# Abstracts of Ghent Methusalem Junior Analysis

## Fall 2021

#### Yevgeny Liokumovich(University of Toronto, Canada)

Finding solutions to PDEs with pictures

One of the important problems in geometry is to find curves (or surfaces) that satisfy a certain 2nd order elliptic PDE.  I will discuss how one can prove existence of solutions to these equations and learn about their properties with very few computations or inequalities.

#### Xueying Yu(MIT, USA)

High-low method of NLS on the hyperbolic space

We prove global existence and scattering for the defocusing cubic nonlinear Schrödinger equation on two dimensional hyperbolic space with subcritical initial data, using a high-low frequency decomposition method. This is a joint work with Gigliola Staffilani.

#### Agnieszka Hejna(University of Wrocław, Poland)

Harmonic analysis in the rational Dunkl setting

Dunkl theory is a generalization of Fourier analysis and special function theory related to root systems and reflections groups. The Dunkl operators, which were introduced by C. F. Dunkl in 1989, are deformations of directional derivatives by difference operators related to the reflection group. The first goal of the talk will be to provide a brief introduction to the Dunkl theory from the point of view of Fourier and harmonic analysis. We will focus on main difficulties and differences between the Dunkl analysis and classical harmonic analysis on Euclidean spaces. Our next goal will be to present some new results. We will focus on two of them: improved estimates of the heat kernel of the Dunkl heat semigroup generated by Dunkl-Laplace operator, and theorem regarding the support of Dunkl translations of compactly supported function (not necessarily radial). This kind of results turn out to be useful tools in studying harmonic analysis in Dunkl setting. We will discuss how our tools can be used to for studying singular integrals of convolution type, Littlewood-Paley square functions, or Fourier-Dunkl multipliers in the Dunkl setting.

This talk is based on the joint articles with J-Ph. Anker and J. Dziubanski.

#### Joel Restrepo(Nazarbayev University, Kazakhstan)

The fractional Laplacian of a function with respect to another function

The theories of fractional Laplacians and of fractional calculus with respect to functions are combined to produce, for the first time, the concept of a fractional Laplacian with respect to a bijective function. The theory is developed both in the 1-dimensional setting and in the general $n$-dimensional setting. Fourier transforms with respect to functions are also defined, and the relationships between Fourier transforms, fractional Laplacians, and Marchaud type derivatives are explored. Function spaces for these operators are carefully defined, including weighted $L^p$ spaces and a new type of Schwartz space. The theory developed is then applied to construct solutions to some partial differential equations involving both fractional time-derivatives and fractional Laplacians with respect to functions, with illustrative examples.

#### Jan Rozendaal(Institute of Mathematics, Poland)

Hardy spaces for Fourier Integral Operators

It is well known that the wave operators $\cos(t\sqrt{-\Delta})$ and $\sin(t\sqrt{-\Delta})$ are not bounded $L^p(\mathbb{R}^n)$, for $n\geq 2$ and $1\leq p \leq \infty$, unless $p=2$ or $t=0$. In fact, for $1 these operators are bounded from $W^{2s(p),p}.$ It is well known that the wave operators $\cos(t\sqrt{-\Delta})$ and $\sin(t\sqrt{-\Delta})$ are not bounded $L^p(\mathbb{R}^n)$, for $n\geq 2$ and $1\leq p \leq \infty$, unless $p=2$ or $t=0.$ In fact, for $1 these operators are bounded from $W^{2s(p),p} (\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ for $s(p):=\frac{n-1}{2}|1/p-1/2|$, and this exponent cannot be improved. This phenomenon is symptomatic of the behavior of Fourier integral operators on $L^p(\mathbb{R}^n)$.
In this talk, I will introduce a class of Hardy spaces $\mathcal{H}^p_{FIO}(\mathbb{R}^n)$ for $p \in [1,\infty]$, on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on $L^p(\mathbb{R}^n)$.
However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on $L^p(\mathbb{R}^n)$. In particular, we shall indicate how one can use this invariance to obtain the optimal fixed-time $L^p$ regularity for wave equations with rough coefficients. We shall also mention the connection of these spaces to the phenomenon of local smoothing.
This talk is based on joint work with Andrew Hassell and Pierre Portal (Australian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat-Sen University).

