Final Workshop (Vila Galé Coimbra, 23 e 24 de julho de 2022)
Program
9h15-09h30 Opening
9h30-10h20 Mark Pollicott, University of Warwick (Part 1/2)
Title: Lyapunov exponents, entropy and fractal dimension: quantifying dynamical systems
Abstract: Chaotic dynamical systems have been studied since the early work of Poincare and Hadamard in the 1890s, via the work of Anosov, Sinai, Smale, Bowen and Ruelle in the 1960s and 1970's. A particularly important aspect of this work was the introduction of numerical values (e.g., Lyapunov exponents, entropy and the fractal dimension of appropriate invariant sets, or repellers).
Unfortunately, it is rare that these quantities have an explicit form. Typically, they need to be estimated numerically (using machinery form the ergodic theory of dynamical systems, e.g., pressure and Gibbs measures). In these lectures we will describe these quantities in specific settings and illustrate them with simple examples. We will motivate their study by showing their role in recent mathematical problems in other areas of mathematics, e.g., Euclidean geometry (Barycentric subdivisions of triangles, circle packings, etc.), Hyperbolic geometry (Random walks, dimension of limit sets), Number Theory (Lagrange spectrum in Diophantine geometry, Zaremba conjecture, etc.) and Analysis (Fractal dimension of sets).
Slides of the presentation.10h25-10h55 Francisco Nascimento, 3.º FCT/Universidade Nova de Lisboa
Title: A bird's-eye view on p-adic analysis
Abstract: In this talk, I will give a brief and accessible introduction to p-adic analysis and will highlight some of its differences when compared to classical real analysis.
Slides of the presentation.11h00-11h20 Coffee Break
11h20-11h50 Carla Morouço, 3.º ano, FCT/Universidade de Coimbra
Title: Split extensions of ordered groups and S-protomodularity
Abstract: Unlike the category of groups, the category of ordered groups is not protomodular. This means that split extensions of ordered groups are more complicated than those of groups. We obtain some characterizations and properties of these extensions, and we see which classes S of extensions have a classifier and form an S-protomodular category.
Slides of the presentation.11h55-12h25 Luís Machado, 3.º ano IST/Universidade de Lisboa
Title: Fall of an elastic rod in a Schwarzschild black hole
Abstract: We will start by presenting a mathematical model of relativistic elasticity for one dimensional bodies (rods) in Lorentzian spacetimes. For the specific case of rigid rods, i.e for which the speed of sound is equal to the speed of light, we will show that the equation of motion is the wave equation for a scalar field related to the stretch factor of the rod. Then, we will analyse the radial motion of a rigid elastic rod for the specific case of the Schwarzschild spacetime, by imposing an arbitrary initial condition for the stretch factor and letting the extremes of the rod be free. Afterwards, we will derive that, in these conditions, the rigid rod starts stretching at every point. Finally, we will observe other cases where we impose different conditions on the extremes of the rigid elastic rod and see how it affects its motion when comparing to the geodesic of a test particle starting at the mean point of the rigid rod.
Slides of the presentation.12h30-14h30 Lunch Break
14h30-15h00 Rodrigo Luís, 2.º ano FC/Universidade de Lisboa
Title: Almost Periodic Functions and Applications to Delayed Differential Equations
Resumo: In many biological systems, periodic and almost periodic variations of variables such as the duration of a life cycle or the amount of available resources are important factors for modeling the evolution of a population over time. We also know that the population size of a particular species of blowfly, at a given instant in time, is well-modeled by a family of Delayed Differential Equations (DDEs) called Nicholson's Blowflies model. In this presentation, we will start with a brief introduction to the definitions of DDE and Almost Periodic Function, mentioning their properties and other related relevant concepts. We will then proceed to study the existence and uniqueness of positive almost periodic solutions of a Nicholson-type blowfly model with arbitrary positive almost periodic coefficients.
Slides of the presentation.15h05-15h35 Nuno Carneiro, 2.º ano, IST/Universidade de Lisboa
Title: Mellin Transform and asymptotic expansions
Abstract: The Mellin Transform provides an effective method to calculate the asymptotic expansion of certain functions, namely when they are defined by series in the positive reals. In this presentation, we will show a few applications of this method, in particular for series whose coefficients are given by arithmetic functions. We will also present results regarding specific trigonometric series which can’t be obtained directly from the Mellin Transform and we will show its relation with the Fourier series coefficients for a class of functions.
Slides of the presentation.15h40-16h10 Pedro Costa Dias, 2.º ano IST/Universidade de Lisboa
Title: Local Existence and Uniqueness of the Non-Linear Schrödinger Equation on the Torus
Abstract: The Non-Linear Schrödinger Equation (NLS) is a Partial Differential Equation with a deep mathematical theory and important applications to Physics.
We will use recent developments by Bourgain to prove local wellposedness of the cubic NLS with periodic conditions and initial value in L2.
