# Matematica | GEOMETRY III

## Matematica GEOMETRY III

 0512300009 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2020/2021

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2018 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 7 56 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 ANNAMARIA MIRANDA T
Objectives
TRAINING PURPOSES

THE COURSE "GEOMETRY III" AIMS TO INTRODUCE STUDENTS TO THE FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY AND REAL PROJECTIVE PLANE GEOMETRY. ITS TRAINING PURPOSE, WORKING EFFICIENTLY FOR DROPPING EUCLIDEAN PREJUDICES, IS TO INTERPRETER THE SAME GEOMETRIC REALITY BY USING THE "TOPOLOGICAL EYE" OR THE “ PROJECTIVE EYE”, BOTH FAR THE “EUCLIDEAN EYE” .

-KNOWLEDGE AND UNDERSTANDING

THE FIRST PART IS CONCERNED WITH THE STUDY OF TOPOLOGICAL SPACES AND THEIR STRUCTURE-PRESERVING FUNCTIONS. THE STUDENTS WILL BE INTRODUCED TO THE GENERAL STRUCTURE OF A TOPOLOGICAL SPACE, THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM OLD, THE TOPOLOGICAL PROPERTIES INVARIANTS UNDER CONTINUOUS MAPPINGS, SUCH AS COMPACTNESS AND CONNECTION.
THE SECON PART OF THE COURSE DEALS WITH THE CONSTRUCTION OF MODELS OF PLANE GEOMETRIES OF ELLIPTIC TYPE SUCH AS THE REAL PROJECTIVE PLANE. TO THESE TOPICS THE STUDY OF CONICS WITH RELATIVE PROJECTIVE, AFFINE AND METRIC CLASSIFICATION IS ADDED.
THE ARGUMENTS, TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.

ON SATISFYING THE REQUIREMENTS OF THIS COURSE, THE STUDENT WILL HAVE THE FOLLOWING KNOWLEDGE AND SKILLS.
• TO UNDERSTAND THE FUNDAMENTAL IDEAS IN GENERAL TOPOLOGY.
• TO UNDERSTAND THE FUNDAMENTAL IDEAS IN PROJECTIVE PLANE GEOMETRY.
• TO BE ABLE TO EXPLAIN CLEARLY THE AQUIRED CONCEPTS AND TO APPLY THEM TO THE PROPOSED PROBLEMS.
•TO BE ABLE TO USE ” EYES” DIFFERENT FROM THE EUCLDEAN ONES .

-APPLICATION SKILLS
THE STUDENT WILL BE ABLE TO:
•DEMONSTRATE CAPACITY FOR MATHEMATICAL REASONING THROUGH ANALYZING PROBLEMS, EXPLAINING CLEARLY CONCEPTS AND PROVING PROPOSITIONS FROM TOPOLOGY AND PROJECTIVE GEOMETRTY.
•SHOW AN EFFICIENT USE OF TOPOLOGY TECHNIQUES, BY APPLYING THEM TO PROBLEM-SOLVING.

Prerequisites
BASIC CONCEPTS ACQUIRED IN THE PREVIOUS COURSES IN ANALYSIS, GEOMETRY AND ALGEBRA.
PARTICULARLY USEFUL ARE THE FOLLOWING TOPICS: CONVERGENCE, CONTINUITY, ALGEBRAIC STRUCTURES, FINITE-DIMENSIONAL VECTOR SPACES. LINEAR EQUATIONS. LINEAR MAPS AND MATRICES. BILINEAR AND QUADRATIC FORMS.
DIAGONALIZATION.
PLANE AND SPACE AFFINITIES AND ISOMETRIES.
Contents
I-GENERAL TOPOLOGY:
1.TOPOLOGICAL SPACES.
2.CONTINUITY AND HOMEOMORPHISMS, TOPOLOGICAL INVARIANCE.
3.CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM OLD (SUBSPACE TOPOLOGY, PRODUCT TOPOLOGY, QUOTIENT TOPOLOGY).
4.SEPARATION PROPERTIES, COUNTABILITY PROPERTIES, COMPACTNESS, CONNECTEDNESS.
II-REAL PLANE PROJECTIVE GEOMETRY:
5.MODELS OF THE REAL PROJECTIVE PLANE.
6.REAL PROJECTIVE CONICS (POLARITY)
7. DEGENERATION OF A CONIC
8.CONICS AND THEIR PROJECTIVE, AFFINE, METRIC CLASSIFICATION.
Teaching Methods
TEACHING METHODS ARE ESSENTIALLY BASED ON LESSONS MADE TO REDUCE COMPLEXITY TO SIMPLICITY AND, CONTEMPORANEOUSLY, DISPLAY AND MAKE GRADUALLY ACCESSIBILE STANDARD DEMONSTRATIVE TECHNIQUES. FURTHER, TO MAKE EASY TO UNDERSTAND AND APPRECIATE HOW FUNDAMENTALS THEOREMS PERFORM SIMPLICITY.
THE TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.
MOREOVER, THE PARTITION INTO WORKING GROUPS REPRESENTS AN USEFUL LEARNING ACTIVITY SUPPORTED BY TUTORS.
Verification of learning
A FINAL EXAMINATION AIMS TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING OF PERFORMED MATHEMATICAL APPROACHES, THE COMMUNICATION SKILLS, THE OPENING IN DISCUSSION, THE ORIGINALITY IN ARGUMENTATION AND THE INVENTION IN DEMONSTRATION.
IT CONSISTS OF TWO STEPS: A SELECTIVE WRITTEN EXAMINATION AND AN ORAL EXAMINATION. THE FIRST ONE CONSISTS OF SIMPLE EXERCISES AND OPEN QUESTIONS, WHILE THE SECOND ONE AIMS TO VALUE NOT ONLY THE ACQUIRED KNOWLEDGES BUT ALSO THE UNDERSTANDING LEVEL AND THE COMMUNICATIONS SKILLS.
THE FINAL MARK WILL BE OBTAINED FROM AN APPROXIMATED GRADE AVERAGE. FULL MARKS WILL BE GIVEN TO THE STUDENTS
ABLE TO APPLY WITH ORIGINALITY THE ACQUIRED KNOWLEDGES.
Texts
[1] V.CHECCUCCI, A.TOGNOLI, E.VESENTINI -"LEZIONI DI TOPOLOGIA GENERALE"- FELTRINELLI

[2] R.ENGELKING -"GENERAL TOPOLOGY"- HELDERMANN VERLAG

[3] E. SERNESI -GEOMETRIA 1 BOLLATI-BORINGHIERI 2000.
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[4] S. WILLARD, GENERAL TOPOLOGY, ADDISON-WESLEY