#### Alessandro Palmieri(Tohoku University, Japan)

On the critical exponent for the semilinear damped wave equation on some unimodular Lie groups

In this talk, I will investigate the critical exponent (the exponent that separates the blow-up region from the global existence region) for the Cauchy problem associated with a semilinear damped wave equation with power nonlinearity. I consider this model on the Heisenberg group and on a general compact (and connected) Lie group. In both cases, the representation theory on the underlying Lie group plays a fundamental role in the establishment of the corresponding energy estimates. Based on a joint work with Prof. Vladimir Georgiev (University of Pisa)

#### Alexander Cardona(University of Los Andes, Colombia)

λ-Structures and Pseudo-differential Operators

λ-Structures (e.g. λ-rings and λ-semi-rings) go back to the work of Grothendieck on Chern classes in algebraic topology, it is a suitable axiomatization of the algebraic properties of exterior power operations on vector bundles; λ-rings were also used by Atiyah and coworkers in the study of representations of groups and K-Theory. During this talk we will present some results on the uses of λ-structures in algebras of pseudo-differential operators, and their relation with index theory.

#### Runlian Xia(University of Glasgow, UK)

Hilbert transforms for groups acting on R-trees

The Hilbert transform H is a basic example of a Fourier multiplier. Riesz proved that H is a bounded operator on $L_p(\mathbb{T})$ for all $1.
We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative L_p spaces.
The pioneering work in this direction is due to Mei and Ricard who proved L_p-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on R-trees. This class of groups includes free groups, amalgamated free products, HNN extensions, totally ordered groups and many others.

#### Divyang Bhimani(Indian Institute of Science Education and Research, India)

Strong ill-posedness (norm inflation) for nonlinear Schrödinger equations

We consider nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation (strong ill-posedness) results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. This talk is based on joint work with Remi Carles and Saikatul Haque.

Yunfeng Zhang
(University of Connecticut, USA)

Nonlinear Schrödinger equation on compact symmetric spaces

Nonlinear Schrödinger equations have been explored intensively on Euclidean spaces and many other Riemanian manifolds. We first review the local well-posedness results on compact manifolds and their proofs. Then we discuss the case of compact globally symmetric spaces. These manifolds bear a commutative ring of differential operators and an explicit harmonic analysis. We provide linear and multi-linear Strichartz estimates on such spaces, and as a consequence local well-posedness of the nonlinear equation for initial data of either subcritical or critical regularity.

On the well-posedness for a  class of pseudodifferential parabolic equations

We present some recent results on the well-posedness of the Cauchy problem for a class of pseudodifferential parabolic equations within the scale of suitable Sobolev spaces. The class of  equations depends on the  choice of symbolic calculus  from which we will extract a certain notion of ellipticity. We give some basic examples of special cases arising in this approach. For instance fractional diffusion and drift diffusion equations on the torus for appropriate exponents can be tackled within the setting introduced by Ruzhansky and Turunen for global symbolic calculus on the torus. Other examples on R^n within  the Weyl-Hörmander calculus are also given.

Federico Santagati
(Politecnico di Torino, Italy)

Analysis on trees with nondoubling flow measures

We will illustrate a Calderón-Zygmund theory on a tree endowed with a locally doubling flow measure. An atomic Hardy space is introduced in this setting. In the particular case of the homogeneous tree, we show that the characterizations of the atomic Hardy space in terms of the heat maximal operator and the Riesz transform fail. We will also give positive and negative boundedness results for the Riesz transform in this setting. This is partially based on joint works with Matteo Levi, Anita Tabacco and Maria Vallarino.

Hardy Chan
(ETH Zürich, Switzerland)

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb{R}^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial \Omega$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.

#### Elisa Affili(University of Deusto, Spain)

A mathematical model for civil wars: a new Lotka-Volterra competitive system

Imagine two populations sharing the same environmental resources in a situation of open hostility. The interactions among these populations are governed not by random encounters, but via the strategic decisions of one population that can attack the other according to different levels of aggressiveness. This leads to a non-variational model for the two populations at war, taking into account structural ecological parameters. The analysis of the dynamical properties of the system reveals several equilibria and bifurcation phenomena. Moreover, we present the strategies that may lead to the victory of the aggressive population, i.e. the choices of the aggressiveness parameter, in dependence of the structural constants of the system and possibly varying in time in order to optimize the efficacy of the attacks, which take to the extinction in finite time of the defensive population. The model that we present is flexible enough to include also technological competition models of aggressive companies releasing computer viruses to set rival companies out of the market. This is joint work with S. Dipierro, L. Rossi and E. Valdinoci.

## Spring 2021

#### Jordy van Velthoven(Ghent University, Belgium)

Density theorems for lattice orbits of discrete series representations

The talk considers the relation between the spanning properties of a lattice orbit of a square-integrable projective representation and the associated lattice co-volume. Under a compatibility condition between the cocycle and the lattice, the density theorem provides a trichotomy that precisely describes the spanning properties of a given lattice orbit in terms of the lattice co-volume. For nilpotent Lie groups, the interplay between the density theorem and Perelomov’s completeness problem for coherent state subsystems will be considered.

