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Math781, 2018 fall

Syllabus


Week-1 possible quiz problems:

  1. With usual notations what kind of objects are f, fg, Xf, X_p, X_p(f), X(fg), \partial/\partial x^i, \partial/\partial x^i|_p? (number? function? vector? vector field? etc)
  2. Define “derivation at the point p\in R^n”. The set of those forms a …………………… It is denoted …………………… Define “derivation of the algebra A”. The set of those forms a …………………… It is denoted ……………………
  3. Problem 2.4 from the Tu-book.


Week-2 possible quiz problems:

  1. Define L_k(V), A_k(V), describe a basis in both. Define the tensor product on L_*(V), the wedge product on A_*(V). Give an 8-term expression for the wedge product of f,g\in A_2(V).
  2. Define differential form. Define df, prove the expansion of df in terms of dx^i.
  3. Tu: problem 3.8 (page 33) [“k-covector” = “linear k-form”].


Week-3 possible quiz problems:

  1. List what operations we have on differentiable forms, and how they interact (e.g. \wedge vs addition of forms, d of a \wedge-product, d of a d of a form, etc).
  2. Adding all compatible charts to an atlas results a maximal atlas. For this claim we needed a lemma. Phrase and prove that lemma.
  3. Describe a complex 1-manifold structure on S^2.


Week-5 possible quiz problems:

  1. Define T_pN. Explain why \partial/(\partial x^i)|_p\in T_pN. Prove the transition matrix formula between the bases \partial/(\partial x^i)|_p and \partial/(\partial y^i)|_p.
  2. Recall the relation between tangent vectors and curves. Prove that for F:N\to M smooth, if X_p \sim c then F_*(X_p) \sim F\circ c.
  3. Prove that the derivative of Lie-group-multiplication is Lie-algebra-addition. What is the derivative of the inverse map in a Lie group?

Week-6 possible quiz problems:

  1. Define regular value. State the regular level set theorem. Define transversality. State the transversality theorem.
  2. State and prove the statement about the tangent space at I of O(n).
  3. State and prove the statement about the tangent space at I of SL_n(R).

Week-8 possible quiz problems:

  1. page 162: 14.10
  2. page 162: 14.12 (the formula was given in class, you need to carry out the calculation to prove it )
  3. page 162: 14.11 (using 14.12)

October 30 possible quiz problems:

  1. Prove that F^*(df)=d(F^*f).
  2. page 162: 14.12 (the formula was given in class, you need to carry out the calculation to prove it)
  3. State the defining axioms of an exterior differentiation on a manifold.

November 6 possible quiz problems:

  1. page 219: 19.8(a).
  2. Prove Cartan’s formula.
  3. page 234: 20.7

November 13 possible quiz problems:

  1. State and prove Key Calculation 1.
  2. State and prove Key Calculation 2.
  3. State and prove Key Calculation 3.

November 20 possible quiz problems:

  1. Prove Stokes theorem for M=\{(x,y)\in R^2: y\geq 0\}.
  2. Calculate the integral \int_{Gr_2C^4} c_1^4.