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INTRODUCTION

The aim of the SIJIMAT seminar is to bring together young researchers from the Centre de Recerca Matemàtica in order to promote the interaction between the different research areas present at the centre. Through informal talks of 45 minutes plus a 15 minutes discussion, PhD students and postdoctoral fellows will have the opportunity to learn about the research done by their colleagues.

All talks are expected to have an introductory section that helps non-expert participants understand the main questions and goals of the field in which the speaker works. A second section might introduce the main tools used by the speaker to tackle those questions. Finally, in the last section, the speaker will provide a brief summary of her current research project.

NEXT SESSION

# On the optimality of effective stability bounds for invariant tori of Hamiltonian Systems

**Date: **Thursday 5th, October 2023.**Time: **12h**Place**: Aula petita CRM.

**Abstract:**

In this talk, I will introduce the concept of effective stability for invariant objects that naturally arise in Hamiltonian Systems, such as invariant tori. We will observe that under specific conditions, invariant tori exhibit a form of “stickiness,” or equivalently, it can be proved that it takes a considerable amount of time for solutions with initial conditions close to these objects to diverge from them. I will then elaborate on how to construct examples to show the optimality of these results.

SPEAKER

#### Gerard Farré Puiggalí

UPC

I am currently a postdoctoral researcher at Universitat Politècnica de Catalunya under the Juan de la Cierva-Formación program. My research interests are related to Dynamical Systems, more specifically in the fields of Hamiltonian dynamics, Ergodic theory and Mathematical Biology.

#### PAST SESSIONS

**21/09/2023 | Cristina Crespo (URJC – CRM) | What can statistics tell us about earthquakes?**

**ABSTRACT:**

In this talk, I will point out the relevance of earthquakes inside the tectonic plates (intraplate), and how thanks to statistical methods, we can shed light on the origin of these poorly understood earthquakes. Throughout my PhD, several statistical methods are applied to clear up the driven mechanism of the seismicity in the intraplate region of NW Iberian Peninsula, and in a small region of Lugo characterized by moderate (5.1 Mw y 4.6 Mw) and continuous seismicity (for more than 25 years). Among other statistical techniques applied, the ETAS model is remarkable, because it is widely used to characterize and identify triggered seismicity. Our results suggested the hypothesis of a mix of tectonic thrust and fluid upward migration as responsible for the seismicity in this region. During my postdoc at the CRM we will apply these models of seismic triggering after larger earthquakes to other examples, such as the 1755 Lisbon earthquake.

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**08/06/2023 | Román Moreno (Universitat Politècnica de Catalunya) | Instabilities in Hamiltonian systems: a study of the original Arnold model**

**ABSTRACT:**

The goal of this talk is to introduce the concept of Arnold diffusion, to explain where it lies inside the area of study of differential equations and to present my specific research topic. First we will contextualize the study of Arnold diffusion. For that, we will trace historically the development of the area of Hamiltonian systems highlighting the question of stability and its importance for physical models. Then, we will look at the more abstract mathematical setting of perturbation theory developed to tackle the problem of stability. We will define Arnold diffusion and we will outline the relevant questions researchers are interested in. Finally, I will focus on my study of Arnold’s original problem using modern techniques of analysis of exponentially small phenomena.

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**25/05/2023 ****| Marvin Weidner (Universitat de Barcelona) ****| About melting ice and pricing American options: Regularity for the obstacle problem**

**ABSTRACT:**

Various phenomena in physics, biology, finance, and other areas can be described mathematically by so-called free boundary problems. Solving the corresponding partial differential equations does not only consist in finding the solution itself, but also in determining an a priori unknown interface. The obstacle problem is a free boundary problem modeling for instance the melting of ice or the pricing of American options.

In the first part of the talk, we will introduce the obstacle problem, explain some of its mathematical features, and study the regularity of the solution and of the free boundary. The second part deals with the nonlocal obstacle problem. This is a related model taking into account long-range interactions. We will discuss a recent result on the regularity for the nonlocal obstacle problem. This talk is based on a joint work with Xavier Ros-Oton and Damià Torres-Latorre.

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**11/05/2023 ****| Mónica Minjares (CRM) | Statistical Analysis of the Madden-Julian Oscillation Events**

**ABSTRACT**: In this talk I will present an analysis of the statistical features of the Madden-Julian Oscilation (MJO) events. MJO is a weather structure that propagates eastward along the globe and near the equator, it has a strong influence on the precipitation in the tropics but it also affects extratropical regions through teleconnection patterns.

Although several studies have widely analysed the MJO, its activation and evolution are not fully understood.

