# Abstracts of Ghent Methusalem Junior Analysis

#### 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.