INSTITUTIONS OF SUPERIOR GEOMETRY

Course objectives

Knowledge and understanding: at the end of the course the student will have acquired the notions and the results basics related to singular homology, to the study of differentiable varieties and a fair knowledge of Riemann's theory of surfaces. Apply knowledge and understanding: at the end of the course the student will be able to solve even complex problems that require the use of techniques related to de Rham's cohomology, the Hurwitz theorem and the Riemann Roch theorem for compact Riemann surfaces; will be able to determine the kind of a Riemann Surface and the size of the linear systems variety cohomology groups. Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between topics that vary between algebraic topology, differential geometry, complex geometry and also algebraic geometry. Communication skills: ability to expose the contents in the oral part of the verification and in the any theoretical questions present in the written test. Learning skills: the knowledge acquired will allow you to devote yourself to more specialized aspects of geometry.

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GABRIELE MONDELLO Lecturers' profile

Program - Frequency - Exams

Course program
Differential manifolds and complex manifolds Vector fields and flows Differential forms, integration, de Rham cohomology, Poincaré duality Singular homology, de Rham theorem Riemann surfaces, algebraic curves in CP^2 and their desingularization, their covers, their field of meromorphic functions Meromorphic differentials, Hurwitz formula Time permitting, a selection of topics among the following ones: Holomorphic 1-forms and Hodge theorem for Riemann surfaces, Riemann existence theorem Uniformization, hyperbolic surfaces, hyperbolic isometries and fundamental polygon Divisors, linear systems, Riemann-Roch theorem, canonical map
Prerequisites
The course requires familiarity with the topics of the fundamental courses in Geometry, Analysis and Algebra of the Bachelor's Degree. This knowledge is mandatory.
Books
Arbarello-Salvati Manni, Dispense di Istituzioni di geometria superiore Jeffrey Lee, “Manifolds and differential geometry” John Lee, “Introduction to smooth manifolds” Bott-Tu, “Differential forms in algebraic topology” Hatcher, “Algebraic topology” Bredon, “Topology and geometry” Beardon, “The geometry of discrete groups” Stillwell, “Geometry of surfaces” Donaldson, “Riemann surfaces”, OUP 2011 Miranda, “Algebraic curves and Riemann surfaces” Springer, "Riemann surfaces"
Frequency
Attendance is stongly encouraged.
Exam mode
The exam aims to evaluate learning through a written test (consisting of the resolution of problems of the same type as those carried out in the exercises) and an oral test (consisting of discussion of the most relevant topics illustrated in the course). The written test will last about 150 minutes. To pass the exam, a grade of not less than 18/30 must be obtained. The student must to prove to have acquired a sufficient knowledge of the topics seen in class and to be able to carry out at least the simplest among the assigned exercises. To achieve a score of 30/30 cum laude, the student must instead prove to have acquired an excellent knowledge of all the topics covered during the course and to be able to link them in a logical and consistent way.
Lesson mode
Traditional lectures and exercises in the classroom
  • Lesson code1031354
  • Academic year2025/2026
  • CourseMathematics
  • CurriculumAlgebra e Geometria
  • Year1st year
  • Semester1st semester
  • SSDMAT/03
  • CFU9