🔗 Algebraic Topology

Course Code: 29013-AUX
Duration: 8 Weeks | 2 Sessions per Week (Theory + Applications)

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📚 Introduction:

Algebraic Topology bridges Algebra and Topology to study the properties of spaces preserved through continuous deformations.
It finds critical applications in computer science (networks, data analysis) and theoretical physics (field theories).

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✨ Course Description:

This course introduces the fundamental tools of algebraic topology, including fundamental groups, homology, and topological spaces analysis.
Students will apply these concepts to solve real-world problems in network theory, robotics, and quantum physics.

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🎯 Course Objectives:

• Understand the fundamental group and compute it for basic spaces.

• Learn about homology groups and their computation.

• Analyze and classify topological spaces using algebraic invariants.

• Explore applications in computational topology and field theory.

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🎯 Target Audience:

• Advanced students of Mathematics, Physics, and Computer Science.

• Researchers interested in Topology, Algebra, Network Analysis, and Theoretical Physics.

• Prerequisites: Familiarity with Basic Algebra and Topology.

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🛠️ Materials and Resources:

• Textbooks:

  • “Algebraic Topology” by Allen Hatcher.

  • “Elements of Algebraic Topology” by James R. Munkres.

  • “Topology and Groupoids” by Ronald Brown.

• Software Tools: SageMath, Python libraries for computational topology (optional).

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🧑‍🏫 Instruction Method:

• 1 Theory Session per week (concepts, theorems, examples).

• 1 Application Session per week (computations, problem-solving, real-world applications).

• Weekly exercises on computing groups and analyzing topological spaces.

• Final project: Analyzing a real-world network or physical system using algebraic topology.

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🗂️ What You Will Learn:

✅ Mastery of fundamental groups and homology theory.
✅ Skills to compute algebraic invariants for topological spaces.
✅ Application of topological methods to problems in networks and physics.
✅ Deep understanding of continuous deformations, connectivity, and holes in spaces.

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🗺️ Detailed Course Outline:

📅 Week 1:

Introduction to Topological Spaces

• Open and closed sets, continuous maps, examples of spaces.

📅 Week 2:

Paths, Loops, and Homotopy

• Definitions, path-connectedness, homotopy between paths.

📅 Week 3:

Fundamental Group π₁(X)

• Construction, examples (circle, torus, graphs).

📅 Week 4:

Applications of Fundamental Groups

• Covering spaces, lifting properties, classification.

📅 Week 5:

Introduction to Homology Theory

• Simplicial complexes, chain complexes.

📅 Week 6:

Computation of Homology Groups

• Examples: simple spaces like spheres, tori, projective spaces.

📅 Week 7:

Advanced Topics: Higher Homotopy Groups

• Basic concepts, relation to homology (optional based on course pace).

📅 Week 8:

Applications to Computer Science and Physics

• Network analysis, topological data analysis (TDA), field theory concepts.

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🏆 Final Outcome:

By the end of the course, you will:

• Compute fundamental and homology groups for basic and some advanced spaces.

• Use topological methods in real-world analysis (such as network connectivity and field theory models).

• Build a strong foundation for further study in Advanced Topology, Algebraic Geometry, or Quantum Topology.

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