Slides of the presentation.9h30-10h20 Mark Pollicott, University of Warwick (Part 2/2)
Title: Lyapunov exponents, entropy and fractal dimension: quantifying dynamical systems
Abstract: Chaotic dynamical systems have been studied since the early work of Poincare and Hadamard in the 1890s, via the work of Anosov, Sinai, Smale, Bowen and Ruelle in the 1960s and 1970's. A particularly important aspect of this work was the introduction of numerical values (e.g., Lyapunov exponents, entropy and the fractal dimension of appropriate invariant sets, or repellers).
Unfortunately, it is rare that these quantities have an explicit form. Typically, they need to be estimated numerically (using machinery form the ergodic theory of dynamical systems, e.g., pressure and Gibbs measures). In these lectures we will describe these quantities in specific settings and illustrate them with simple examples. We will motivate their study by showing their role in recent mathematical problems in other areas of mathematics, e.g., Euclidean geometry (Barycentric subdivisions of triangles, circle packings, etc.), Hyperbolic geometry (Random walks, dimension of limit sets), Number Theory (Lagrange spectrum in Diophantine geometry, Zaremba conjecture, etc.) and Analysis (Fractal dimension of sets).
Slides of the presentation.10h25-10h55 André Guimarães, 3.º ano IST/Universidade de Lisboa
Title: Paired kernels and their properties
Toeplitz operators are, due to their properties and applications, an important family of operators defined on the Hardy subspace of the space of square-integrable complex-valued functions defined on the unit circle, L^{2}(T). They can be seen as the compression of multiplication operators on L^2(T) to the Hardy space.
Model spaces are an important family of subspaces of the Hardy space. The compression of Toeplitz operators to these spaces yields the notion of (asymmetric) truncated Toeplitz operators, which have been extensively studied in recent years.
The study of these operators naturally leads us to consider the properties of paired operators, yet another family of operators defined on L^{2}(T). We present some properties of the kernels of paired operators and their projections into the Hardy space, in comparison to known and similar properties for the kernels of Toeplitz operators.
This talk is based on joint work with M. Cristina Câmara and Jonathan Partington.
Slides of the presentation.11h00-11h20 Coffee Break
11h20-11h50 Ricardo Marques, 2.º ano FC/Universidade do Porto
Title: Factorization of Hurwitz Quaternions
Abstract: Exposition on Hurwitz quaternions and their arithmetic, which we use to prove Lagrange's four square theorem. Using this arithmetic, we simplify proofs of theorems on integer quaternions by Dickson and Pall that required the theory of quadratic forms.
Slides of the presentation.11h55-12h25 João Camarneiro, 3.º ano, IST/Universidade de Lisboa
Title: Interacting particle systems on manifolds
Abstract: Macroscopical physical systems are usually composed of a very large number of microscopic constituents. The study of such systems cannot be carried out by a deterministic analysis of each individual component, since this would be both computationally and empirically impossible. One solution to this problem is to introduce randomness and look at the stochastic behaviour of the system. In particular, one question of interest is to derive, from the probabilistic description of the microscopic dynamics, a partial differential equation which describes the evolution in time of certain macroscopic quantities. This is known as the hydrodynamic limit of an interacting particle system.
In my work, I am trying to define various interacting particle systems on compact Riemannian manifolds and Minkowski space, which in the hydrodynamic limit lead to PDEs such as the heat equation, the porous medium equation, and the fractional diffusion equation.
Slides of the presentation.12h30-14h30 Lunch Break
14h30-15h00 Pedro Roque da Costa, 1.º ano IST/Universidade de Lisboa
Title: Mathematical models for biologically plausible neural networks
Abstract: One of the most successful algorithms for training artificial neural networks, Backpropagation, has certain qualities which distance it from the reality of living neurons. As such, in 2017, Equilibrium Propagation (EP) was proposed, an alternative learning algorithm using an energy-based neural model. However, energy-based models still present some biologically implausible aspects (e.g. requiring symmetric feedback weights). To address these issues, other models were proposed generalizing EP to vector field dynamics and abandoning the need for a global energy function. One of these models was DirEcted Equilibrium Propagation (DEEP).
In this project I am studying vector field dynamics models. A generalization to include leakage in non-input neurons is proposed, solving the stability issues found in DEEP. Furthermore, conditions for convergence of the neuronal dynamics are investigated and an alternative learning rule is explored.
Slides of the presentation.15h05-15h55 Josef Urban, Czech Institute of of Informatics, Robotics and Cybernetics
Title: AI for Automated and Interactive Theorem Proving
Abstract: The talk will give a brief overview of today's methods that combine deductive automated and interactive theorem proving with inductive AI approaches such as machine learning. I would also like to mention topics such as automated formalization, i.e., training semantically-assisted AI systems to produce formal computer-understandable math/code from informal writings.
16h00- Closure