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8 Nov 2019: Since every z S1 is in some such U, it follows that pis a covering map. ... We make some remarks on what Definition 3.1 means for exis-tence and uniqueness.

22 Feb 2019: Euclidean domain. Could there be some other Euclidean function φ making Z[3]. ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written asra sb for some r,s R.

4 Apr 2019: Thus we can write b = v/2 for some v Z and hence α = u. ... factor |det(B)| for some B GLn(Z); and this factor equals 1.Now let d Z be a negative square-free integer, and let L = Q(.

11 Mar 2019: Lecture 4. Aside on group commutators. We recall some general group theoretic facts. ... Since ω(G{1}) is a discrete subset of R (Lemma 3.13(b))there is some t < s maximal such that we can find x = gλ X with xGt =

10 Jan 2019: b) [Needs some topological knowledge.] First project S2 onto its equatorial plane by(x,y,z) 7 ζ = xiy. ... Show that χ(g) = 0 for some g G. [Hint: partition G into equivalence classes bycalling two elements of G equivalent if they generate the same

12 Apr 2019: x. In some sense, X(A) corresponds to the tangent bundle of X. ... 1 =ri=1. bki di,. for some d1,. ,dr D. Thus, b =ri=1 b.

9 Aug 2019: for some set Λ, where the fibres are considered with respect to f|B. ... for some set Λ. Proof. Apply Lemma 4.6 and Lemma 4.9.

9 Aug 2019: Hence,in some sense, algebraic de Rham cohomology goes back to the source of modern cohomologytheories. ... the case where R =k [x1,. ,xn] /I for some ideal I of k [x1,.

10 Jan 2019: Now P(z) = 0 if and only if z = ajfor some 1 j m. ... Thus there must exist an M suchthat EM has non-empty interior, so EM (a,b) for some non-empty interval(a,b).

17 Apr 2019: Factorisation of polynomials in Fp[T] is effective, so Proposition 2.4.7 gives analgorithm to compute pOK for some values of p. ... A binary quadratic form is a map q : Z2 Z of theform q(x,y) = ax2 bxy cy2 for some (a,b,c) Z3.