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15 Jan 2014: Hence find the complete character table of S5. Repeat, replacing S4 by the subgroup 〈(12345), (2354)〉 of order 20 in S5.

27 Oct 2014: 5. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 1 1 00 3 20 1 0.

18 Sep 2014: 20. Application: SL2(Z) = Z/4 Z/2 Z/6. Recall that Isom(H2) = PSL2(R), so SL2(Z)acts on H2 in a natural way, with kernel {1}.G =

18 Sep 2014: 20. 1. 1 Introduction. Let M be a compact surface with a geometric structure. ... 20. 2. If χ(M) = 0 then M is a torus and admits a Euclidean structure.

18 Sep 2014: So all leaves in C(Y ) havecyclic fundamental group. 20. Now let L be a closed vertical loop in an annulus C(Y ). Then the image ofπ1(L) in G is

3 Oct 2014: 2g(V ) 2 = n(2g(W) 2) PV. (eP 1). 20. This result enables us to interpret the genus topologically (non-examinable).

19 Dec 2014: M[H] such that M[H] |= (20 = λ and there is a Suslin tree).(iv) Suppose that M |= (P is a c.c.c. ... MM requires a large cardinal axiomfor its consistency. 20 Optional. Normal Functions and Mahlo Cardinals.

29 Apr 2014: γV θ = µ λS0, (1.20). which is solved by takingθ = γ1V 1(µ λS0). ... If one unit of time corresponds to one year, then typical values for α would be of theorder of 10% 30% and for σ of the order of 20% 80%.

18 Sep 2014: 2.3 Alexandrov’s Lemma. Lemma 2.20. Suppose the triangles 1 (x,y,z1), 2 (x,y,z2) satisfythe CAT(0) condition and y [z1,z2]. ... 20. Cube complexes Winter 2011. Definition 3.12. Consider the cube [0, 1]V (Σ) and identify Σ with a subcomplexof Lk(0) in

16 Apr 2014: Automorphy lifting for residually reducible l-adic Galois. representations. Jack A. Thorne. April 16, 2014. Abstract. We prove automorphy lifting theorems for residually reducible Galois representations in the settingof unitary groups over CM fields.