### Projective Geometry (MATH 526)

2019 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
10.00
Michel Lavrauw -mlavrauw@sabanciuniv.edu,
English
Doctoral, Master
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### CONTENT

Homogeneous coordinates, projective spaces, the principle of duality, projective planes and the configurations of Desargues and Pappus, collineations and correlations, perspectivities, the projective groups, polarities, algebraic varieties, classical polar spaces, Plücker coordinates, the Klein quadric, Segre varieties, Veronese varieties.

### LEARNING OUTCOME

By the end of the course the students

1. should be able to work with homogeneous coordinates, quotient spaces, and projections in projective spaces,

2. explain and apply the principle of duality, and reproduce the definitions of collineations, correlations, elations, homologies, perspectivities,

3. should be able to state and prove the Fundamental Theorem of Projective Geometry,

4. should be able to work with the action of the classical groups on projective and polar spaces,

5. should be able to state and prove the classical theorems of Desargues and Pappus for projective planes over fields,

6. should be able to give examples non-Desarguesian projective planes,

7. should be able to state and prove combinatorial properties for finite projective planes,

8. should be able reproduce the coordinatization method for projective planes, explain the connection between the algebraic structures of ternary rings (in particular quasifields, semifields, nearfields, division rings, skewfields) and the geometric and group theoretic properties of the corresponding projective planes (in particular, the different types of translation planes),

9. should also be able to define the basic concepts related to the study of algebraic varieties in affine and projective spaces, give the classification of sesquilinear and quadratic forms and the connection with quadratic and hermitian varieties in projective spaces as well as their relation with the classical groups; define the Klein correspondence and the Klein quadric, the Segre variety and the Veronese variety.

### PROGRAMME OUTCOMES

1. Develop and deepen the current and advanced knowledge in the field with original thought and/or research and come up with innovative definitions based on Master's degree qualifications 4

2. Conceive the interdisciplinary interaction which the field is related with ; come up with original solutions by using knowledge requiring proficiency on analysis, synthesis and assessment of new and complex ideas. 1

3. Evaluate and use new information within the field in a systematic approach. 4

4. Develop an innovative knowledge, method, design and/or practice or adapt an already known knowledge, method, design and/or practice to another field; research, conceive, design, adapt and implement an original subject. 1

5. Critical analysis, synthesis and evaluation of new and complex ideas. 5

6. Gain advanced level skills in the use of research methods in the field of study. 4

7. Contribute the progression in the field by producing an innovative idea, skill, design and/or practice or by adapting an already known idea, skill, design, and/or practice to a different field independently. 2

8. Broaden the borders of the knowledge in the field by producing or interpreting an original work or publishing at least one scientific paper in the field in national and/or international refereed journals. 1

9. Demonstrate leadership in contexts requiring innovative and interdisciplinary problem solving. 1

10. Develop new ideas and methods in the field by using high level mental processes such as creative and critical thinking, problem solving and decision making. 4

11. Investigate and improve social connections and their conducting norms and manage the actions to change them when necessary. 1

12. Defend original views when exchanging ideas in the field with professionals and communicate effectively by showing competence in the field. 2

13. Ability to communicate and discuss orally, in written and visually with peers by using a foreign language at least at a level of European Language Portfolio C1 General Level. 3

14. Contribute to the transition of the community to an information society and its sustainability process by introducing scientific, technological, social or cultural improvements. 1

15. Demonstrate functional interaction by using strategic decision making processes in solving problems encountered in the field. 1

16. Contribute to the solution finding process regarding social, scientific, cultural and ethical problems in the field and support the development of these values. 1

1. Develop the ability to use critical, analytical, and reflective thinking and reasoning 5

2. Reflect on social and ethical responsibilities in his/her professional life. 1

3. Gain experience and confidence in the dissemination of project/research outputs 3

4. Work responsibly and creatively as an individual or as a member or leader of a team and in multidisciplinary environments. 2

5. Communicate effectively by oral, written, graphical and technological means and have competency in English. 3

6. Independently reach and acquire information, and develop appreciation of the need for continuously learning and updating. 4

1. Design and model engineering systems and processes and solve engineering problems with an innovative approach. 1

2. Establish experimental setups, conduct experiments and/or simulations. 1

3. Analytically acquire and interpret data. 1

1. Employ mathematical methods to solve physical problems and understand relevant numerical techniques. 1

2. Conduct basic experiments or simulations. 1

3. Analytically acquire and interpret data. 1

4. Establish thorough understanding of the fundamental principles of physics. 1

1. Use advanced Math (including probability and/or statistics), advanced sciences, advanced computer and programming, and advanced Electronics engineering knowledge to design and analyze complex electronics circuits, instruments, software and electronic systems with hardware/software. 1

2. Analyze and design advanced communication networks and systems, advanced signal processing algorithms or software using advanced knowledge on diff. equations, linear algebra, complex variables and discrete math. 1

1. Apply software, modeling, instrumentation, and experimental techniques and their combinations in the design and integration of electrical, electronic, control and mechanical systems. 1

2. Interact with researchers from different disciplines to exchange ideas and identify areas of research collaboration to advance the frontiers of present knowledge and technology; determine relevant solution approaches and apply them by preparing a research strategy. 1

3. Take part in ambitious and highly challenging research to generate value for both the industry and society. 1

1. Apply knowledge of key concepts in biology, with an emphasis on molecular genetics, biochemistry and molecular and cell biology. 1

2. Display an awareness of the contemporary biological issues in relation with other scientific areas. 1

3. Demonstrate hands-on experience in a wide range of biological experimental techniques. 1

1. Establish a strong theoretical background in several of a broad range of subjects related to the discipline, such as manufacturing processes, service systems design and operation, production planning and control, modeling and optimization, stochastics, statistics. 1

2. Develop novel modeling and / or analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy, also using an interdisciplinary approach whenever appropriate. 1

3. Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 1

4. Acquire skills to independently explore and tackle problems related to the discipline that were not encountered previously. Develop appropriate modeling, solution, implementation strategies, and assess the quality of the outcome. 1

1. Assess and identify developments, strategies, opportunities and problems in energy security and energy technologies. 1

2. Define and solve technical, economic and administrative problems in energy businesses. 1

3. Establish knowledge and understanding of energy security, energy technologies, energy markets and strategic planning in energy enterprises. 1

4. Demonstrate an awareness of environmental concerns and their importance in developing engineering solutions and new technologies. 1

5. Acquire a series of social and technical proficiencies for project management and leadership skills. 1

1. Apply a broad knowledge of structure & microstructure of all classes of materials, and the ability to use this knowledge to determine the material properties. 1

2. Apply a broad understanding of the relationships between material properties, performance and processing. 1

3. Apply a broad understanding of thermodynamics, kinetics, transport phenomena, phase transformations and materials aspects of advanced technology. 1

4. Demonstrate hands-on experience using a wide range of materials characterization techniques. 1

5. Demonstrate the use of results from interpreted data to improve the quality of research, a product, or a product in materials science and engineering. 1

1. Apply knowledge of mathematics, science, and engineering in computer science and engineering related problems. 1

2. Display knowledge of contemporary issues in computer science and engineering and apply to a particular problem. 1

3. Demonstrate the use of results from interpreted data to improve the quality of research or a product in computer science and engineering. 1