Since the conception of quantum mechanics, Uncertainty Principles have played a central role in Analysis and Partial Differential Equations. Indeed, from Heisenberg, through Hardy, and all the way to unique continuation, this rich topic has expanded to several new, exciting direction throughout the past century.
One of the most notable of such directions is the emerging area of Fourier Uniqueness Pairs. In spite of the celebrated contribution of M. Viazovska to the field, inspiring many questions with her Fourier analysis solution to the sphere packing problem in dimensions 8 and 24, a preceding work by H. Hedenmalm and A. Montez-Rodríguez had incidentally already touched upon the same sort of results years prior, by establishing a link between Fourier analysis, the Klein-Gordon equation and dynamical systems.
In this talk, we will explore this seminal controbution by Hedenmalm and Montes-Rodríguez, its relationship to Fourier interpolation, and some of the conjectures posed by those authors.
This is based on a joint work with D. Radchenko.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
Pêdra D. S. Andrade, Paris Lodron Universität Salzburg
In this talk, we provide an introductory overview of fully nonlinear partial differential equations (PDEs), emphasizing the regularity of solutions and key analytical techniques. Fully nonlinear equations play a crucial role in diverse areas such as geometry, physics, and finance. They are distinguished from linear and semilinear equations by their intricate structure and the absence of the superposition principle. This nonlinearity introduces unique challenges in understanding the existence, uniqueness, and regularity of solutions. We explore foundational concepts, such as viscosity solutions and a priori estimates, which are central to the modern theory.
The Hardy Uncertainty Principle states that if both a function f and its Fourier transform decay faster than the Gaussian function with a specific weight, then f = 0. This result can be reformulated in terms of solutions to the free Schrödinger equation. In a series of work Escauriaza, Kenig, Ponce and Vega extended this result to the Schrödinger equation with potential and to NLS by the use of Carleman estimates. More precisely, it was proven that if u is a solution to the Schrödinger equation with potential, which at two times has Gaussian decay, and given the right conditions on the potential, then u = 0.
The proof is based on Carleman estimates, which formally relies on calculus and convexity arguments. However, going from a formal level to a rigorous one is not straight forward, and if we do not justify the computations rigorously, we can prove wrong results.
We have adapted these techniques to the hyperbolic Schrödinger equation, which is physically relevant in for example Fluid mechanics. We will go through the main ideas of the proof with the Carleman estimates, and where the main changes are when going from the Laplace operator to the hyperbolic Laplace operator in the Schrödinger equation.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
The goal of this talk is to give a friendly introduction to several questions related to the remarkable fact that linear differential equations on the complex plane are actually of topological nature. This result, known as the Riemann-Hilbert correspondence, says that a complex linear ODE is fully characterized by its monodromy, a topological object which describes what happens when one follows the solutions of the equation along a loop around one of its singularities.
An interesting consequence of this arises when we consider deformations of linear ODEs: looking for deformations which keep the monodromy constant amounts to an integrable system of nonlinear PDEs. Many equations ubiquitous in mathematical physics, for example all Painlevé equations, arise in this way. These PDEs themselves admit a topological version, consisting of the action of a discrete group on the space parametrizing all possible monodromy data.
I will try to explain the main ideas involved in this rich picture, and, if time permits, also say a few words about the case of irregular singularities and some of my work.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
María Cueto Avellaneda, Instituto Superior Técnico, Universidade de Lisboa
In the framework of the functional analysis, this talk aims to illustrate the research we have developed during the last years: from the study of geometric and algebraic properties of triple structures to a one-to-one correspondence with TKK Lie algebras. We shall highlight the last novelties achieved in the Lie setting.
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Room P3.10, Mathematics Building
Alfilgen N. Sebandal, Linnaeus University (Sweden) and RCTP (Philippines)
In the 1960's, W. Leavitt studied a class of universal algebras which do not have a well-defined rank, i.e., algebras for which as -modules with , later known as the Leavitt algebra . In two simultaneuous but independent studies by G. Abrams and G. Pino, and P. Ara et al., an algebra arising from a directed graph and a field has been introduced called the Leavitt path algebra . This algebra turned out to be the generalization of . In fact, where is the graph having one vertex and loops.
In 2013, R. Hazrat formulated the Graded Classification Conjecture for Leavitt path algebras which claims that the so-called talented monoid is a graded Morita invariant for the class of Leavitt path algebras. In this talk, we will see Leavitt Path algebra modules as a special case of quiver representation with relations and how it will take a role in proving the conjecture.
More concretely, we will provide evidences and a confirmation in the finite-dimensional case of the conjecture.
This is a compilation of joint works with Wolfgang Bock, Roozbeh Hazrat, and Jocelyn Vilela.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
Serena Benigno, Università degli Studi della Campania Luigi Vanvitelli
The goal of this talk is to study a reaction diffusion problem with anisotropic diffusion and mixed Dirichlet-Neumann boundary conditions on for , . First, we prove that the parabolic problem has a unique positive, bounded solution. Then, we show that this solution converges as to the unique nonnegative solution of the elliptic associated problem. The existence of the unique positive solution to this problem depends on a principal eigenvalue of a suitable linearized problem with a sign-changing weight. Next, we study the minimization of such eigenvalue with respect to the sign-changing weight, showing that there exists an optimal bang-bang weight, namely a piece-wise constant weight that takes only two values. Finally, we completely solve the problem in dimension one.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
Leland Brown, Instituto Superior Técnico, Universidade de Lisboa
Visualization of mountainous topography is commonly done using a shaded relief algorithm that combines an imaginary light source with a Lambertian reflectance model. The resulting images are intuitive to understand, but they also suffer certain drawbacks. Among these is anisotropy, or directional dependence, whereby the choice of lighting direction favors certain terrain features at the expense of others. Another common issue is lack of visual hierarchy, since small terrain features tend to clutter the image and obscure larger landforms that are actually more prominent. Most attempts to mitigate these problems have stayed within the general paradigm of illumination models.
