28377 - Geometry 3

Learning outcomes

At the end of the course, the student has the correct mathematical interpretation of curves and surfaces in space and the foundations of the theory of functions of a complex variable, with particular emphasis on the geometric viewpoint. They know how to use the acquired knowledge to analyze fundamental concepts and classic examples. They are capable of applying such knowledge to other mathematical disciplines and to solving simple problems posed by applied sciences. They possess learning skills and a high standard of knowledge and competence, such that it allows them to access lectures and programs of second-level degree courses, particularly in the study of more advanced topics in differential and complex geometry.

Course contents

First semester (module 1)

Covering spaces: lifting properties, universal cover, classification of covering spaces, deck transformations.

Cell complexes. Classification of topological surfaces.

Differentiable curves in R^3. Differentiable surfaces in R^3: first and second fundamental form, curvature, Theorema Egregium, Gauss-Bonnet theorem. Notes on abstract differentiable manifolds.

 

Second semester (module 2)

Complex analysis in one variable: power series and analytic functions; holomorphic functions and Cauchy-Riemann equations; differential forms and integrations; Cauchy's integral formula and fundamental theorems of the theory of holomorphic functions of one complex variable.

Elements of Riemann surfaces theory: abstract surfaces; complex structures on surfaces; holomorphic forms and fields on surfaces; Riemann-Hurwitz formula; genus-degree formula; elliptic curves; hyperbolic surfaces.

Readings/Bibliography

• Hatcher, Algebraic Topology
• Manetti, Topologia
• Do Carmo, Differential Geometry of Curves & Surfaces
• Pressley, Elementary Differential Geometry
• Gallier & Xu, A Guide to the Classification Theorem for Compact Surfaces
• J. Milnor, Topology from the Differential Viewpoint
• S. Donaldson, Riemann Surfaces
• Benedetti & Petronio, Lectures on Hyperbolic Geometry
• Freitag Busam, Complex Analysis
• Stein & Shakarchi, Complex Analysis
• Miranda, Algebraic curves and Riemann surfaces

Teaching methods

Lectures with blackboard.

Assessment methods

The assessment is conducted independently for the two semesters of the course. Once both semesters have been passed with grades of at least 18/30, the exam can be recorded (with the grade being the average of the grades obtained in the two semesters).

The grade for a single semester is valid for 12 months. In other words, the two semesters must be passed within 12 months of each other.

For students who attended the Geometry 3 course before the academic year 2023/24, the grades obtained in the individual semesters with Professors Manaresi or Idà remain valid. In this case, the validity is exceptionally extended to 24 months.

 

Examination methods for each of the two semesters

There is a written exam followed by an oral exam. Both exams must be taken in the same exam session. Access to the oral exam is granted by scoring at least 16/30 in the written exam. The final grade is determined by the oral exam.

Handing in your work at the end of a written exam invalidates any previously obtained grade in a written or oral exam. The oral exam cannot be repeated without first passing the written exam again.

The written exam lasts 2 hours. During the written exam, you are allowed to have with you two A4 sheets, handwritten in a reasonable size (no more than one line of text per 1/2 cm), containing any result deemed useful for the exam. No other reference materials, such as books or notes, are allowed.

Office hours

See the website of Giovanni Paolini

See the website of Maria Beatrice Pozzetti