https://us02web.zoom.us/meeting/register/tZYtf-6qqzsoH9LZzG5PiA5ypw_US1b5gcGF
Dopo l'iscrizione, verrà inviata una email di conferma con le informazioni necessarie per entrare nella riunione.
Contatto di riferimento: Maria Teresa BORGATO
Per informazioni scrivere a: federica.ferretti5@gmail.com
Date dei seminari:
Titolo del primo seminario: Infinitesimal bialgebras, Rota-Baxter operators and related structures.
Abstract: In the first seminar we will present the concepts of pre-Lie algebra, dendriform algebra, (quasitriangular) infinitesimal bialgebra, Rota-Baxter operator, and various relations between these concepts. For instance, we will show how to obtain a pre-Lie algebra from an infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra.
Titolo del secondo seminario: Properties and examples of Rota-Baxter operators.
Abstract: In the second seminar we will recall the definition of Rota-Baxter operators (introduced in the first seminar) and present some of their properties, some examples and structure theorems.
Titolo del terzo seminario: Examples and properties of infinitesimal bialgebras.
Abstract: In the third seminar we will present examples of infinitesimal bialgebras (mainly, the path algebra of a quiver) and, time permitting, some properties of infinitesimal bialgebras.
Titolo del quarto seminario: More properties of infinitesimal bialgebras .
Abstract: In the fourth seminar we will continue to present properties of infinitesimal bialgebras, particularly properties of quasitriangular infinitesimal bialgebras.
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- Lunedì 29 Maggio, ore 11
- Martedì 30 Maggio, ore 11
- Giovedì 1 Giugno, ore 11
- Lunedì 5 Giugno, ore 11
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Abstract:
A classical tool to get numerical invariants of a curve singularity is the study of its value
semigroup. In case of a one branch singularity this semigroup is a numerical semigroup
(i.e. a submonoid of N with finite complement in it); in case the singularity has h branches,
this semigroup is a subsemigroup of Nh, belonging to the class of the so-called ”good semi-
groups”. Despite their name, the combinatoric of good semigroups is quite problematic;
moreover, for h ≥ 2, it is an open problem to understand which good semigroups can be
realized as value semigroups.
In case of a plane singularity with one branch, an old result of Ap ́ery shows that there
is a particularly strict connection between the value semigroups of the singularity and
of its blowup; this connection is obtained using a particular set of generators of the
semigroup, named ”Ap ́ery set”. In fact, using that result, it is possible to show very
easily, that the equisingularity classes given by the multiplicity sequence and by the
value semigroup coincide. In particular, this method allows to reconstruct the numerical
semigroup associated to a plane branch singularity starting from the multiplicity sequence.
When the singularity has more than one branch, in order to generalize the Ap ́ery result,
two main problems arise: firstly, the Ap ́ery set becomes an infinite set; secondly, in the
process of blowing-up it is necessary to deal with semilocal rings, that cannot be presented
as quotients of a power series ring in two variables, as it happens in the local case. These
problems where partially solved twenty years ago in the two branch case, but the general
case is still open.
In my talk, after describing some key definitions and results on value semigroups and
good semigroups of a curve singularity with h branches, I will explain explaining the
Ap ́ery process for a plane branch and then, I will present some recent results obtained in
a joint project with F. Delgado de la Mata, L. Guerrieri, N. Maugeri and V. Micale, that
are a significant progress toward a complete solution of this problem.
Dipartimento di Matematica e Informatica - University of Catania
V.le A. Doria 6, 95125
Catania
Italy
E-mail address: marco.danna@unict.it