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5 Nov 2006: 6. Modify the construction of the Hopf bundle given in Lectures, replacing C everywhere by R toobtain a rank one real vector bundle over S1. ... Show that the tangent bundle TS2 induces on Y a structureof a principal S1-bundle over S2.(ii) Show further

13 Mar 2006: on SU2 = {(. a bb̄ ā. )| aā bb̄ = 1}, and normalize it so that the integral over the group is one. ... 4.8 Question. Let G = {(. a b0 1. )| a F p , b Fp}, with multiplication given by matrix multiplication.

30 Nov 2006: of the first part. 1. The square matrices A and B over the field F are congruent if B = P tAP for some invertible matrixP over F. ... Which, if any, over Q?) [Try to get away with the minimum calculation.]. (.

25 Oct 2006: 4. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices. ... . . Which of the matrices are diagonalizable over C? Which over R?

13 Nov 2006: Show that the minimum polynomial of A,over the complex numbers, has real coefficients. ... Verify that (DS) = SD and (SD) = DS. 17. For A an n m and B an m n matrix over the field F, let τA(B) denote trAB.Show that, for

10 May 2006: d. dt. t=0. log Jut dν. where ν runs over all φ-invariant Borel probability measures and hν (φ) is the measuretheoretic entropy of φ with respect to ν. ... Then. (12). N. X(f) Θ =. N. f LX Θ. Integrating the Pestov identity over SM against the

18 Sep 2006: Thosematrices which are partition regular (over N+) were completely characterizedby Rado in 1933 [7]. ... Let S′ be theamalgamation of S over Su for some u [n]d0.

1 Jun 2006: The proof over an alge-braically closed field has already appeared in [12]. ... Let A1, A2 and B be n nmatrices over a ring R.