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13 Dec 2017: If H(M) is free over Z, we can choose a basis ofH2n(M) and write (a,b) = aTAb for some symmetric matrix A with entries in Z. ... H(Xφ) = 〈1,xn,x2n〉, where xi Hi(Xφ).Thus x2n = H(φ)x2n for some H(φ) Z.

10 Mar 2017: 3. Prove Cauchy’s mean value theorem: let f,g : [a,b] R be continuous functionswhich are differentiable on the open interval (a,b); show that for some c ... Show thatif f ′(a) < y < f ′(b) for some a < b in I and y R, then there exists x I witha <

19 Jan 2017: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

22 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that

4 Oct 2017: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

4 Oct 2017: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z.

14 Feb 2017: 6. Let (An : n N) be a sequence of events in some probability space (,F,P). ... Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}.

18 Jan 2017: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

6 Oct 2017: ii) m n a b (n m) [(a = 1) (b = 1) (ab 6= n)]Is (ii) true? ... 5. How large can the product of some (not necessarily distinct) natural numbers be, if their sumis 100?

10 Jan 2017: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.