#### Ujue Etayo(University of Cantabria, Spain)

On Determinantal point processes

Among the different random point processes, the so-called Determinantal present several characteristics that make them strong candidates for solving equidistribution problems: they exhibit local repulsion, they tend to be uniformly distributed, and they are easily computable. In this talk I will present the main characteristics of these processes, how to define them in different types of spaces and two specific applications in spheres of arbitrary dimension.

#### David Beltran(University of Wisconsin-Madison, USA)

L^p bounds for the helical maximal function

A natural 3-dimensional analogue of Bourgain’s circular maximal function theorem in the plane is the study of the sharp L^p bounds in R^3 for the maximal function associated with averages over dilates of the helix (or, more generally, of any curve with non-vanishing curvature and torsion). In this talk, we present a sharp result, which establishes that L^p bounds hold if and only if p>3. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

#### Marcello Malagutti(University of Bologna, Italy)

A Crash Introduction to Non-Commutative Harmonic Oscillators

The purpose of this talk is to introduce the study of Non-Commutative Harmonic Oscillators (NCHOs) i.e. of a class of pseudodifferential systems given by the Weyl-quantization of matrix-valued symbol operators with homogeneous polynomial of degree 2 entries in the phase variables. More in detail, we focus on the spectral properties of this class investigating a particularly important subclass introduced by A. Parmeggiani and M. Wakayama. Indeed, among the main results we prove a theorem of diagonalization of NCHOs under a spacing condition of eigenvalues of the principal symbol. Then, we show the Weyl asymptotic for our subclass by the use of the previous diagonalization theorem. Finally, we state results where we show that the spectral zeta function associated to a NCHO is meromorphic and, in particular, we will see the Ichinose-Wkayama Theorem.

#### Cesar Ceballos (Institute of Geometry, TU Graz, Austria)

Hopf Algebras and Diagonal Harmonics

The theory of Hopf algebras is a fundamental area in mathematics which was originated in the 1940’s and 1950’s motivated by work of Hopf on algebraic topology and of Diedonné on algebraic groups. Diagonal harmonics, on the other hand, is a more recent and apparently unrelated area initiated by Garsia and Haiman in the early 1990’s, which has remarkable connections to Macdonald polynomials, algebraic geometry, representation theory, knot theory, and mathematical physics.

In this talk, I will give an insight to these fascinating areas, mainly through a series of examples and without many technicalities. The main purpose is to present some unexpected connections arising in the study of a Hopf algebra structure on pipe dreams, certain discrete objects that provide a combinatorial understanding of Schubert polynomials.

The talk is addressed to a general mathematical audience and no previous knowledge of Hopf algebras or diagonal harmonics will be assumed.

#### Andreas Debrouwere(Ghent University, Belgium)

Gabor frame characterizations of generalized modulation spaces

In this talk, we discuss modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions. Most importantly, we show how these spaces can be characterized in a discrete fashion via Gabor frames. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the classical modulation spaces $M^{p,q}_w$ (or, more generally, in coorbit theory) fail in our setting. Inspired by the theory of projective representations, we present a new approach based on the twisted convolution. This talk is based on work in collaboration with B. Prangoski.

#### Lorenzo Ruffoni(Florida State University, Florida)

Projective structures, representations, and ODEs on surfaces

In one of its easiest formulations, Hilbert’s XXI problem deals with the relationship between linear ODEs on a surface and representations of its fundamental group. When a complex structure on the surface is fixed, a classical theory is available. However, not much is understood in the complementary case, i.e. when the type of the ODE is fixed, while the complex structure is allowed to vary. Projective structures have been known since Poincare’s times to be a geometric bridge between the analytic and the algebraic side of this picture. In this talk we will present how their geometric deformation theory can be used to explore the space of ODEs associated with a fixed representation, including some recent results.

Standing waves on a flower graph

In this talk we consider positive single-lobe solutions to the the cubic nonlinear Schrödinger equation on a particular type of metric graphs. We use the period function for second-order differential equations towards understanding the symmetries and bifurcations of standing waves. The positive single-lobe symmetric state (which is the ground state of energy for small fixed mass) undergoes exactly one bifurcation for larger mass, at which point $(N−1)$ branches of other positive single-lobe states appear: each branch has $K$ larger components and $(N−K)$ smaller components, where $1\leq K\leq N−1$. We show that only the branch with$K=1$ represents a local minimizer of energy for large fixed mass, however, the ground state of energy is not attained for large fixed massif $N≥2$.This is a joint work with Robert Marangell (University of Sydney), DmitryPelinovsky (McMaster University) and Ke Xiao (McGill University).

