Colloquium

Past sessions

Imaginary time flows to reality

The mathematical expressions of the idea of quantization are a source of rich interdisciplinary relations between different areas in geometry and other subjects such as analysis and representation theory. In this colloquium, after a gentle description of some structures of modern geometry and of the quantization problem, we will describe “flows in imaginary time” and will give an idea of their role in quantization (in particular, in so-called real polarizations) and Kahler geometry. Finally, we will give a light description of some recent results.

Moduli Spaces and their Polynomial Invariants

The idea of symmetry, present in ancient civilizations, became part of our mathematical tools with the introduction of groups and group actions by Galois and Lie. Understanding the geometry of the spaces of orbits, and the algebra of the invariant functions on them is extremely useful both in algebraic and in geometric classification problems. These problems were greatly unified by the notion of a moduli space, introduced by Riemann and developed by Mumford.

In this colloquium, we present some classification problems in algebra and geometry which give rise to interesting moduli spaces — polygon spaces, character varieties, Higgs bundles —, and show some of the tools used in their study, such as polynomial invariants (named after Euler, Poincaré, Hodge, etc.). In some simple cases, there are explicit formulas for these polynomials (some just recently computed), and we end up announcing an interesting form of Langlands duality for character varieties of free groups.

Testing General Relativity with Gravitational Waves

This year marks the centenary of a pivotal breakthrough: the confirmation that gravity can be described as spacetime curvature. Among the most outrageous predictions of the theory are the existence of black holes and gravitational waves.

Gravitational waves offer a unique glimpse into the unseen universe in different ways, and allow us to test the basic tenets of General Relativity, some of which have been taken for granted without observations: are gravitons massless? Are black holes the simplest possible macroscopic objects? Do event horizons and black holes really exist, or is their formation halted by some as yet unknown mechanism? Do singularities arise in our universe as the outcome of violent collisions? Can gravitational waves carry information about the nature of the elusive dark matter?

I will describe the science encoded in a gravitational wave signal and what the upcoming years might have in store regarding fundamental physics and gravitational waves.

Poster

Mimetic Discretization Methods

Mimetic discretizations or compatible discretizations have been a recurrent search in the history of numerical methods for solving partial differential equations with variable degree of success. There are many researches currently active in this area pursuing different approaches to achieve this goal and many algorithms have been developed along these lines. Loosely speaking, "mimetic" or "compatible" algebraic methods have discrete structures that mimic vector calculus identities and theorems. Specific approaches to discretization have achieved this compatibility following different paths, and with diverse degree of generality in relation to the problems solved and the order of accuracy obtainable. Here, we present theoretical aspects for a mimetic method based on the extended Gauss Divergence Theorem as well as examples using this method to solve partial differential equations using the Mimetic Operators Library Enhanced (MOLE).

Poster

On ARL-unbiased Control Charts

A control chart is a graphical device used to monitor a parameter of a measurable characteristic. An observation of the plotted control statistic beyond the control limits suggests a change in the parameter being monitored. The control limits of most charts tend to be set ignoring the skewness character of the control statistic and this may dramatically affect their average run length (ARL) performance. We derive several ARL-unbiased control charts whose ARL profiles attain a pre-specified maximum when the parameter is on-target. R is used to provide striking illustrations of how ARL-unbiased charts work in practice.

Cryptography after Quantum Computation

Due to Shor's algorithm and the imminence of quantum computers, all cryptographic standards are being rethought. In this talk, we address a security functionality associated with privacy, namely oblivious transfer. We present some cryptographic solutions proposed within the Security and Quantum Information Group at Instituto de Telecomunicações. Specifically, we discuss the proof of security for methods based on quantum information (quantum cryptography) and for classical methods, based on hardness assumptions conjectured to be robust to quantum attacks (post-quantum cryptography). We conclude by examining the advantages and disadvantages of each approach, as well as their implications in cryptography and technology.

