# Colloquium

## Past sessions

### 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.

Poster

### Para além de 12 e 24

A geometria simpléctica e a matemática discreta estão fortemente interligadas devido à existência de acções hamiltonianas de toros. Estas acções estão associadas a uma aplicação (denominada aplicação momento) que transforma uma variedade simpléctica compacta num polítopo convexo. Nesta palestra vamos concentrar-nos numa classe de polítopos, definida por Batyrev no contexto de simetria-espelho e que tem atraído muita atenção recentemente: os polítopos reflexivos. Em particular, vamos ver como as famosas propriedades "12 e 24" em dimensão 2 e 3 podem ser generalizadas com a ajuda da geometria simpléctica.

Poster

### Random Matrices and Number Theory

I will review conjectural connections between some important problems in analytic number theory, such as the Riemann Hypothesis, and random matrix theory, which plays a significant role in many areas of mathematical physics. These connections have had a major impact on our understanding of number-theoretic $L$-functions (e.g. the Riemann zeta-function).

### Univalent Foundations of Mathematics

I will outline the main ideas of the new approach to foundations of practical mathematics which we call univalent foundations. Mathematical objects and their equivalences form sets, groupoids or higher groupoids. According to Grothendieck's idea higher groupoids are the same as homotopy types. Therefore mathematics may be considered as studying homotopy types and structures on them. Homotopy type theories, the underlying formal deduction system of the univalent foundations allows one to reason about such objects directly.

### Stability results for sumsets in $\mathbb{R}^n$

Given a Borel set $A$ in $\mathbb{R}^n$ of positive measure, one can consider its semisum $S=(A+A)/2$. It is clear that $S$ contains $A$, and it is not difficult to prove that they have the same measure if and only if $A$ is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of $S$ is close to the one of $A$, is $A$ close to his convex hull? More in general, one may consider the semisum of two different sets $A$ and $B$, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When $n=1$, one can approximate a set with finite unions of intervals to translate the problem onto $\mathbb{Z}$, and in the discrete setting this question becomes a well studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I'll review some results in the one-dimensional discrete setting, and show how to answer to this problem in arbitrary dimension.

### Tight-frame Approach for Image Processing

In many practical problems in image processing, the observed data sets are often incomplete in the sense that features of interest in the image are missing partially or corrupted by noise. The recovery of missing data from incomplete data is an essential part of any image processing procedures whether the final image is utilized for visual interpretation or for automatic analysis. In this talk, we present our tight-frame algorithm for missing data recovery. Tight-frames are extension of wavelets. They generalize orthonormal wavelet systems and give more flexibility in filter designs. We begin our talk with an introduction of tight-frames. Then we illustrate how to apply the idea to different image processing applications such as inpainting, impulse noise removal, super-resolution image reconstruction and video enhancement.

Joint work with X. M. Yuan (Hong Kong Baptist University) and J. F. Yang and M. Tao (Nanking University).

### Topology of representation varieties of surface groups

This will be a survey talk on some aspects of the geometry and topology of moduli spaces of representations of surface groups into Lie groups. I will discuss recent generalizations of the techniques of Atiyah and Bott on equivariant Morse theory. These extend results on stable bundles to Higgs bundles and associated moduli spaces, which correspond to representation varieties into noncompact Lie groups.
Partially funded by LARSyS through CAMGSD.

### Operator Theory and Complex Geometry

One approach to the study of multivariate operator theory on Hilbert space is the study of algebras of operators. Many algebras of operators act on natural Hilbert spaces of holomorphic functions defined on some complex domain in ${ℂ}^{m}$. Examples are the algebras of bounded multipliers on the Hardy and Bergman spaces for the unit ball and polydisk in ${ℂ}^{m}$. One approach to the study of such algebras is to adapt the complex geometric methods of M. Cowen and the author to the context of Hilbert modules over the polynomial algebra in several variables. In this talk, I will describe this line of study with an emphasis on concrete examples as well as a focus on several results in operator theory whose proof rests on concepts and techniques from algebraic and complex geometry.

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The Mathematics Colloquium is a series of monthly talks organized by the Department of Mathematics of IST, aiming to be a forum for the presentation of mathematical ideas or ideas about Mathematics. The Colloquium welcomes the participation of faculty, researchers and undergraduate or graduate students, of IST or other institutions, and is seen as an opportunity of bringing together and fostering the building up of ideas in an informal atmosphere.

Organizers: Conceição Amado, Lina Oliveira e Maria João Borges.