedness

16:15-16:45.  Ashish Pathak (Banaras Hindu University, Vanarasi, India)

Title: Characterizing wavefront sets of solutions to the time-dependent Schrödinger equation using the Stockwell transform

Abstract: In this talk, we will explore a new perspective on microlocal analysis of the Schrödinger equation with sub-quadratic perturbations. We begin by revisiting the notions of the usual and Sobolev wavefront sets, clarifying their roles in detecting singularities of distributions. We then introduce a novel framework for representing the Schrödinger equation using the Stockwell transform, which blends localization in both time and frequency. This approach offers a convenient way to track the propagation of singularities and provides new insights into the microlocal structure of solutions. Finally, we demonstrate how this method can be applied to determine the wavefront set of solutions to the perturbed Schrödinger equation, highlighting both theoretical implications and potential avenues for further research.

16:45-17:15.  Alibek Yeskermessuly (Altynsarin University, Arkalyk, Kazakhstan)

Title: Well-Posedness of the Singular Klein-Gordon Equation on a Bounded Domain

Abstract: This talk considers the well-posedness of the singular Klein-Gordon equation on bounded domains with singular coefficients, initial data and fractal boundaries. We use weak solutions for regular cases and very weak solutions via regularization for singular cases, using Galerkin approximations, energy estimates, and trace theory. 

17:15 – 17:45  Imtiaz Waheed (National University of Sciences and Technology, Islamabad, Pakistan)

Title: Generalized Integral Transform Methods and Their Applications 

Abstract:  In this talk, we begin by introducing the framework of generalized fractional calculus, emphasizingits theoretical foundations and its relevance in modern mathematical analysis. We then examinegeneralized Fourier transform methods and Laplace transform methods, discussing their keyproperties and demonstrating how they extend classical techniques. Some examples are presentedto showcase the practical effectiveness of these transforms in solving mathematical and physicalproblems.In the final part of the talk, we address specific problems in fractional calculus, including thedevelopment of operational calculus for the Hadamard fractional derivative and its application tofractional partial differential equations.

This Microlocal Day continues the tradition of Microlocal Days that we have been organising at Imperial College London, then continued at Ghent University. So, this Microlocal Day can be viewed as Microlocal Day #14.

Organisers: Vishvesh Kumar, Michael Ruzhansky (Ghent Analysis and PDE Center). 

Previous Microlocal Days: #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13