Abstracts of Ghent Methusalem Junior Analysis

Spring 2023

Anthony Baptista
(Queen Marry University of London and Alan Turing Institute, UK)

Zoo guide of Network embedding

Networks are intrinsically combinatorial objects (i.e., interconnected nodes, where certain pairs of nodes are connected by links), with no a priori ambient space, nor node geometric information such as `coordinates’. Network embedding (a.k.a representation learning) is the process of assigning such a latent space (a.k.a embedding space) to a network. This is typically done by mapping the nodes to a geometric space, such as a Euclidean space, while preserving some properties of the nodes, links, and/or network. Overall, network embedding methods are used for learning a low-dimensional vector representation from a high-dimensional network. The relationships between nodes in the network are represented by their distance in the low-dimensional embedding space. Then, the low-dimension vector representation can be used for visualisation, and in a wide variety of downstream analyses, from network inference or link prediction to node classification or community detection. Over the past few years, there has been a significant surge in the number of embedding methods, making it challenging to navigate this fast-evolving field. I will present an overview of network embedding methods and introduce a new taxonomy that captures the latest developments in the field. Additionally, I will present a groundbreaking embedding technique that is able to project in the same latent space heterogeneous information. Finally, I will conclude by highlighting open questions and challenges that require further investigation in the field.


Stefano Bucceri
(University of Vienna, Italy)

Viscosity solutions for nonlocal equations with space-dependent operators

In this seminar we will introduce a fractional operator with variable integration domain. We start with some motivations for the study of such an object and then present some results about the well posedness of the associated boundary value problem and principal eigenvalue problem.


Louis Yudowitz
(Queen Mary University of London, UK)

Bubble Tree Convergence of Shrinking

ISince the late 1900s, parabolic PDEs have had a massive impact in geometry and topology. In particular, Ricci flow has been used to solve a variety of problems in these fields, such as the Poincare conjecture.  A vital part of such proofs is a good understanding of finite time singularities. While we have such an understanding in dimensions 2 and 3, singularity models in higher dimensions still pose issues. This is partially due to the existence of singularity models which are singular themselves. In this talk, we will prove bubble tree convergence of certain shrinking singularity models, which involves a detailed analysis of the singular set when it consists of isolated points. As a consequence, we will prove a local diffeomorphism finiteness result, as well as an identity which gives information about how much total curvature/topology is lost due to the formation of the singular points. The latter involves new L^p estimates and an improved Kato inequality for shrinkers. This is all based on a joint work with Reto Buzano.


Mirco Piccinini
(University of Parma, Italy)

Nonlinear fractional equations in the Heisenberg group

In the sub-Riemannian setting of the Heisenberg group we state some classical regularity properties of weak solutions to the Dirichlet problem related to a wide class of integro-differential operators, whose prototype is the conformally invariant fractional subLaplacian; see Branson, Fontana and Morpungo, Ann. of Math. 2013. We show that weak solutions belong to the intrinsic fractional Hölder space, extending the classical results by De Giorgi-Nash-Moser to the nonlocal framework in the Heisenberg group. Moreover, we state some Harnack–type inequalities which are the analog in the Heisenberg setting of the results proven by Di Castro, Kuusi and Palatucci ( J. Funct. Anal. 2014) and in turn we generalize them to the non–homogeneous case. In the linear case when p = 2 the robustness of these inequalities is investigated as the differentiability exponent goes to 1.


Tobias König
(Goethe University Frankfurt, Germany)

Stability of the Sobolev inequality: best constants and minimizers

Since the ground-breaking inequality of Bianchi and Egnell (1991), which bounds the ‘Sobolev deficit’ of a function in terms of a constant c_{BE} > 0 times its squared distance to the manifold of optimizers, it has been an open problem to determine the optimal value of c_{BE} and, if it is achieved, its optimizer. 
In this talk, I will present some recent partial progress on this problem. The main result is that c_{BE} admits an optimizer for every dimension d \geq 3. The proof relies on new strict upper bounds on c_{BE}, which exclude that the optimal value c_{BE} is attained by sequences which are asymptotically equal to one or two Talenti bubbles (i.e. optimizers of the Sobolev inequality).


Gisel Mattar
(University of Göttingen, Germany)

Lagrangian distributions with complex phase

Lagrangian distributions with complex phases arise naturally when dealing with parametrices for operators that have non-real principal symbols. It is then useful for the analysis of PDEs to have a complete theory for this type of distributions. A systematic approach was proposed by Melin and Sjöstrand (1975). They use almost analytic machinery to construct a complex-valued analogue of the standard real-valued theory.  In this talk we will review both the real- and complex- valued theory, with a special emphasis on the principal symbol map for Lagrangian distributions.


