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Tuesday, 9th Nov 2010 | 12:25 pm

Domani e dopodomani non c'e' lezione perche' sono a Bari.
Ci sara' di nuovo lezione di recupero venerdi' alle 14.
La mia prossima assenza prevista e' la settimana del 30 novembre, quando saro' di nuovo a Bari.
Per Natale, sicuramente non faccio lezione il 23 dicembre.

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Lecture 6

Friday, 5th Nov 2010 | 12:19 pm

Definition of subspace with K functions. proof of the universal property.

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Lecture #5

Thursday, 4th Nov 2010 | 12:10 pm

Definition of space with K functions. Let K be a field. A space with K functions is a pair (X,O_X) where X is a topological space, and for every U open subset we are given a subset O_X(U) of all functions from U to K (called regular functions on U), such that
1) if f,g are regular on U, then so are f+g and fg;
2) constant functions are regular:
3) if f is regular on U and for each p in U f(p) isn't zero, then 1/f is also regular in U;
4) if f is regular on U, then the locus of points in U where f is zero is closed in U;
5) if f is regular on U and V\subset U is open, then f restricted to V is regular on V;
6) if f:U->K is a function and there is an open cover U_i of U such that f restricted to each U_i is regular, then f is regular on U.

Example. X any topological space, K any topological field such that points are closed, regular functions = continuous unctions
Example X a manifold, K=reals, regular =C^\infty.

There are notes about this on my website.

da finire.

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Lecture 4

Thursday, 28th Oct 2010 | 09:05 am

Definition of manifold structure on P^n.
Exercise 1: show that lines in R^2 are a 2-dimensional manifold, with one chart being y=mx+q.
Exercise 2: figure out one chart for lines in R^3.

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Lecture 3

Wednesday, 27th Oct 2010 | 11:02 am

Inverse function theorem.
Local coordinates for R^n near a point.
Implicit function theorem.
Local coordinates version of impl fcn thm
Corollary: another proof that S^n is a manifold

PS I'll try to post notes on this lecture next week.

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Informazioni importanti

Wednesday, 13th Oct 2010 | 12:12 pm

1) Questa settimana non c'è kezione causa blocco della didattica in tutta la facoltà.
2) la settimana prossima non c'è lezione perché sono via da Trieste.
3) lezioni regolari il 27 e 28 ottobre. In quella settimana vorrei discutere della possibilità di introdurre qualche ora di recupero, visto che dovrò fare altre assenze per motivi di lavoro, sia convegni che un concorso in cui sono in commissione.
4) niente lezione il 3 novembre (san Giusto).

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