#### Prashanta Garain(Ben-Gurion University of the Negev, Israel)

On a degenerate singular elliptic problem

In this talk, we will discuss some qualitative properties of a purely singular quasilinear elliptic problem. To be more precise, we focus on the existence, uniqueness, and regularity results for a class of weighted $p$-Laplace equations with purely singular nonlinearity. We work on a class of Muckenhoupt weights that captures the degenerate behavior of the equations.

#### Pritam Ganguly(Indian Institute of Science Bangalore, India)

An uncertainty principle for spectral projections on rank one Riemannian symmetric spaces

An Uncertainty principle due to Ingham provides the best possible decay of the Fourier transform of a function on \mathbb{R} which vanishes on a nonempty open set.  In this talk, we investigate similar results in more general context. To be precise, given a function which vanishes on an open set, we investigate the best possible decay of its spectral projections associated to Laplacian on \mathbb{R}^n.  Also we prove this Ingham type result for the spectral projections associated to the Laplace-Beltrami operators on rank one compact and noncompact Riemannian symmetric spaces.

#### Haonan Zhang(Institute of Science and Technology, Austria)

Around noncommutative Ricci curvature lower bounds

The lower bound of Ricci curvature has many applications in analysis. In the classical setting the lower bound of Ricci curvature can be characterized via the $\Gamma$-calculus using Bakry-Émery theory, or via the geodesic semi-convexity of entropy with respect to 2-Wasserstein metric following Lott-Sturm-Villani. In this introductory talk I will present some attempts in recent years to generalize Ricci curvature lower bounds to noncommutative setting. These different notions of noncommutative Ricci curvature lower bounds have many useful applications to noncommutative analysis and quantum information theory. In particular, one can deduce a number of noncommutative functional inequalities from a strictly positive Ricci curvature lower bound. Time permitting, I will also speak about some recent work with Melchior Wirth (IST Austria).

#### Liliana Esquivel(Gran Sasso Science Institute, Italy and Universidad de Pamplona, Colombia)

An introduction to initial boundary value problems for some nonlinear dispersive models on the half-line

In the last years, the study of initial boundary value problems for nonlinear dispersive equations on the half-lines has given attention of many researchers. In this talk, we review some of the main results about this topic, such as local and global well posedness, and asymptotic behaviour of small solutions for these equations.

#### Louise Gassot(Université Paris-Sud, France)

On the Schrödinger equation on the Heisenberg group

In this talk, we introduce the cubic Schrödinger equation on the Heisenberg group, which is a model for totally non-dispersive evolution equations. As the lack of dispersion causes difficulties to solve the Cauchy problem even locally in time, we present two alternative approaches. First, we construct a family of ground state traveling waves parametrized by their speed in (-1,1). When the speed is close to 1, we establish the uniqueness up to symmetries of the ground state and study its stability properties. Finally, we consider the randomized Cauchy problem for the associated Grushin-Schrödinger equation.

#### Vanessa Hurtado (Sorbonne Université, France)

An introduction to heat kernel techniques on manifolds and some applications

In this talk, I am going to give an introduction to the study of Heat Kernels for compact surfaces without boundary. We are going to study some examples applied to the quasi-geostrophic Ocean models and also, I am going to do, from this context, an invitation to the apparent horizon of Black Holes.

#### Effie Papageorgiou(University of Crete, Greece)

An introduction to multipliers on non-compact symmetric spaces and locally symmetric spaces

The purpose of this talk is to give a brief introduction to Fourier multipliers on symmetric spaces of non-compact type, which are non-positively curved manifolds, including hyperbolic space. Their rich structure induces some remarkable phenomena related to multiplier theory. We also consider quotients of a symmetric space by good enough subgroups of its isometry group, that is, locally symmetric spaces. This talk intends to compare some of the results on settings above with their euclidean analogues. Finally, we examine a family of multipliers that falls outside the scope of standard theory.

#### Juan Pablo Velasquez Rodriguez(Ghent University, Belgium)

An introduction to Vilenkin groups

The purpose of this talk is to give a quick overview of the history and motivation of a particular class of topological groups called Vilenkin groups. Through the exposition of important particular examples, this talk intends to exhibit some of the similarities and differences between the analysis on these groups and the analysis on connected topological groups. Specially, we would like to describe the unitary dual of a Vilenkin group and show how the usual Fourier analysis looks like in this setting.

#### Hong-Wei Zhang(Orleans University, France, and UGent)

An introduction to spherical Fourier analysis on noncompact symmetric spaces

This introductory talk intends to present some preliminaires about the harmonic analysis on Riemannian symmetric spaces of non-compact type for the junior researchers who are interested in such nice negatively curved Riemannian manifolds. By comparing with real hyperbolic spaces (which are rank one symmetric spaces), this talk will focus on the higher rank analysis, I will also share some recent progress on this topic.