Francisco Gomes Teixeira and the internationalization of Portuguese mathematics

Francisco Gomes Teixeira (1851-1933) was a remarkable Portuguese mathematician, one of the greatest of the nineteenth and twentieth centuries and certainly the most prolific in that period. He maintained regular and intense correspondence with the greatest mathematicians of his time. In this lecture we will present some aspects of this correspondence.

Poster

Dynamical systems for arithmetic schemes

We construct a natural infinite dimensional dynamical system whose periodic orbits come in compact packets $P$ which are in bijection with the prime numbers $p$. Here each periodic orbit in $P$ has length $\log p$. In fact a corresponding construction works more generally for finitely generated normal rings and their maximal ideals or even more generally for arithmetic schemes and their closed points. Moreover the construction is functorial for a large class of morphisms. Thus the zeta functions of analytic number theory and arithmetic geometry can be viewed as Ruelle type zeta functions of dynamical systems. We will describe the construction and what is known about these dynamical systems. The generic fibres of our dynamical systems are related to an earlier construction by Robert Kucharczyk and Peter Scholze of topological spaces whose fundamental groups realize Galois groups. There are many unproven conjectures on arithmetic zeta functions and the ultimate aim is to use analytical methods for dynamical systems to prove them.

Poster

From randomness to determinism

In this seminar I will describe how to derive rigorously the laws that rule the space-time evolution of the conserved quantities of a certain stochastic process. The goal is to describe the connection between the macroscopic equations and the microscopic system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly. Depending on the choice of the transition probability that particles obey, we will see that the macroscopic laws can be of different nature.

Poster

Mathematics and Cardiovascular Diseases

Mathematical modeling and simulations of the human circulatory system is a challenging and complex wide-range multidisciplinary research field.

In this talk we will consider some mathematical models and simulations of the cardiovascular system and comment on their significance to yield realistic and accurate numerical results, using stable, reliable and efficient computational methods.

Poster

Mathematical Relativity and the Cosmic Censorship Conjecture

Einstein's general theory of relativity has always been a great catalyst for mathematical development, from Riemannian geometry to partial differential equations. In this talk we give a mathematical history of the subject, aiming to explain one of its most important open problems, the strong cosmic censorship conjecture.

Poster

Qualitative Theory of Differential Equations — Morse-Smale Evolution Processes

The qualitative theory of differential equations aims at the description of the asymptotic behavior of solutions of differential equations.

We survey aspects of the Morse-Smale theory, from dynamical systems generated by ordinary differential equations to evolution processes generated by non-autonomous partial parabolic differential equations.

Poster

The Interplay Between Dispersive Partial Differential Equations and Fourier Analysis

Partial differential equations have always been a subject of fruitful interaction with Fourier analysis, starting precisely with the study of the heat equation. In the last few decades, nonlinear partial differential equations of hyperbolic and dispersive type, in particular, have been at the center of a significant new interplay and mutual progress between these two fields, through the works of prominent mathematicians like Tosio Kato, Charles Fefferman, Jean Bourgain, Carlos Kenig and Terence Tao.

In this talk, we will review some of the basic concepts and ideas that play a central role in this connection between techniques from Fourier analysis and properties of solutions of dispersive PDEs, covering topics like Strichartz estimates, smoothing effects, local and global well posedness of initial value problems at low regularity, among others.

Poster

The World is Incomplete, Reducible and Real

Some results and reduction techniques for proving decidability of mathematical theories and completeness of logics are presented. The crucial role of the theory of real closed ordered fields is explained. Selected illustrations from Euclidean Geometry to Quantum Logic are discussed.

Poster

Sums and products of equivalence orbits

Similarity and equivalence of matrices over fields are well-understood relations, and that understanding is elementary, specially in the case of equivalence. For matrices over rings – e.g. the integers – the situation is different. Equivalence is still simple and its study leads to the concept of invariant factors of a matrix. About these, several interesting basic questions can be raised. This talk will address two of those questions: how do invariant factors behave under matrix addition and multiplication? Some things are known about these problems, and the second one – already completely solved for certain classes of rings – has deep relations with other parts of mathematics.