Adolfo Arroyo Rabasa 
(UCLouvain, Belgium)

The theory of constant-rank operators (homological properties, Lp estimates, and open questions)

In this talk, I would like to introduce the constant-rank condition, which is a generalization of ellipticity for constant-coefficient operators. First, I will discuss their homological properties, building a bridge between a generalized Poincare Lemma and homological algebra. Then, I will recall what is known about the Lp regularity for constant-rank operators, how it compares with elliptic regularity theory, and the subtle differences arising between considering them as operators over the torus, euclidean space, or its subdomains. If time permits, I will end my talk by sharing with you some interesting open questions. 


Felipe Ponce Vanegas
(University of Southern California, USA)

Regularity of envelopes swept by rigid bodies

Turbine blades and other aircraft components are usually manufactured by 5-axis flank CNC machining, in which a cutting tool removes material from the workpiece leaving behind an envelope swept by a rotational symmetric rigid body. In the talk I will recall the classical theory of envelopes, and I will present two results about the regularity of envelopes depending on the smoothness of motion and cutting tool profile. In general, we need two derivatives to get some regularity, but we will see that even after dropping derivatives at some points, the envelope can still retain some smoothness. 


Fall 2022

Grigalius Taujanskas
(University of Cambridge, UK)

Conformal Scattering of Maxwell Potential

The conformal approach to scattering is a combination of the ideas of Penrose’s conformal compactification, the classical scattering theory of Lax and Phillips, and Friedlander’s work on radiation fields, all developed in the 1960s. Recently there has been a resurgence of interest in the development of precise scattering theories, in particular on curved spacetimes, due to their importance for asymptotics, stability of spacetimes, and potential applications to quantum gravity. In this talk I will review the general setup of these ideas and show how to construct a scattering theory for Maxwell potentials on a non-trivial class of curved spacetimes, called Corvino-Schoen-Chrusciel–Delay spacetimes, where the combination of spacetime curvature and gauge freedom in the Maxwell potential have implications for the regularity of the initial and scattering data. Based on joint work with J.-P. Nicolas (Brest).

Katrina Morgan
(Northwestern University, USA)

Wave propagation on rotating cosmic string backgrounds

Rotating cosmic string spacetimes are solutions to the Einstein field equations which exhibit a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. These spacetimes have a notable unusual feature: they admit closed timelike curves near the so-called “string” and are thus not globally hyperbolic. This presents challenges to studying the existence of solutions to the wave equation on cosmic string geometries via conventional energy methods. In recent work with Jared Wunsch, we show that forward in time solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities which provides a microlocal energy estimate that allows us to establish existence of solutions. In this talk we will discuss the dynamics of null geodesics on cosmic string spacetimes, propagation of singularities along these paths, and the utility of the resulting estimate. No expertise in microlocal analysis will be assumed


Zongyuan Li
(Rutgers University, USA)

Asymptotic of harmonic functions near rough boundaries

We discuss asymptotic expansions of harmonic functions with zero Dirichlet boundary conditions. Domain are rough in the sense that there is no control over normal directions. Some relations with unique continuation properties and counterexamples will also be discussed. Joint work with D. Kriventsov (Rutgers).


Nicolas Camps
(Université Paris-Sud, France)

Some probabilistic approaches for NLS in the Euclidean space

Following the seminal work of Bourgain in 1996, and Burq and Tzvetkov in 2008, a statistical approach to nonlinear dispersive equations has developed in various contexts. 
We are interested here in Schrödinger equations with cubic nonlinearity (NLS) in R^d. We first recall the relevant probabilistic Cauchy theory  developped by Bényi, Oh and Pocovnicu in 2015 in supercritical regimes, before specifying the norm inflation instability that occurs in this context.
The second part is dedicated to long-time dynamics for solutions initiated from these randomized initial data. We demonstrate a scattering result that relies on a probabilistic version of the I-method and that allows to solve statistically the scattering conjecture for NLS in dimension 3.
Finally, we present recent developments in quasi-linear regimes, which were initiated by Bringmann in 2019 and which we exploit to exhibit strong solutions to some weakly dispersive equations. This last result is in collaboration with Louise Gassot and Slim Ibrahim.


Yu Deng
(University of Southern California, US)

Mathematical wave turbulence theory: the full range of scaling laws

The wave turbulence theory has been a subject of great interest to mathematicians and physicists in the last few decades. In this talk I present recent joint works with Zaher Hani (University of Michigan), which establishes the mathematical foundation of wave turbulence theory, in the full range of physically relevant scaling laws. This includes a rigorous derivation of the wave kinetic equation, justification of the propagation of chaos assumption, and associated evolution of densities.  

