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6 Mar 2014: a) for almost all finite v, χv is unramified;. (b) if x K, thenv χv(x) = 1;. ... a) if v is real, χv(x) = xmv for all positive x Kv = R;(b) if v is complex, χv(z) = znvz̄n.

4 Dec 2014: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an

16 Oct 2014: 6. Show that R(s,t) 6(st2s1. )for all s, t > 2. ... Suppose that (i) |A| = 3 for each A A; and (ii) |AB| = 1for all distinct A, B A.

22 Sep 2014: all n-connected maps W Bwhich are lifts of τW along the fibration θ (and are specified on W). ... Let BDiffθ(W) be defined as above,but using all bundle maps, not just the n-connected ones.

18 Sep 2014: Example 1.2 Every nitely generated free. group embeds into F, so all free groups are limit. ... components. Consider a map. : 1()! Z=pZ. for some prime p that maps all the boundary.

17 Oct 2014: a) The set C of continuous functions. (b) The set {f C : |f(t)| 1 for all t [0, 1]}.(c) The set {f C : f(t) 0 as t }.(d) ... i Ui. Give an example where Ui Uj = {0} for all i 6= j, yet U1.

19 Feb 2014: ii) DefineẐ = {(xm)m1 | xm Z/mZ, πm,d(xmd) = xm for all d,m}. ... a) Replacing K by a finite extension, assume that f has all its roots in K.

10 Oct 2014: 3. For a, b, c, d G, use the associativity axiom to show that ((a b) c) d = a (b (c d)).You will find similarly that all possible ways to ... 6. (a) Show that the rotations of a regular triangle form a subgroup of all symmetries of thetriangle.

21 Oct 2014: a) the set {(j, k) : 1 j < k n} of all transpositions in Sn;(b) the set {(j, j 1) : 1 j < n};(c) the set {(1, k) : 1 < k ... Find a (natural) bijection between the set of all leftcosets and the set of all right cosets of H in G.

10 Oct 2014: 3. Show that the set of functions on R of the form f(x) = ax b, where a and b are realnumbers and a 6= 0, forms a group under composition ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

27 Jan 2014: By considering a suitable subgroup of thegroup G of all maps from Zp to itself of the form x 7 ax b, where a, b Zp with a 6= 0,show ... Without knowledge of what this group is, whyis it obvious that the group of all symmetries of the dodecahedron cannot

5 Mar 2014: In particular,. prove that L(ab) = L(a) L(b) for all positive a and b. ... 2)n cos(θx)dx. Prove that θ2In =. 2n(2n1)In1 4n(n1)In2 for all n 2, and hence that θ2n1In(θ) = n!(Pn(θ) sin θ Qn(θ) cos

15 Jan 2014: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible

3 Nov 2014: denote the vector space of all (Riemann) integrable functions on [0, 1]. ... sequences (xk) such that xk = 0 for all but a finite number of k.).

13 Oct 2014: a) The set C of continuous functions. (b) The set {f C : |f(t)| 1 for all t [0, 1]}.(c) The set {f C : f(t) 0 as t }.(d) ... i Ui. Give an example where Ui Uj = {0} for all i 6= j, yet U1.

25 Nov 2014: 13. Let. n=1xn be convergent. If xn > 0 for all n, show that. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T?

19 Dec 2014: f g f(n) g(n) for all but finitely many n ω. ... 14 (i) Suppose that 01 = 1. Prove for all non-zero n < ω,0n = n.

1 Aug 2014: M. Gross and B. Siebert, “From real affine geometry tocomplex geometry,” 2007.Scattering diagrams in all dimensions. ... LiftQ(m0) =. γ. Mono(γ). where the sum is over all broken lines for m0 with endpoint Q.

26 Nov 2014: ni=1 θi. 5. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases.

19 Dec 2014: A κ-filtration of A is an indexed sequence{Aα : α < κ} such that for all α,β < κ. ... Show there exists a club Cof λ+ such that for all δ C , the restriction map π δ is a bijection from δ ontoλδ.