A newer algorithm called texture shading attempts to address the shortcomings of conventional shaded relief by taking a different approach, not derived from a lighting model. The underlying mathematics uses a fractional Laplacian operator adapted to work on discrete gridded data (in this case, a digital elevation model) by means of a discrete cosine transform. The algorithm is isotropic, and scale-invariant in a particular sense, which gives a clear visual hierarchy to the images, and it highlights the network of ridges and canyons in the terrain. Fractional derivatives appear to be well-suited to terrain data due to the fractal nature of many topographic structures over a wide dynamic range.
The presentation will cover the motivation and goals of the algorithm, the implementation and mathematical details, example results, potential applications, and future work.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
In this talk, we investigate the existence of infinitely many non-constant weak solutions to the quasilinear elliptic equation for every , where is a bounded domain with a Lipschitz continuous boundary.
Our approach is variational; however, the associated energy functional is differentiable only along directions .
We discuss the notion of weak slope and the critical point theory for continuous functionals, and we apply these concepts to our problem to establish the existence of multiple solutions.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
Pablo dos Santos Corrêa Junior, Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo
What we are calling Emden-Fowler-Hénon (EFH) type equations are elliptic partial differential equations that have the main structure as , usually considering a radial function and with a subcritical exponent in the sense of Sobolev embeddings, that is, . Taking , the resulting equation is called Lane-Emden equation which is used in astrophysics to model stellar structures. If we consider , we obtain the so called Hénon equation, proposed in [1] as a perturbation of Lane-Emden equation to study the stability of spherical stellar systems. In mathematics, the Hénon equation plays many important roles in the theory of qualitative analysis of PDE, for instance, by providing a nontrivial example to theorems and working as a guide to develop new tools to fortify the theory. Significant developments have been made by [2, 3, 4, 5] to understand the qualitative properties of the Hénon equation and to set a process to analyze EFH type equations. In this setting, we propose an EFH type equation considering , which is a weight that presents a behavior unexplored in literature and, we believe, necessary to comprehend the full picture of EFH type equations. We are also interested in understanding how well the developed technics apply to this new configuration.
Hénon, M. (1974). Numerical Experiments on the Stability of Spherical Stellar Systems. Symposium - International Astronomical Union, 62, 259-259. https://doi.org/10.1017/s0074180900070662
Smets, D., Willem, M. & Su, J. Non-radial ground states for the Hénon equation. Communications In Contemporary Mathematics. 4, 467-480 (2002,8)
Cao, D., Peng, S. & Yan, S. Asymptotic behaviour of ground state solutions for the Hénon equation. IMA Journal Of Applied Mathematics. 74, 468-480 (2008,12)
Silva, W. & Santos, E. Asymptotic profile and Morse index of the radial solutions of the Hénon equation. Journal Of Differential Equations. 287 pp. 212-235 (2021,6)
Mercuri, C. & Santos, E. Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations. Nonlinearity. 32, 4445-4464 (2019,10).
We would like to express our gratitude to CAPES for the financial support.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
In this talk we consider differential operators of the form where the coefficients of are measurable and bounded functions and the eigenvalues of are positive and uniformly bounded from . A solution on a domain is a function that satisfies
We examine boundary behavior of the gradient of solutions to such differential operators. The classical case, where the differential operator in question is the Laplacian, is well understood for domains due to Havin and Mozolyako [1] and later also for Lipschitz domains due to Müller and Riegler [2]. This talk mentions techniques to obtain estimates on the radial variation of positve harmonic functions and also gives an outlook of how to adapt the existing techniques to general uniformly elliptic divergence-form operators or solutions on more complicated domains.
Pavel A. Mozolyako and Viktor P. Khavin. Boundedness of variation of a positive harmonic function along the normals to the boundary. Algebra i Analiz 28.3 (2016), pp. 67–110. issn: 0234-0852. doi: 10.1090/spmj/1454. url: https://doi.org/10.1090/spmj/1454.
Paul F. X. Müller and Katharina Riegler. Radial variation of Bloch functions on the unit ball of Rd. Ark. Mat. 58.1 (2020), pp. 161–178. issn: 0004-2080,1871-2487. doi: 10.4310/ARKIV.2020.v58.n1.a10. url: https://doi.org/10.4310/ARKIV.2020.v58.n1.a10.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
In this talk, we introduce a parametric bilinear fractional integral operator and obtain uniform bounds via Marcinkiewicz interpolation. This operator is motivated by a reformulation of Euler-Riesz systems, where the nonlocal term in the momentum equation is expressed as the divergence of a tensorial integral operator. This reformulation is relevant as it leads to an a priori higher integrability estimate for the density of finite-energy solutions. The uniform bounds obtained for the parametric fractional integrals then yield bounds for the tensorial integral operator. This is based on joint work with L. Grafakos and A. E. Tzavaras.
– Europe/Lisbon
Room P3.10, Mathematics Building — Online
Marina Garcia Romero, Universitat Politècnica de Catalunya
The replicator equation, originally used in evolutionary game theory, has been widely applied in evolutionary biology and ecology to model the complex dynamics of multi-type interacting species, such as multi-species ecological ensembles or multi-strain microbial pathogens. In this seminar we explore aspects of the competitive dynamics encoded in the replicator equation, paying special attention to invader-driven systems, in which all species are proactive. We focus on relating properties of the fitness matrix to the quality of species dynamics and to the diversity of these systems at equilibrium, and we find a mechanism for the selection of the final set of surviving species.