INTRODUCTION TO MANIFOLDS

Prerequisites & course information

Requisites: MATH 522 , ( MATH 521 and 561 ), graduate/professional standing, or declared in Mathematics VISP Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Fall 2023 Learning Outcomes: 1. Use the techniques of multilinear algebra and apply them to differential forms in Euclidean space and on abstract differentiable manifolds. This includes being able to compute with the exterior derivative. Audience: Both Grad & Undergrad 2. Identify and/or construct differentiable manifolds. This includes being able to identify spaces as differentiable manifolds and to endow abstract topological spaces with differentiable manifold structures. Audience: Both Grad & Undergrad 3. Extend calculus concepts from Euclidean space in order to apply them to abstract differentiable manifolds. This includes being able to identify and work with differentiable maps between differentiable manifolds in computations and proofs Audience: Both Grad & Undergrad 4. Use the essential definitions and theorems from calculus on manifolds in the context of computations and proofs. This includes taking Lie derivatives of vector fields on differentiable manifolds, computing integrals of curves, integrating differential forms, and Stokes’ theorem. Audience: Both Grad & Undergrad 5. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Use the techniques of multilinear algebra and apply them to differential forms in Euclidean space and on abstract differentiable manifolds. This includes being able to compute with the exterior derivative. Audience: Both Grad & Undergrad 2. Identify and/or construct differentiable manifolds. This includes being able to identify spaces as differentiable manifolds and to endow abstract topological spaces with differentiable manifold structures. Audience: Both Grad & Undergrad 3. Extend calculus concepts from Euclidean space in order to apply them to abstract differentiable manifolds. This includes being able to identify and work with differentiable maps between differentiable manifolds in computations and proofs Audience: Both Grad & Undergrad 4. Use the essential definitions and theorems from calculus on manifolds in the context of computations and proofs. This includes taking Lie derivatives of vector fields on differentiable manifolds, computing integrals of curves, integrating differential forms, and Stokes’ theorem. Audience: Both Grad & Undergrad 5. Identify applications of course content in current areas of research. Audience: Graduate