Minhyun Kim
(Universität Bielefeld, Germany)

Regularity theory for nonlocal operators on manifolds

In this talk, we study the Krylov–Safonov theory for nonlocal operators of fractional-order on Riemannian manifolds. We establish the Harnack inequality and Hölder estimates which are robust in the sense that the constants in the estimates remain uniform as the order of differentiability approaches 2. These results partially extend the classical results by Cabre (CPAM ’90) and Wang–Zhang (Adv. Math. ’13) for second-order operators on Riemannian manifolds. This talk is based on joint works with Jongmyeong Kim and Ki-Ahm Lee.  

Cristiana de Filippis
(University of Parma, Italy)

Quasiconvexity meets nonlinear potential theory

A classical problem in the regularity theory for vector-valued minimizers of multi- ple integrals consists in proving their smoothness outside a negligible set, cf. Evans (ARMA ’86), Acerbi & Fusco (ARMA ’87), Duzaar & Mingione (Ann. IHP-AN ’04), Schmidt (ARMA ’09). In this talk, I will show how to infer sharp partial reg- ularity results for relaxed minimizers of degenerate/singular, nonuniformly elliptic quasiconvex functionals, using tools from nonlinear potential theory. In particu- lar, in the setting of functionals with (p,q)-growth – according to the terminology introduced by Marcellini (Ann. IHP-AN ’86; ARMA ‘89) – I will derive optimal local regularity criteria under minimal assumptions on the data. This talk is partly based on joint work with Bianca Stroffolini (University of Naples Federico II). 

Spring 2022

Ricardo Grande Izuierdo
(University of Michigan – Ann Arbor, US)

Large Deviation Principle for the Cubic NLS Equation

In this talk we will explore the weakly nonlinear cubic Schrödinger equation with random initial data as a model for the formation of large waves in deep sea. First we will prove a large deviation principle for the solution of the equation, i.e. we derive the top order asymptotics for the probability of seeing a large wave at a certain time as the height of the wave tends to infinity. Then we will study a related problem: if we do see a large wave, what is the most likely initial datum that produced it? We answer this question in the weakly nonlinear regime by giving a probabilistic characterization of the set of rogue waves. This is joint work with M. Garrido, K. Kurianski and G. Staffilani. 

José Ramón Madrid Padilla
(University of California, Los Angeles, US)

On classical inequalities for autocorrelations and autoconvolutions

We will discuss some convolution inequalities on the real line, the study of these problems is motivated by a classical problem in additive combinatorics about estimating the size of Sidon sets. We will also discuss many related open problems. This talk will be accessible for a broad audience.

José Ramón Madrid Padilla
(University of California, Los Angeles, US)

On classical inequalities for autocorrelations and autoconvolutions

We will discuss some convolution inequalities on the real line, the study of these problems is motivated by a classical problem in additive combinatorics about estimating the size of Sidon sets. We will also discuss many related open problems. This talk will be accessible for a broad audience.

Daniel Campos
(Universidad de Costa Rica, Costa Rica)

A magnetic Schrödinger inverse problem in a cylindrical setting

This talk will be a brief introduction to a class of inverse problems, with the intention to motivate the use of Carleman estimates and certain techniques of pseudodifferential calculus for these questions. We consider the problem of recovering the magnetic field of a Schrödinger operator from boundary measurements in a constructive way, a procedure which relies on the construction of many special solutions (typically called “complex geometric optics solutions” or CGO’s). To obtain these solutions, we prove a global Carleman estimate for the magnetic Schrödinger operator by conjugating the magnetic operator essentially into the Laplacian using pseudodifferential operators.

Joshua Flynn
(University of Connecticut, US)

Some Sharp Uncertainty Principles and Related Geometric Inequalities

The Heisenberg uncertainty principle is a fundamental result in quantum mechanics. Related inequalities are the hydrogen and Hardy uncertainty principles and all three belong to the family of geometric inequalities known as the Caffarelli-Kohn-Nirenberg inequalities. In this talk, we survey recent results pertaining to uncertainty principles and CKN inequalities with a particular focus on higher order derivatives and vector-valued cases.

Cody Stockdale
(Clemson University, US)

A different approach to endpoint weak-type estimates for Calderón-Zygmund operators

The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.

Federico Castillo
(Catholic University of Chile, Chile)

Lineup polytopes in physics

Motivated by an instance of the quantum marginal problem in physics, we define the r-lineup polytope of P as a polytope parametrizing all possible linear orders on the vertices of P. We focus on the concrete case when P is a hypersimplex. This example sits in between the sweep polytopes of A. Padrol and E. Philippe and the theory of symmetric polytopes. This is based on joint work with JP. Labbe, J. Liebert, A. Padrol, E. Philippe and C. Schilling.

