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SUMMARY:André Oliveira (CMUP & UTAD\, Portugal)
DTSTART:20210519T150000Z
DTEND:20210519T160000Z
DTSTAMP:20260422T230716Z
UID:What_is_seminars_at_CMA-UBI/1
DESCRIPTION:Title: What is… a moduli space?\nby André Oliveira (CM
UP & UTAD\, Portugal) as part of What is...? [CMA-UBI seminar]\n\n\nAbst
ract\nMathematicians like to classify and organize mathematical objects\,
up to some fixed equivalence relation. Sometimes the objects in question d
o not admit continuous variations and so the classification is given by di
screte invariants. But many other times\, especially for objects coming fr
om algebraic geometry\, the objects admit such variations. Then they are c
lassified by what is known as a moduli space. It turns out that many modul
i spaces are usually themselves algebraic varieties with a very rich geome
try and topology\, under current intensive research. Moduli space theory i
s 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 give
n 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.\n
LOCATION:https://researchseminars.org/talk/What_is_seminars_at_CMA-UBI/1/
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SUMMARY:Yuri Lima (Universidade Federal do Ceará\, Brazil)
DTSTART:20210317T150000Z
DTEND:20210317T160000Z
DTSTAMP:20260422T230716Z
UID:What_is_seminars_at_CMA-UBI/2
DESCRIPTION:Title: What is...Symbolic Dynamics?\nby Yuri Lima (Univer
sidade Federal do Ceará\, Brazil) as part of What is...? [CMA-UBI semin
ar]\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/What_is_seminars_at_CMA-UBI/2/
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SUMMARY:Manuel Silva (Universidade Nova de Lisboa\, Portugal)
DTSTART:20210623T150000Z
DTEND:20210623T160000Z
DTSTAMP:20260422T230716Z
UID:What_is_seminars_at_CMA-UBI/3
DESCRIPTION:Title: What is... Ramsey theory?\nby Manuel Silva (Univer
sidade Nova de Lisboa\, Portugal) as part of What is...? [CMA-UBI semina
r]\n\n\nAbstract\nIn 1928 Frank P. Ramsey\, motivated by philosophical con
siderations\, proved a theorem in his paper “On a problem of formal logi
c”. This result can be viewed as a powerful generalization of the pigeon
hole principle and implies that every large combinatorial structure contai
ns some regular substructure. Since then\, Ramsey Theory has become an imp
ortant area of combinatorics with connections to other fields of mathemati
cs 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 exampl
es of Ramsey-type results\, trying in each case to find some order in a la
rge combinatorial system.\n
LOCATION:https://researchseminars.org/talk/What_is_seminars_at_CMA-UBI/3/
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SUMMARY:Sérgio Mendes (ISCTE – IUL & CMA-UBI\, Portugal)
DTSTART:20210721T150000Z
DTEND:20210721T160000Z
DTSTAMP:20260422T230716Z
UID:What_is_seminars_at_CMA-UBI/4
DESCRIPTION:Title: What is… the Langlands Program?\nby Sérgio Men
des (ISCTE – IUL & CMA-UBI\, Portugal) as part of What is...? [CMA-UBI
seminar]\n\n\nAbstract\nLet f ∈ Z[x] be an irreducible monic polynomial
of degree n > 0 with integer coecients. Given a prime p\, reducing the co
ecients of f modulo p\, gives a new polynomial which can be reducible. A r
eciprocity law is the law governing the primes modulo which f factors comp
letely. The celebrated quadratic reciprocity law\, introduced by Legendre
and completely solved by Gauss\, is the case when f has degree two. Many o
ther reciprocity laws due to Eisenstein\, Kummer\, Hilbert and others lead
to the general Artin’s reciprocity law and (abelian) class eld theory i
n the early 20th century.\n\nIn 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 conjec tures\, connecting number theory\, represent
ation theory (harmonic analysis) and algebraic geometry. It contains all t
he abelian class eld theory as a particular case\, and another special cas
e plays a crucial role in Wile’s proof of Fermat’s Last Theorem.\n\nTh
ere is a vast amount of number theory problems than can be studied in the
framework of the Langlands Program\, namely: (i) non-abelian class eld the
ory\; (ii) several conjectures regarding zeta-functions and L-functions\;
(iii) and an arithmetic parametrization of smooth irreducible representati
ons of reductive groups.\n\nIn this talk we will give an elementary introd
uction to the Langlands Program\, dedicating special attention to the loca
l Langlands correspondence and explain how it can be seen as a general non
-abelian class eld theory. We\nshall concentrate more on examples\, avoidi
ng general and long denitions.\n\nIf time permits\, an application to nonc
ommutative geometry will also be presented.\n
LOCATION:https://researchseminars.org/talk/What_is_seminars_at_CMA-UBI/4/
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