CUWB-V: From resonant scattering to tectonics: recent advances in exactly solvable models and their applications
The conference will focus on recent results in the area of solvable models given by boundary value problems for partial differential equations. These have been central to the research project CF 2019 304005 "The evolution of active zones in tectonic plates: a mathematical approach based on low-frequency flexural vibrations". The conference will bring together researchers working in the field and on related topics both in the CUWB community and in other institutions, aimed at consolidating the ongoing collaborations and generating new connections. Solvable models are useful in the analysis of complex settings that are difficult to treat directly. However, they also involve simplifications that require their validation against the realistic setting they represent, which necessitates putting them to a fitting test. In the case of seismo-gravitational oscillations, the signal from gravimetric stations play an important role in this test. There are plans for experts in gravimetry to present talks and interact with the mathematical community.
Workshop lectures:
Miguel Ballesteros (IIMAS-UNAM)
Title: A new population model inspired by statistical physics, quantum mechanics, and artificial intelligence
Abstract: In this talk, we will address problems related to amphibian population distribution in a jungle area of Mexico, based on a mathematical model inspired by quantum mechanics and statistical physics. Using partial information of the mathematical model about the spatial distribution, we reconstruct the model's parameters and obtain the distribution of the amphibian population in the region.
Kirill Cherednichenko (University of Bath)
Title: Exactly solvable models in wave propagation through composite media via operator asymptotics
Abstract: I will start with an introduction to "effective" descriptions of the behaviour of highly inhomogeneous media. It will start with an overview of what can now be referred to as "classical homogenisation", a range of mathematical techniques aimed at providing a set of material constants that (in some rigorous sense) represent the response of the medium to long waves or loads varying at length scales that are much larger than the size of the heterogeneity. Through examples, we will then see how the limitations of such techniques emerge, how they can be overcome, and what kind of "generalised" effective descriptions one arrives at upon relaxing the assumptions of classical homogenisation. Finally, I will address some links of the new approaches to the design of novel materials with "unusual" properties.
Alfredo Esparza (CENAM)
Title: Gravimetric measurements
Abstract: This talk presents the different methods for accurately measuring gravitational acceleration. It includes an overview of measurement theory and all the physical phenomena that affect the precise measurement of gravitational acceleration. Finally, it discusses the instruments currently available for making gravimetric measurements.
Eduardo Gómez (UASLP)
Title: Atomic ans MEMS gravimeters
Abstract: Gravimeters are devices sensitive to the amount of mass surrounding them by measuring the local acceleration due to gravity. Gravity maps are useful in underground exploration, as well as in monitoring various types of hazards. Two outstanding technologies in gravimetry are atomic gravimeters and those based on Micro Electro-Mechanical Systems (MEMS). I will describe our progress in implementing these mutually complementary sensors.
Yi Sheng Lim (Texas A&M University)
Title: An overview of wave propagation in periodic composites
Abstract: In this talk, we shall discuss the problem of homogenization of the wave equation, which is the study of approximating the effective transport properties of a highly heterogeneous medium by a homogeneous one. While one measures the goodness of an approximation in the elliptic setting solely by its accuracy (ε), the error for the wave equation is measured in accuracy (ε) and time (t) simultaneously. However, achieving high accuracy and long times are competing objectives in the hyperbolic setting, and it turns out that the classical two-scale ansatz is only good to an O(ε^{-2+δ}) timescale. We shall explain the reason behind this fundamental limitation, and several approaches to overcome this, namely, a "criminal ansatz" (Conca-Orive-Vanninathan 2022) and a "spectral ansatz" (Benoit-Duerinckx-Gloria-Ruf 2023).
Iván Naumkin (IIMAS-UNAM)
Title:
Abstract:
Danila Prikazchikov (Keele University)
Title: Modelling the elastodynamics of layered structures involving aerogel layers
Abstract: Aerogels are extremely lightweight porous materials known for their multifunctional properties, having wide engineering applications, with their mechanical properties mostly studied within the context of elastostatics. In this presentation we will aim at clarifying some aspects of their dynamic behaviour. We will also consider specific multi-parametric scaling arising in a sandwich structure involving the aerogel interlayer, presenting another
example of a strongly inhomogeneous laminate. It is known that the presence of strong inhomogeneity in layered structures high contrast in layered elastic structures involves emergence of two phenomena not typical of a mild contrast scenario, namely, the extra low-frequency vibration spectra, as well as the slowly decaying degenerated boundary layer solutions. The relation between these two phenomena will also be discussed.
Julien Ricaud (IIMAS-UNAM)
Title: On the inverse problems for Love waves in a layered, elastic half-space
Abstract: In this talk, I will present recent results on Love waves in a layered, elastic half-space, with the goal of recovering the parameters of the medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves. To that end, I will first present the direct problem and the characterization of Love waves' existence through the dispersion relation, then I will address the inverse problem and show how one can recover parameters of the medium. This is joint work with Maarten de Hoop, Josselin Garnier, and Alexei Iantchenko, and is based on [1].
[1] M. V. de Hoop, J. Garnier, A. Iantchenko, J. R., Inverse problem for
Love waves in a layered, elastic half-space, e-prints (2023).
Stephen Shipman (Louisiana State University)
Title: Resonant scattering in wave mechanics
Abstract: Various physical systems that admit classical waves, such as in optics or elasticity or electromagnetics, can exhibit sharp resonant scattering. Particularly, this can occur in a layered medium with defects. One mathematical mechanism is the perturbation bound states at frequencies within the continuum of radiation states. We discuss the creation and properties of the associated resonance in different physical scenarios.
Luis O. Silva (IIMAS-UNAM)
Title: The evolution of active zones in tectonic plates: a mathematical approach based on low-frequency flexural vibrations
Abstract: Tectonic plates can be modelled as thin membranes floating on a liquid surface. The oscillations of these membranes can be partially described using boundary value problems involving partial differential equations. Boundary conditions encode the phenomena between plates. Based on the results of this mathematical model, it is concluded that studying the temporal evolution of the oscillations of these membranes is relevant for inferring information about changes occurring in active plate interaction zones. Plate oscillations are low-frequency, low-amplitude oscillations (seismogravitational oscillations) and can be studied using highly precise gravimetric measurements. However, recording changes in these oscillations is very complex, and new mathematical methods are required for analyzing the signals from gravimetric stations.
| Time | Friday 9 | Monday 12 | Tuesday 13 | Wednesday 14 |
|---|---|---|---|---|
| 10:30 | Lim | Gómez | ||
| 11:30 | Shipman | Esparza | Naumkin | |
| 12:30 | Discussions | Ballesteros | Ricaud | Silva |
| 13:30 | Prikazchikov | Cherednichenko | Closure |
Friday 9: Room 200
Monday 12: Room 200
Tuesday 13: Room 201
Wednesday 14: Room 201
Pre-recorded lectures:
Edgar Migueles Pérez (IIMAS-UNAM)
Title: Analytical sampling in spaces arising from Jacobi operators
Abstract:
Mario Alberto Ruiz Caballero (IIMAS-UNAM)
Title: Spectral stability of operators with singular perturbations
Abstract:
Participants
| UNAM | Bath | Other |
|---|---|---|
|
Miguel Ballesteros
Diego Iniesta
Iván Naumkin
Luis Silva
Julien Ricaud
|
Kirill Cherednichenko
|
Alfredo Esparza (CENAM)
Eduardo Gómez (UASLP)
Yi Sheng Lim (Texas A&M)
Danila Prikazchikov (Keele)
Stephen Shipman (Louisiana State)
|