Undergraduate Seminar on Topology and Physics


Figure 64: Derivative of a mapping [Arnold]

Abstract: The goal of the semester will be to understand the concept of a smooth manifold and how it can be used in geometric descriptions of physical problems. The first half will be a leisurely introductions to the basic objects of differential topology: manifolds, vector fields, and differential forms. Once we’re comfortable with these new concepts we’ll learn how to describe Lagrangian and Hamiltonian mechanics in these terms. If time permits we will also discuss electromagnetism.
Prerequisites: A strong background in multivariable calculus and linear algebra. Basic knowledge of point-set topology (open sets, continuous maps, etc) will be helpful.
Time: TBA
Location: TBA

Schedule:

  1. September 16: (Felix Roz) Examples of smooth manifolds.
    A manifold is a space which can be given linear (Euclidean) coordinates near every point, but might be non-linear as a whole. They are constructed by ``gluing'' together open discs along smooth functions of real variables. In the first talk I will make these ideas precise and give lots of examples including spheres, tori, and projective spaces. References: [Tu, Ch 2.5]. Notes.
  2. September 23: (Speakers) Vector fields and tangent spaces.
    Abstract. References: [?]. Notes.
  3. September 30: (Speakers) Differential forms.
    Abstract. References: [?]. Notes.
  4. October 7: (Speakers) Title.
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  5. October 14: (Speakers) Title.
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  6. October 21: (Speakers) Title.
    Abstract. References: [?]. Notes.
  7. October 28: (Speakers) Title.
    Abstract. References: [?]. Notes.
  8. November 4: (Speakers) Title.
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  9. November 11: (Speakers) Title.
    Abstract. References: [?]. Notes.
  10. November 18: (Speakers) Title.
    Abstract. References: [?]. Notes.
  11. December 2: (Speakers) Title.
    Abstract. References: [?]. Notes.
  12. December 9: (Felix Roz) Title.
    Abstract. References: [?]. Notes.

References:

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