Gian Maria Dall’Ara
(Scuola Normale Superiore, Italy)

Schrödinger operators and complex analysis

In this talk I will recount how Schrödinger operators and related ideas from mathematical physics appear and can be of use in complex analysis. In one complex variable the connection is somewhat more direct and has been successfully exploited by various authors (B. Berndtsson, M. Christ, S. Fu, E. Straube …) to prove new theorems about Bergman kernels and the d-bar Neumann problem. While there are several stumbling blocks preventing an easy extension of this connection to several complex variables, I will show that it can still be of help in understanding, e.g., exponential decay of Bergman kernels and regularity properties of the d-bar Neumann problem in two or more dimensions. The original part of the talk is in part the result of a collaboration with Samuele Mongodi of the University of Milano-Bicocca.

Rajula Srivastava
(University of Wisconsin- Madison, US)

The Korányi Spherical Maximal Function on Heisenberg groups

In this talk, we will consider the problem of obtaining sharp (up to endpoints) L^p\to L^q estimates for the local maximal operator associated with averaging over dilates of the Korányi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Korányi sphere (in particular, the flatness at the poles) and the non-isotropic dilation structure encapsulated by a new type of Knapp example. We shall see that despite the non-Euclidean setting, the theory of Fourier Integral Operators can be applied to establish our estimates.

Esther Bou Dagher
(Imperial College London, UK)

Coercive Inequalities and U-Bounds

In the setting of step-two Carnot groups, we prove Poincaré and $\beta$-Logarithmic Sobolev inequalities for probability measures as a function of various homogeneous norms. To do that, the key idea is to obtain an intermediate inequality called the U-Bound inequality (based on joint work with B. Zegarlinski). Using this U-Bound inequality, we show that certain infinite dimensional Gibbs measures- with unbounded interaction potentials as a function of homogeneous norms- on an infinite product of Carnot groups satisfy the Poincaré inequality (based on joint work with Y. Qiu, B. Zegarlinski, and M. Zhang).
We also enlarge the class of measures as a function of the Carnot-Carathéodory distance that gives us the q-Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, we use the Hamilton-Jacobi equation in that setting to prove the p-Talagrand inequality and hypercontractivity.

Bae Jun Park
(Sungkyunkwan University, Korea)

Multilinear rough singular integrals

In this talk we will study m-linear rough singular integral operator \mathcal{L}_{ \Omega} associated with rough functions \Omega on the sphere \mathbb{S}^{mn-1} with mean value zero. We prove boundedness for \mathcal{L}_{\Omega} from L^{p_1}(\mathbb{R}^n) \times \cdots \times L^{p_m}(\mathbb{R}^n) to L^p(\mathbb{R}^n) when 1<p_1,\dots,p_m<\infty and 1/p=1/p_1+\cdots+1/p_m in the largest possible open set of exponents when \Omega\in L^q(\mathbb{S}^{mn-1}) and q\geq 2. Some related open problems will be also discussed at the end of this talk.
This is based on joint work with Grafakos, He, and Honzík.

Birgit Schörkhuber
(University of Innsbruck, Austria)

Non-trivial self-similar blowup for supercritical nonlinear wave equation

Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions.  Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable (modulo symmetries). The talk is based on joint works with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). 

Chenmin Sun
(CY Cergy-Paris Université, France)

Revisit the damped wave equation

The damped wave equation is widely used to describe propagation phenomena for waves in viscoelastic materials where the energy is dissipated from some part the domain or some portion of the boundary. Determining the optimal energy decay rate is a classical problem in PDE and control theory. It turns out that the geometry of the underlying background and the damped region play crucial roles for the optimal decay rate. In this talk, I will overview some classical and recent results for the damped wave equation with internal damping, and explain how the state of the art of microlocal analysis (semiclassical analysis) enters in these problems.

Matthew Schrecker
(University of London, UK)

Self-similar gravitational collapse for the Euler-Poisson equations

The Euler-Poisson equations give the classical model of a self-gravitating star under Newtonian gravity. It is widely expected that, in certain regimes, initially smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In this talk, I will present recent work with Yan Guo, Mahir Hadzic and Juhi Jang that demonstrates the existence of smooth, radially symmetric, self-similar blow-up solutions for this problem. At the heart of the analysis is the presence of a sonic point, a singularity in the self-similar model that poses serious analytical challenges in the search for a smooth solution.

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

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<p<\infty 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<p<\infty 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<p<\infty.
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.

Julio Delgado
(Universidad del Valle, Colombia)

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.

Adilbek Kairzhan
(University of Toronto, Canada)

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.

Soledad Villar
(Johns Hopkins University, USA)

Equivariant machine learning structured like classical physics

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality d.  The key observation is that nonlinear O(d)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars — scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.