CICLO DI SEMINARI SU TEORIA DI MORSE
Il prof. Andrei PAJITNOV (Univ. DI Nantes), invitato come professore visitatore dell’Ateneo, sarà presente nel nostro Dipartimento nel periodo 9 aprile - 8 maggio 2004 e terrà un ciclo di seminari sulla Teoria di Morse, della durata di circa 12 ore.
Tali seminari avranno luogo due giorni a settimana: nel primo, la durata del seminario sarà di un’ora e mezzo, con intervallo, e vi saranno svolti gli argomenti indicati qui di seguito; il secondo, della durata di un’ora, sarà dedicato ad approfondimenti degli argomenti svolti precedentemente, dando chiarimenti e fornendo esempi agli interessati.
Il primo appuntamento è previsto per mercoledì 14 oppure giovedì 15 aprile, ad esempio alle ore 14-15.45. In tale occasione si deciderà il calendario definitivo del ciclo, assieme agli interessati.
Gli interessati possono mettersi in contatto con Stefano Marchiafava (marchiaf@mat.uniroma1.it, tel. 06-49913246).
1) Morse functions and their
gradients.
2) The Morse complex.
These two sections are devoted to the classical Morse theory, especially
to the aspects which are less well known, and usually not included in the
standard courses on Morse theory.We give a detailed construction of the chain complex associated
with a Morse function and its gradient (the Morse complex).
3) Circle-valued Morse functions
and the Novikov complex.
We give the construction of the
Novikov complex, prove its properties, and deduce the Novikov inequalities for
the number of the critical points of a circle-valued Morse function.
4) Cellular gradients and
generic rationality of the Novikov complex.
For any circle-valued Morse function f there is a particular class
of gradients of f, for which the
gradient descent map has remarkable
properties, resembling the properties
of cellular maps. For this gradients the Novikov complex is defined over
the ring of rational functions of one
variable.
We shall define this class of gradients, and prove the properties cited
above in the general context of the Novikov exponential growth conjecture.
5) Circle-valued Morse theory for knots and links.
A knot is called fibred if its complement in the three-dimensional
sphere is the total space of a fibration over a circle. In other words, a knot
is called fibred, if there is a circle-valued function on its complement
without critical points. If the knot is not fibred, any circle-valued Morse
function on its complement has a non-empty set of critical points, and their
number can be estimated from below by the Morse-Novikov theory. The invariants
which appear here are related to the Alexander polynomials and twisted
Alexander polynomials of the knot. The recent progress in the software related
to knot theory makes these invariants computable. We shall discuss this new
branch of the knot theory, which is actively developing at present.
6) Applications to geometry and physics.
We shall present the applications
of the circle-valued Morse theory, in particular, applications to the problem
of existence of periodic solutions of the equations of Kirchhof type
(S.P.Novikov) and the (generalized) Arnol'd conjecture about Lagrangian
Intersections.