2017-2018 Sessions
9th of March, 2018 - Room 2.5, DMat of University of Coimbra
11h30 - 12h00
Speaker: António Goucha¹
Title: Phaseless rank and amoebas
Abstract: The phaseless rank of a nonnegative matrix M is defined to be the least k for which there exists a complex matrix N such that |N|=M, entrywise speaking. In optimization terms, it is the solution to the rank minimization of a matrix under phase uncertainty on the entries. This concept has a strong connection with some algebraic objects, called amoebas. An algebraic amoeba is the image of an algebraic variety under the absolute value map. In this talk we state some results related to phaseless rank and explore its connection with amoebas.
Lunch Break
13h30 - 14h15
Speaker: Rúben Sousa²
Title: Product formulas, generalized convolutions and integral transforms
Abstract: It is well-known that the ordinary convolution is closely related with the Fourier transform. It is therefore natural to ask: for other important integral transforms, can we define generalized convolution operators having analogous properties? Actually, the answer depends on the existence of a product formula for the kernel of the integral transform. In this talk, I will explain the general connection between product formulas, generalized convolutions and integral transforms. I will report on recent progress in constructing the product formula and convolution associated with the index Whittaker transform. Some applications will be presented, and the probabilistic motivation behind this work will be discussed.
14h15 - 15h00
Speaker: Jorge Soares³
Title: A tour in Extreme Value Laws
Abstract: In this talk, we analyse the stochastic process that arises from a dynamical system by evaluating an observable function along a given orbit of the system. Our goal is to give sufficient conditions for the existence of an Extreme Value Law for the considered process. We will present an example where the observable function is maximized in a Cantor Set and we will prove the existence of an Extreme Value Law with Extremal Index less than 1.
¹ António Goucha is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Optimization, under the supervision of Professor João Gouveia.
² Rúben Sousa is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in Analysis, under the supervision of Professor Semyon Yakubovich (Univ. Porto) and Professor Manuel Guerra (Univ. Lisboa).
³ Jorge Soares is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in Dynamical Systems, under the supervision of Professor Jorge Milhazes de Freitas.
18th of October, 2017 - Room 0.30, DMat of University of Porto
11h00 - 12h00
Speaker: Peter Lombaers¹
Title: Integers and Ideals: There and Back Again
Abstract: In number theory, when you try to solve an equation in a number field, it is often more convenient to work with ideals than with integers. This stems from the fact that ideals have unique factorization, but integers may not. I will explain the advantages and difficulties of this method using concrete examples.
Lunch Break
13h30 - 14h30
Speaker: Willian Silva²
Title: Introducing (T,V)-categories
Abstract: In this seminar we introduce the concept of (T, V)-categories through its fundamental examples. In order to do so, we explore the concepts of monads and quantales, also with examples. We finish relating to the work on cartesian closed categories.
14h30 - 15h00
Speaker: Mina Saee Bostanabad³
Title: SOS versus SDSOS polynomial optimization
Abstract: It is NP-hard to decide whether a polynomial is nonnegative, however, semide nite programming can be used to decide whether a polynomial is a sum of squares of polynomials (SOS) in a practically e cient manner. In the context of polynomial optimization, it has become usual to substitute testing for nonnegativity with testing for SOS. Since there are much fewer sums of squares than nonnegative polynomials, we get only a relaxation and one that does not scale very well with the number of variables and degree of the polynomial. Recently, Ahmadi and Majumdar introduced a more scalable alternative to SOS optimization that they refer to as scaled diagonally dominant sums of squares (SDSOS). The idea is searching for sums of squares of binomials, instead of general polynomials, which leads to a more scalable SOCP problem. In this presentation, we investigate the quantitative relationship between sums of squares of polynomials and scaled diagonally dominant polynomials. More speci cally, we use techniques established by Blekherman to bound the ratio between the volume of the cones of these two classes of polynomials, showing that there are signi cantly less SDSOS polynomials than SOS polynomials. This drawback can be circumvented by using a recently introduced basis pursuit procedure of Ahmadi and Hall that iteratively changes the polynomial basis to a more suitable relaxation. We illustrate this by presenting a new application of this technique to an optimization problem.
¹ Peter Lombaers is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in Number Theory, under the supervision of Professor António José Machiavelo.
² Willian Silva is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Category Theory, under the supervision of Professor Maria Manuel Clementino.
³ Mina Saee Bostanabad is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Optimization, under the supervision of Professor João Eduardo da Silveira Gouveia.