66731 - Algebra and Geometry for applications

Learning outcomes

The student is supposed to acquire some advanced knowledge  in algebra and geometry and to be able to apply them to problems .

Course contents

Module I (24 hours, Mirella Manaresi)
Polynomials in one variable with coefficients in a field, resultant of two polynomials.
Polynomials in several variables with coefficients in a field and their properties. Monomial orders. Groebner basis of an ideal of the polynomial ring and Buchberger algorithm.
Systems of polinomial equations and elimination theory. Resultants and elimination ideals.
Application of Groebner basis and elimination theory to the study of  singular points of curves and surfaces, impliticization problems, interpolations problems, cinematic problems of robotics.

Module II (24 hours, Francesca Cagliari):

1. Classification of compact surfaces (14 hours):
Compact surfaces with and without boundary. Fundamental group and van Kampen's theorem. Euler characteristic for a triangulated manifold.
Classification of compact surfaces with and without boundary.

2. Curves and surfaces (10 hours):
Planar Curves: Cycloid, Epicycloid and hypocycloid, Lemniscate, Cardioid, Catenary, Cissoid.  Examples. 
S pace curves: cylindrical helix, helicoid, asteroid.
Parametric surfaces in R³. Study of some remarkable surfaces, such as, for example: the ball, the ellipsoid, the torus, the paraboloid and the hyperboloid.

The two modules will be in parallel. Some topics of one module will be used in the other one.

Readings/Bibliography

For Module I:
D.Cox - J.Little - D.O'Shea: Ideals, Varieties and Algorithms. 3rd Ed.  Undergraduate Texts in Mathematics. Springer Verlag, New York 2007

For Module II:
L.Christine Kinsey: Topology of Surfaces . Springer Verlag
R. Caddeo, A. Gray: Lezioni di geometria differenziale. Curve e Superfici. vol. 1 Cooperativa Universitaria Editrice Cagliaritana (2001)

Teaching methods

Lectures and laboratory exercise sessions. Sheets of exercises will be handed out during the lectures (see http://www.dm.unibo.it/~manaresi/ and http://www.dm.unibo.it/~cagliari/) , in addition to the ones available in the suggested textbooks. 

During laboratory sessions some of these exercises will be discussed, some others must be prepared by  students before the exam.

  In the office hours students will be coached individually.

Assessment methods

Oral exam, starting from the discussion of the exercises that students must solve and give to the teachers at least a week before the exam. For the solution of exercises of Module I students need to use COCOA or Singular or some other software for symbolic computation available in the Laboratories.

The date of the exam must be fixed with the teachers of the course.

Teaching tools

Lectures and laboratory exercise sessions. Sheets of exercises will be handed out during the lectures (see http://www.dm.unibo.it/~manaresi/ and http://www.dm.unibo.it/~cagliari/) , in addition to the ones available in the suggested textbooks.  In the office hours students will be coached individually.

For Module one will be used the free software COCOA (see ftp://cocoa.dima.unige.it/cocoa) and Singular (see http://www.singular.uni-kl.de/).

In the students Laboratories of the Department of Mathematics students can be use also Macaulay, Maple, Reduce,  Mathematica.

In the office hours students will be coached individually.

Links to further information

http://www.dm.unibo.it/~manaresi/

Office hours

See the website of Mirella Manaresi

See the website of Francesca Cagliari