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Upon completion of the course, successful students will understand the hierarchy of mathematical structures (sets, topological spaces, topological manifolds, smooth manifolds and metric manifolds) that underlies modern geometry and will be able to understand in which structural layer a particular question or application is situated. They will understand key ideas, definitions, techniques and theorems at each level and gain the necessary practice required to solve a variety of prototypical problems at each structural layer. In the course of a final project, students will explore a meaningful application of the acquired techniques to either architecture, physics or artificial intelligence and will practice how to present the results in written formUpon completion of the course, successful students will understand the hierarchy of mathematical structures (sets, topological spaces, topological manifolds, smooth manifolds and metric manifolds) that underlies modern geometry and will be able to understand in which structural layer a particular question or application is situated. They will understand key ideas, definitions, techniques and theorems at each level and gain the necessary practice required to solve a variety of prototypical problems at each structural layer. In the course of a final project, students will explore a meaningful application of the acquired techniques to either architecture, physics or artificial intelligence and will practice how to present the results in written form.
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The following topics will be developed in two sessions a week, with the first session each week focusing on the theoretical structure and the second one on practical calculations. The last three weeks will be devoted to assigned project work.
Week 1: Tour d’horizon
Week 2: Sets, maps and algebraic structures
Week 3: Topological spaces
Week 4: Topological manifolds
Week 5: Smooth manifolds
Week 6: Tensor fields
Week 7: Riemannian manifolds
Weeks 8-10: Project Work: Applications in Architecture, Computer Science, Physics
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