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Syllabus ( MATH 575 )


   Basic information
Course title: Algebraic Topology
Course code: MATH 575
Lecturer: Assoc. Prof. Dr. Ayşe SÖNMEZ
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Third Cycle (Doctoral)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: None
Professional practice: No
Purpose of the course: To discuss basic concepts and methods of Algebraic Topology
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Comprehend the basic principles and methods of Algebraic Topology

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Acquire scientific knowledge and work independently,
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Homework assignment
  2. Compute fundamental group, homology and cohomology using the main methods of homological algebra

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Understand relevant research methodologies and techniques and their appropriate application within his/her research field,
    3. Acquire scientific knowledge and work independently,
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.
    5. Support his/her ideas with various arguments and present them clearly to a range of audience, formally and informally through a variety of techniques

    Method of assessment

    1. Written exam
    2. Homework assignment
    3. Term paper
  3. Construct topological spaces and maps with given invariants

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Gain original, independent and critical thinking, and develop theoretical concepts and tools,
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
   Contents Up
Week 1: Fundamental Groups and its applications
Week 2: Van-Kampen Theorem, Covering Spaces
Week 3: Singular complexes and Singular Homology
Week 4: Simplicial complexes and Simplicial Homology
Week 5: Exact Sequences and excision
Week 6: The equivalence of Simplicial and Singular Homology
Week 7: CW-complexes and Cellular Homology
Week 8: Mayer-Vietoris Sequences
Week 9: Axioms for Homology
Midterm Exam
Week 10: Kohomoloji Grupları
Week 11: Universal Coefficient Theorem
Week 12: Cross product, cup product
Week 13: Künneth Theorem
Week 14: Homology and cohomology of manifolds
Week 15*: Duality Theorem
Week 16*: Final Exam
Textbooks and materials: Algebraic Topology, Allen Hatcher
Recommended readings: A concise course in Algebraic Topology, J.Peter May
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 8 25
Other in-term studies: - 0
Project: 14 20
Homework: 3,5,7,9,11,13 20
Quiz: - 0
Final exam: 16 35
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 2 14
Practice, Recitation: 0 0
Homework: 4 6
Term project: 3 14
Term project presentation: 2 1
Quiz: 0 0
Own study for mid-term exam: 10 2
Mid-term: 3 1
Personal studies for final exam: 10 2
Final exam: 3 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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