What is…? Seminar

Seminário O que é…? / What is…? seminar

Upcoming seminars | Previous | What is the What is…? seminar? | Organizers

 

Upcoming seminars

 

This seminar series will return in the next academic year 2021/2022. For more information soon.

Previous seminars

Previous seminar are available at CMA-UBI Youtube channel.

> What is… the Langlands  Program? Sérgio Mendes (ISCTE – IUL & CMA-UBI, Portugal)

July, 21th, 2021 | 4 pm (Portugal time, GMT/UTC+0)

Let f ∈ Z[x] be an irreducible monic polynomial of degree n > 0 with integer coefficients. Given a prime p, reducing the coefficients of f modulo p, gives a new polynomial which can be reducible. A reciprocity law is the law governing the primes modulo which f factors completely. The celebrated quadratic reciprocity law, introduced by Legendre and completely solved by Gauss, is the case when f has degree two. Many other reciprocity laws due to Eisenstein, Kummer, Hilbert and others lead to the general Artin’s reciprocity law and (abelian) class field theory in the early 20th century.

In 1967, in a letter to André Weil, Robert Langlands paved the way for what is known today as the Langlands Program: a set of far reaching conjectures, connecting number theory, representation theory (harmonic analysis) and algebraic geometry. It contains all the abelian class field theory as a particular case, and another special case plays a crucial role in Wiles’s proof of Fermat’s Last Theorem.

There is a vast amount of number theory problems than can be studied in the framework of the Langlands Program, namely: (i) non-abelian class field theory; (ii) several conjectures regarding zeta-functions and L-functions; (iii) and an arithmetic parametrization of smooth irreducible representations of reductive groups.

In this talk we will give an elementary introduction to the Langlands Program, dedicating special attention to the local Langlands correspondence and explain how it can be seen as a general non-abelian class field theory. We shall concentrate more on examples, avoiding general and long definitions.

If time permits, an application to noncommutative geometry will also be presented.




 

> What is… Ramsey theory? Manuel Silva (Universidade Nova de Lisboa, Portugal)

 

June, 23th, 2021 | 4 pm (Portugal time, GMT/UTC+0)

In 1928 Frank P. Ramsey, motivated by philosophical considerations, proved a theorem in his paper “On a problem of formal logic”. This result can be viewed as a powerful generalization of the pigeonhole principle and implies that every large combinatorial structure contains some regular substructure. Since then, Ramsey Theory has become an important area of combinatorics with connections to other fields of mathematics such as number theory, ergodic theory, mathematical logic, and graph theory. In the same spirit, Van der Waerden proved in 1927 a regularity result about partitions of the natural numbers. We will see several examples of Ramsey-type results, trying in each case to find some order in a large combinatorial system.

 

> What is… a moduli space? André Oliveira (CMUP & UTAD, Portugal)

May, 19th, 2021 | 4 pm (Portugal time, GMT/UTC+0)

Mathematicians like to classify and organize mathematical objects, up to some fixed equivalence relation. Sometimes the objects in question do not admit continuous variations and so the classification is given by discrete invariants. But many other times, especially for objects coming from algebraic geometry, the objects admit such variations. Then they are classified by what is known as a moduli space. It turns out that many moduli spaces are usually themselves algebraic varieties with a very rich geometry and topology, under current intensive research. Moduli space theory is indeed a vast and intricate topic, whose origins go back to Riemann and which has been behind several Fields Medals (like Mumford, Donaldson or Mirzakhani, just to name a few). Even their rigorous definition is not a trivial matter and, somehow contradicting the title, it will not be given in this talk. The aim is just to provide a general idea of what a moduli space is supposed to be and mainly focus on basic examples.

 

> What is… Symbolic Dynamics? Yuri Lima (Universidade Federal do Ceará, Brazil)

March, 17th, 2021 | 3 pm  (Portugal time, GMT/UTC+0)

Symbolic dynamics is a very useful tool that, in a nutshell, allows to represent dynamical systems by a combinatorial model of sequences of symbols, in a way that many dynamical and statistical properties of the original system are obtained by the study of the combinatorial model. In this talk, we will give a gentle introduction to this topic and explain how a geometrical configuration known as Markov partition allows to construct symbolic dynamics.




 

 

 


What is the What is…? seminar?

 

The Center of Mathematics and Applications of University of Beira Interior (CMA-UBI) is promoting a What is…? seminar series.

The What is…? seminars aim to expose a variety of mathematical themes for a wide audience. The topics go from mathematical culture issues to new and challenging ideas in distinct areas of Mathematics. In the audience we expect mathematicians and graduate students which are not necessarily familiar with the theme. These topics will certainly enrich the mathematical culture of researchers, teachers and students, and spark their curiosity to new subjects.

Due to Covid-19 these seminars will run for the next sessions in an on-line format (ZOOM).

Moreover,

– talks will have from 30 to 45 minutes, plus some additional minutes to questions and comments;

– talks will take place preferably on Wednesdays, 3pm (Portugal time zone: UTC+0/GMT+0, wintertime), but adjustments on schedule can be made;

– talks will not be recorded, unless the speaker suggests; it is also possible to share the notes on the website of the seminar.


Organizers

Rui Pacheco and Helder Vilarinho