Poster

From Toeplitz matrices to black holes, and beyond

What do Toeplitz matrices, random matrix models, orthogonal polynomials, Painlevé transcendents, the KdV equation, and black holes, seemingly very unrelated subjects, have in common? These, and a variety of other mathematical problems, can be studied by means of the so called Riemann-Hilbert method. In this talk we briefly describe what a Riemann-Hilbert problem is and present several recent applications, from the spectral properties of Toeplitz operators to exact solutions of Einstein field equations.

Poster

Decide to Win

In the finance world there are many problems related with the optimal time to undertake some action. One of the most common problems is the derivation of the exercise time of an American option. But also in decisions regarding investments this question is essential. Questions like: when to adopt a new technology? When to invest in a new airport? When should suspension out of production occur? These problems have a real impact in the economy, and therefore one needs a proper mathematical formulation and solution for them.

In this talk we address such problems. They are known in the literature as optimal stopping problems, closely related with free boundary problems. One way to solve the optimisation problem is to use a variational inequality, known as the Hamilton-Jacobi-Bellman equation (HJB, for short). In the first part of the talk we present briefly the mathematical formulation and tools to solve such problems, and in the second part we show some applications, providing solution and discussion.

The Mathematics of the Brain

With an extremely large number of functional units (neurons) and an even larger number of connections between them (synapses) the brain is maybe the most complex system that Science has ever tried to explain and simulate. Neuroscience is nowadays a multidisciplinary field which mobilizes all over the world thousands of scientists having different profiles, from medical doctors to computer engineering. including mathematicians. The mathematical tools of Neuroscience are getting more and more complex, giving rise to new branches, such as Mathematical Neuroscience or Computational Neuroscience. In this talk we will visit some of the most well-known mathematical models, emphasizing the role that mathematical topics such as Differential Equations, Numerical Analysis or even Algebraic Topology play in the modelling of the brain and the nervous system.

Poster
Colóquio_PLima

O Poder dos Sistemas Analógico-Digitais

Introduz-se um modelo de computação analógico-digital em que a componente digital é o modelo padrão (e.g. máquina de Turing) e a componente analógica é o resultado de uma medição, e.g. obtida através de sensor de grandeza física. A medição atua como oráculo e a troca de informação entre a componente digital e a componente analógica decorre no tempo intrínseco ao processo físico. As medições realizadas pelo acoplamento analógico-digital podem ser executadas através de diferentes protocolos, de estocástico a determinístico, nomeadamente fazendo variar a precisão da medição entre finita e infinita. Discute-se a natureza das medições que podem ser efetuadas pelos sistemas analógico-digitais. Estabelece-se que o poder computacional destes sistemas que operam num número polinomial de passos é o das classes computacionais $\mathit{BPP//}\log^{(k)}\!\star$. Por fim discutem-se os limites da simulação computacional de sistemas físicos e comparam-se os conceitos de número mensurável segundo a abordagem de Geroch e Hartle e a nossa.

Poster
Extended abstract
Apresentação

O mais impenetrável ABC

Neste colóquio tratar-se-á de um famoso problema em aberto em Teoria de Números, conhecido como conjectura ABC. Mostra-se por que razão esta Conjectura é provavelmente o problema mais importante da área — a seguir à Hipótese de Riemann —, evidenciando as suas fortíssimas e surpreendentes consequências. Como exemplo destas consequências fornecer-se-á uma demonstração do Teorema de Fermat (módulo ABC). Finalmente, detalhar-se-á a bizarra situação actual do problema, com uma possível demonstração que a comunidade matemática se tem esforçado por decifrar ao longo da última década — sem grande sucesso até hoje.