## COLLOQUIA PATAVINA |

Prof. Ciro Ciliberto (Univ. Roma Tor Vergata)

Limiting cases

January 21, 2014

Limiting cases

January 21, 2014

Abstract

A classical tool, used by the algebraic geometers of the so-called
''Italian school'' (in particular by Guido Castelnuovo and Federigo Enriques)
was the study of the so-called ''limiting cases''.
The idea is to consider geometric objects varying in a family and
''degenerating'', in the limit, to very particular configurations. These can
be on one side very complicated, on the other may be simpler than the objects
one started with. The point is to deduce properties of the general object of
the family from those, hopefuly simpler, of the ''limit''. This technique,
classically proved to be very fruitful, but was also very much criticized from
the viewpoint of mathematical rigour. However in recent times it was revived
and put on a firm basis, and it is today an essential tool for the resolution
of difficult problems. In this talk I intend to start from very simple and
concrete examples, at the level of secondary school, and, as such, already
considered by Emma Castelnuovo (who inherited them from her father and uncle).
These examples will illustrate the usefulness of these methods, but also the
conceptual difficulties which they present. I will then try to illustrate some
open research problems which can be attacked with these techniques and the
interesting results which is possible to prove using them.

Short bio

Ciro Ciliberto is Full Professor in Higher Geometry at University of Rome ``Tor Vergata" since 1985.
He obtained the Degree in Mathematics at the University of Naples in 1973. He was: Assistant
professor at the University of Naples from 1974 to 1980; Professor of Mathematics from 1977 to
1978 and of Algebraic Geometry from 1978 to 1980 at the University of Naples; Full professor of
Higher Mathematics at the University of Lecce in 1980 where he has been working from 1980 to
1981; Full professor of Algebraic Geometry at the University of Naples from 1981 to 1985.
Prof Ciliberto was Vice-president of the Istituto Nazionale di Alta Matematica ``F. Severi" in the years 1990-1995, a member of the scientific committee of the same Institute in the years 1995-1999, and ``Professore distaccato" at the ``Centro Interdisciplinare B. Segre" of the Accademia Nazionale
dei Lincei in the years 1993-1996.
He was Director of the Doctorate School in Mathematics at the University of Roma Tor Vergata in the
years 1990-1994, and a member of the scientific committee of the same school afterwards.
He is currently President of the ``Unione Matematica Italiana" since June 1, 2012.

He is member of the Editorial board of Note di Matematica (University of Lecce), Rendiconti dell'Accademia dei Lincei (edited by the European Mathematical Society), Ricerche di Matematica (edited by Springer Verlag), Lettera Matematica Pristem (edited by Springer Verlag), the book series Unitext (Springer Verlag Italia, see the web site http://www.springer.com/series/5418), Heritage of European Mathematic (edited by European Mathematical Society). Editor in chief of UMI Lecture Notes in Matehmatics (starting on Jan. 1, 2013). Co-editor in chief of Rendiconti del Circolo Matematico di Palermo (edited by Springer Verlag). Director of the Bollettino della Unione Matematica Italiana and of Matematica nella Societa' e nella cultura, Rivista della Unione Matematica Italiana (starting on Jan. 1, 2013).

His home page is at http://www.mat.uniroma2.it/~cilibert/

He is member of the Editorial board of Note di Matematica (University of Lecce), Rendiconti dell'Accademia dei Lincei (edited by the European Mathematical Society), Ricerche di Matematica (edited by Springer Verlag), Lettera Matematica Pristem (edited by Springer Verlag), the book series Unitext (Springer Verlag Italia, see the web site http://www.springer.com/series/5418), Heritage of European Mathematic (edited by European Mathematical Society). Editor in chief of UMI Lecture Notes in Matehmatics (starting on Jan. 1, 2013). Co-editor in chief of Rendiconti del Circolo Matematico di Palermo (edited by Springer Verlag). Director of the Bollettino della Unione Matematica Italiana and of Matematica nella Societa' e nella cultura, Rivista della Unione Matematica Italiana (starting on Jan. 1, 2013).

His home page is at http://www.mat.uniroma2.it/~cilibert/