Coloquio Oaxaqueño
Dimitri Leemans
Abstract: The classification of finite simple groups, achieved in the eighties, is one of the most spectacular achievements in mathematics, combining the efforts of hundreds of researchers. Its proof amounts to roughly 15,000 pages. Current revisions of this result aim to bring back to less than 10,000 pages the proof of this amazing result. Simple groups play in group theory the role of prime numbers in number theory. They are the building blocks of groups as the primes are the building blocks of numbers.
The classification gives 18 infinite families and 26 groups that are called sporadics, the largest one being the Monster or Friendly Giant, a group of order 808,017,424,794,512,875,886,
Geometric interpretations of the finite simple groups have been discovered over decades and Jacques Tits' theory of buildings is so far the most unified geometric way of seeing these groups. Only the alternating and the sporadic groups are not covered by Tits' theory.
In this talk, we will focus on polyhedra and polytopes. These very natural geometric objects have been studied for millennia. The most famous ones are probably the five platonic solids, competing with the truncated icosahedron, also known as the buckminsterfullerene.
We will introduce polytopes in an abstract way. We will then focus on two classes of polytopes. Those that have maximum level of symmetry (the regular ones) and those that have all rotational symmetries but no mirror symmetries (the chiral ones).
We will then make the link between these objects and special sets of generators of groups. A regular polytope has an automorphism group generated by involutions. This group is a smooth quotient of a Coxeter group. Similary a chiral polytope has a group generated by elements (not necessarilyinvolutions) and therefore these geometric objects can be classified using their automorphism group.
The finite simple groups that can appear as automorphism groups of regular polytopes have been classified. We will talk briefly about that classification. In the chiral case, this classification is not complete yet. We will explain where we stand nowadays.
The talk will be accessible to a large audience. We do not intend to enter into technical details.
https://sites.google.com/im.unam.mx/coloquioax/
Dr. Bruno Cisneros y Dr. Francisco Delgado Jueves 13:00 hrs. (Horario de la CDMX) El objetivo del coloquio es generar un espacio en donde se planteen temas de investigación actuales en distintas áreas de las matemáticas, de preferencia que se desarrollen en Oaxaca de tal manera que sean asequibles a estudiantes de los últimos semestres de la licenciatura, posgrado y a todos los investigadores en matemáticas. Es por ello que recomendamos a los expositores que las pláticas sean mucho más básicas que las pláticas de un seminario. Si tiene alguna duda recomendamos la página Cómo dar una plática de coloquio de Mónica Clapp y Michael Barot.
Coloquio Oaxaqueño
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