MJO is of great importance as it provides a major source of tropical and extratropical predictability on subseasonal time scales (two weeks to three months). To monitor MJO we use a Real-time Multivariate (RMM) index which consists of the first and second principal components (RMM1 and RMM2) obtained from the empirical orthogonal functions (EOF) that combine latitudinal averages of outgoing longwave radiation (OLR), and winds at lower and higher atmospheric levels.

In our framework an event takes place when the index amplitude (the modulus of the vector (RMM1,RMM2)) exceeds a certain threshold. With this, we define the observables of an event; these are, the maximum amplitude, duration, the period in which the amplitude is above the threshold and size, which is the sum of the amplitudes along the duration of an event. We further explore whether the tails of the event distributions are heavy tailed. A distribution is heavy-tailed if the tail (right or left) decays more slowly than those of any exponential distribution. They are of interest because they can be used to model processes which have a tendency to produce extreme outcomes. Of particular interest are extreme MJO events and for this we use extreme-value theory in concrete, the peak-over-threshold framework. We then make a composite analysis of these extreme events considering the phase (geographical location) of initiation of the events to see how they affect different regions away from the equator.

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**27/04/2023 ****| Lucy Auton (Centre de Recerca Matemàtica) ****| Macroscopic Contaminant Transport & Removal in Soft & Heterogeneous Filters**

**ABSTRACT: **Two key challenges that if tackled can drastically improve sustainability are removing pollutants from waste products of industrial processes to prevent them contaminating our air or water and decontaminating our existing resources especially in regions of high demand but low availability. Despite the importance of contaminant transport and removal in such filters, the effect of microstructure on flow and transport remains poorly understood and improving our understanding of this remains a major challenge. I consider fluid flow through a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium, allowing for anisotropy. To analyse the dispersive effects, I use multiple temporal scales and a nontrivial extension to classical homogenisation theory via the method of multiple scales to rigorously upscale the problem involving cells of varying area. Subsequently, I perform a drift transformation to account for the three spatial scales intrinsically present in dispersive problems. This results in simple effective continuum equations for macroscale flow and dispersive transport where the effect of the microscale geometry on the macroscopic transport and removal is encoded within these simple macroscale equations via effective parameters such as an effective local anisotropic diffusivity and an effective local adsorption rate. I determine these effective parameters numerically for a simple example geometry.

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**13/04/23 | Jessica Elisa Massetti (Research fellow at Università Roma Tre) | Recursive phenomena and lifespan of solutions in Hamiltonian PDEs**

**ABSTRACT: **Many physical phenomena have ondulatory nature: the motion of the sea, the deflection of a beam, electromagnetic waves (light, radio waves)… Many of these are modelled by nonlinear Hamiltonian PDEs and their mathematical description is often extremely complicated, characterized by a non trivial interplay between stable and chaotic behaviours. Of course, the typical behaviour of the solutions is strongly related to the boundary conditions: in this talk I shall deal with (very simple) compact domains. Two fundamental questions arise naturally: the existence of recursive solutions (periodic, quasi-periodic or almost periodic in time) and long time stability of initial data. I shall describe some of these models and discuss the main difficulties in dealing with these questions, through a dynamical system approach. Finally, I shall describe some very recent results in collaboration with R. Feola, concerning the long time behavior of Sobolev norms, via Normal Form and Energy methods.

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**30/03/23 ****| Giulia Binotto (Postdoctoral researcher at UAB) ****| ****Can the memory loss help to study the propagation of a disease?**

**ABSTRACT: **Mathematical models applied to the field of biology and, in particular, epidemiology are currently of great interest due to the pandemic generated by COVID-19. However, their study to describe the spread of a disease has a long history. It dates back to the research on smallpox inoculation carried out by Bernoulli in the 18th century. Since then, both deterministic and stochastic models have been considered and many factors have been taken into account: infectious agents, mode of transmission, latency periods, temporary or partial immunity, quarantine periods, etc.

The aim of this talk is to present a SIR-type stochastic model, which is an extension of the one proposed by Tuckwell and Williams in 2007. It is a discrete-time Markovian model in which the total population remains constant and individuals meet a random number of other individuals at each time step. One of its innovative aspects is the time dependence of this parameter.

An analytic description of the model and its dynamics will be provided.

In order to better comprehend them, a brief introduction to Markov chains will be given and their connection with memory loss will be explained. Finally, with the aid of some simulations, it will be shown how the evolution of a disease is affected by the time dependence of the number of daily encounters.

These results are part of a work carried out with M. Besalú.

