ADVANCED GEOMETRY B
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Teaching methods
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Elementary differential geometry of surfaces. Some basic notions of differentiable varieties
At the end of the course, the student will give a seminar on a topic agreed with the teacher for the preparation of which the concepts and methods developed in the course will be used in an essential way.
The maximum score Of 30/30 e lode is obtained if the student, in answering questions from the teacher during his seminar, proves to have understood not only the topic of the seminar, but also the concepts and methods used to prepare it, and the related mathematical proofs.
TRAINING OBJECTIVES.
The course aims to provide advanced theoretical training on the theory of complex algebraic varieties, through a differential and complex analysis based approach.
EXPECTED LEARNING RESULTS.
The student will be able to orient himself in the recent mathematical literature on the subject in question. The acquired notions can be useful for an introduction to some current research topics in complex or algebraic geometry or theoretical physics.
1) Introduction through examples of algebraic curves in the projective plane. Low genus curves, Bezout’s theorem.
2) Introduction to complex and algebraic varieties by sheaves. Affine and projective varieties. Vector fields,, differential forms.
3)Elements of sheaf theory.
4) Sheaf cohomology. Reviews of De Rham theorem, Poincarè duality, Lefschetz fixed point formula.
5) Riemann surfaces qnd their projective models
6) Elements of Hodge theory for complex manifolds.
7) Introduction to Kahler manifolds
8) Weak and Hard Lefschetz theorems
9) Coherent sheaves cohomology
10) Computation of Hodge numbers of projective spaces and projective hypersurfaces
11) Review of some famous open problems, for example the Hodge conjacture
1) Introduction through examples of algebraic curves in the projective plane. Low genus curves, Bezout’s theorem.
2) Introduction to complex and algebraic varieties by sheaves. Affine and projective varieties. Vector fields,, differential forms.
3)Elements of sheaf theory.
4) Sheaf cohomology. Reviews of De Rham theorem, Poincarè duality, Lefschetz fixed point formula.
5) Riemann surfaces qnd their projective models
6) Elements of Hodge theory for complex manifolds.
7) Introduction to Kahler manifolds
8) Weak and Hard Lefschetz theorems
9) Coherent sheaves cohomology
10) Computation of Hodge numbers of projective spaces and projective hypersurfaces
11) Review of some famous open problems, for example the Hodge conjacture
Frontal lessons with direct involvement of students in reading parts of the reference text and question and answer sessions with the teacher. Common elaboration of shared notes on the course topics. Development at home of some of the omitted mathematical proofs.
The teacher is available to receive the students for clarifications and to help them in the development of the final seminar, by appointment by e-mail
riccardo.re@uninsubria.it
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Degree course in: MATHEMATICS