2. Professors
  3. VITAGLIANO Luca
  4. Teaching

HOMOLOGICAL METHODS IN DIFFERENTIAL GEOMETRY

Luca VITAGLIANO HOMOLOGICAL METHODS IN DIFFERENTIAL GEOMETRY

8803000053
DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
P.H.D. COURSE
MATHEMATICS, PHYSICS AND APPLICATIONS
2021/2022

YEAR OF COURSE 1
YEAR OF DIDACTIC SYSTEM 2021
FULL ACADEMIC YEAR
CFUHOURSACTIVITY
420LESSONS
LUCA VITAGLIANO T Curriculum
Objectives
THE AIM OF THE COURSE IS TO ILLUSTRATE SOME COMPUTATIONA TECHNIQUES FOR DE RHAM COOHOMOLOGY OF SMOOTH MANIFOLDS.

- KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL LEARN THE NOTION OF SPECTRAL SEQUENCE A HOW DOES THIS CONSTRUCTION HELP IN COMPUTING THE COHOMOLOGY OF A COCHAIN COMPLEX EQUIPPED WITH A FILTRATION. THEY WILL ALSO LEARN THAT THERE IS A SPECTRAL SEQUENCE NATURALLY ASSOCIATED TO A FIBER BUNDLE THAT IN MANY CASES ALLOWS ONE TO COMPUTE THE DE RHAM COHOMOLOGY OF THE TOTAL SPACE.

- APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL BE ABLE TO APPLY THEIR KNOWLEDGE TO THE COMPUTATION OF THE DE RHAM COHOMOLOGY OF THE TOTAL SPACE, THE FIBER OR THE BASE OF A FIBER BUNDLE.
Prerequisites
THE PREREQUISITES ARE:

ALGEBRA AND GEOMETRY OF THE M.SC. COURSE IN MATHEMATICS.
Contents
1. SPECTRAL SEQUENCES
2. DIFFERENTIAL CALCULUS ON A FIBER BUNDLE
3. LERAY-SERRE SPECTRAL SEQUENCE
4. GAUSS-MANIN CONNECTION
5. APPLICATIONS
6. FOLIATED COHOMOLOGY
Teaching Methods
THE INSTRUCTOR WILL PROVIDE THE VERY FIRST ELEMENTS OF THE THEORY AND THE TEACHING MATERIAL AT THE BEGINNING OF THE COURSE IN THE FORM OF FRONT LECTURES. MOREOVER, HE WILL ASSIGN A NUMBER OF EXERCISES. THE STUDENTS WILL AUTONOMOUSLY METABOLIZE THIS MATERIAL, AND WILL SOLVE THE EXERCISES. LATER, DURING THE LECTURES, THEY WILL REPORT ON THE NOTIONS THEY HAVE LEARNT, ON THEIR LEVEL OF UNDERSTANDING AND THE APPLICATIONS THAT THEY MANAGED TO ACCESS, IN THE FORM OF INTERACTIVE SEMINARS AND GROUP DISCUSSIONS. A SIGNIFICANT TIME WILL BE DEVOTED TO THE GROUP ELABORATION OF PROOF STRATEGIES FOR THE MOST IMPORTANT RESULTS OF THE COURSE.
Verification of learning
THE FINAL EXAM WILL CONSIST OF A SEMINAR ON A TOPIC RELATED TO THE MAIN THEME OF THE COURSE BUT NOT EXPLICITLY TREATED DURING THE CLASSES. THE STUDENT WILL HAVE TO PROVE TO BE ABLE TO AUTONOMOUSLY APPLY WHAT HE LEARNT TO A NEW SITUATION AND TO EFFECTIVELY ILLUSTRATE THEIR LEVEL OF CONTROL ON THE COURSE SUBJECT. THE LEVEL OF PARTICIPATION TO THE GROUP ACTIVITIES PROPOSED DURING THE CLASSES WILL CONTRIBUTE TO THE FINAL ASSESSMENT.
Texts
R. BOTT, AND L. W. TU, DIFFERENTIAL FORMS IN ALGEBRAIC TOPOLOGY, SPRINGER-VERLAG.
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