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22 Feb 2019: X4 2X 2, X4 18X2 24, X3 9, X3 X2 X 1, X4 1, X4 4.

2 Nov 2019: This provesthe uniqueness, i.e. that if d exists then it must be expressed by (1.24) in local coordinates. ... So let d′ denote the exterior differentiation constructed asin (1.24), but using different choice of local coordinates.

11 Nov 2019: 24 alexei kovalev. with the local complex coordinate z = z2/z1 on U1, and ζ = z1/z2 on U2, and ζ = 1/z whenz 6= 0.

24 Dec 2019: These arecopies of X0(m) for m = 19, 20, 21, 24 and 32. ... Proof. This is [AR, Theorem 24.1]. The proof works by analysing the exact se-.

1 May 2019: 8.2. Pushforward along an open embedding. We now recall from [8] thatweak holonomicity is generally not stable under pushforward either.For this, recall from [24, Definition 13.1.1] that

11 Mar 2019: We note that by Lemma 2.24 each component grλ G is an abelian group. ... Since Gt/Gt+ iscentral in G/Gt+ (Lemma 2.24) we obtain. gλµGt+ = gλgµGt+ = hGt+.

8 Nov 2019: Theorem 2.24. If X is a path-connected, locally simply connected2 topolog-ical space, then there exists a universal cover p X̃ X. ... 24. Example 2.28. Example 2.20 tells us that π1(S1, 1) Z.

3 Nov 2019: Date: October 24, 2019.1. In Section 2, we begin by proving the analogue of the theorem of Poonen and Voloch forhypersurfaces in products of projective spaces:.

27 Oct 2019: Lemma 3.24. The sphere S = colim Sn is contractible. Sketch proof.

4 Apr 2019: 24. 8 Cyclotomic fields 27. These are Jack Thorne’s notes for the Part II Number Fields course, lectured in Lent term 2019 at theUniversity of Cambridge.