Research Seminar Program (RSP) - UC|UP Joint PhD Program in Mathematics
 
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Description: 11h00 - 12h00

Speaker: Peter Lombaers

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 Silva

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 Bostanabad

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.

 


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.

 

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.

 

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.

 

Place:   Room 0.30, DMat of University of Porto
URL:  https://sites.google.com/view/rspucupphdmath201720188/p%C3%A1gina-inicial
Start Date:   2017-10-18
Start Time:   11:00
See more:   <Main>   <Research Seminar Program - UC|UP MATH PhD Program>  
 
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