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**16/03/23 | Blai Vidiella (Postdoctoral researcher at CSIC – Institut de Biologia Evolutiva) | Complex Systems to understand the Biosphere’s Past, Present, and Future**

**ABSTRACT: **Since the dawn of humanity, scientists have been captivated by the intricate and complex systems found in nature. From the movements of celestial bodies to the behavior of microorganisms, we have sought to understand and explain the phenomena we observe in the natural world. Today, as we face increasing challenges such as climate change and ecosystem degradation, our understanding of these systems has never been more important. In this talk, I will explore the power of mathematical modeling in helping us to gain a deeper understanding of the biological systems that make up the biosphere. Specifically, I will focus the application of mathematical rules to understand evolutionary questions, such as punctuated equilibria; the role of mathematical modeling in studying anthropogenic ecology; and the concept of ‘ghost ecosystems’ resulting from climate change. Through this discussion, I hope to shed light on the exciting possibilities that lie ahead for those working at the intersection of mathematics and biology.

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**02/03/2023 ****| Manuel Molano (CRM) ****| Investigating how rats make decisions**

**ABSTRACT**: In this talk I will briefly explain how neuroscientists investigate the strategies used by animals to make decisions and the neural mechanisms underlying such strategies. I will then provide an example in which we have studied the behavior of rats performing a task in which they have to listen to an auditory stimulus and decide between two alternatives basing their decision on the information provided by the stimulus. By tracking the position of the rats using a machine learning tool we were able to characterize how they combine their previous experience with the incoming stimulus information, and show that they flexibly update an initial decision by incorporating new incoming information.

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**19/01/23 | Giovanni Dalmasso (CRM) | 4D reconstruction of developmental trajectories using spherical harmonics**

**ABSTRACT**: Normal organogenesis cannot be recapitulated in vitro for mammalian organs, unlike in species including Drosophila and zebrafish. Available 3D data in the form of ex vivo images only provide discrete snapshots of the development of an organ morphology. Here, we present a computer-based approach to recreate its continuous evolution in time and space from a set of 3D volumetric images. This method is based on the remapping of shape data into the space of the coefficients of a spherical harmonics expansion where a smooth interpolation over time is simpler. This approach had been tested on mouse limb buds and embryonic hearts. A key advantage of this method is that the resulting 4D trajectory can take advantage of all the available data while also being able to interpolate well through time intervals for which there are little or no data. This allows for a quantitative, data-driven 4D description of mouse limb morphogenesis.

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**15/12/22 | Alfonso Garmendia (CRM) | Deformation Quantization**

**ABSTRACT**: The goal of quantization is to associate a C*-algebra (a quantum space) to any Poisson manifold (a classical space) in a way that preserves the symmetries. This talk tries to convey the motivation and a summary on deformation quantization with examples.

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**01/12/22 | Angelica Torres (CRM) | Algebraic Geometry and line reconstruction in Computer Vision**

**ABSTRACT**: The 3D image reconstruction problem aims to create a 3D model of a scene or object starting from 2D images. This process is done in four stages: Feature identification in the images, point and line matching, camera estimation and triangulation, and construction of the 3D model. Stages two and three of the process deal mainly with geometric information that can be studied algebraically. This is precisely where Algebraic Geometry comes into play. In this talk I will introduce the 3D image reconstruction problem and the algebraic tools that allow to model the cameras and image features, I will also define the point and line Multiview varieties, and finally mention some algebraic results on the geometry of points and lines that can be reconstructed effectively.

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#### **18/11/22 | Mar Giralt-Miron (UPC) | Chaotic dynamics, exponentially small phenomena and Celestial Mechanics**

**ABSTRACT**: A fundamental problem in dynamical systems is to prove that a given model has chaotic dynamics. One of the methods employed to prove this type of motions is to verify the existence of transversal intersections between the stable and unstable manifolds of certain objects. Then, there exists a theorem (the Smale-Birkhoff homoclinic theorem) which ensures the existence of chaotic motions.

In this talk we present a method to analyze the distance and transversality between certain stable and unstable manifolds when a small perturbation is added to an integrable system. In particular, we consider the case where the distance between manifolds is exponentially small. This implies that this difference cannot be detected by expanding the manifolds into a series with respect to the small perturbation parameter. Therefore, classical perturbation theory cannot be applied.

Finally, we apply these techniques to a celestial mechanics problem. In particular, we study the Lagrange point L 3 in the restricted planar circular 3-body problem.

organizers

Lucía Arancibia | CRM PhD Student | Computational Neuroscience | |

Giovanni Dalmasso | CRM Postdoctoral Researcher | Cancer Modelling Lab | |

Dídac Gil | CRM PhD Student | Dynamical Systems | |

Roser Homs | CRM Postdoctoral Researcher | Computational & Mathematical Biology |