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21 Nov 2008: GROUPS EXAMPLES 4. G.P. Paternain Michaelmas 2008. The questions on this sheet are not all equally difficult and the harder ones are marked with ’s. ... 10. Let X be the set of all 2 2 real matrices with trace zero.

12 Mar 2008: Here a, b, c, d are arbitrary constants. The others 44 equations can all be solved in terms of elementary equations or in terms of thetranscendents. ... The other 5 equations can all be derived from (5) by allowing the fixed singular points to coalesceor

10 Oct 2008: If we think of the planeas C you may assume that all isometries have the form z 7 az b or z 7 az̄ b, where a and bare complex numbers ... Hint: is there a point in the plane fixed by all theelements in the group?

7 Nov 2008: Let G be the set of all 3 3 matrices of the form 1 x y0 1 z. ... If we think of the plane as C you may assumethat all isometries have the form z 7 az b or z 7 az̄ b, where a and b are complex

24 Nov 2008: GROUPS EXAMPLES 4. G.P. Paternain Michaelmas 2008. The questions on this sheet are not all equally difficult and the harder ones are marked with ’s. ... 10. Let X be the set of all 2 2 real matrices with trace zero.

13 Oct 2008: If we think of the plane. as C you may assume that all isometries have the form z 7 az b or z 7 az̄ b, where a and bare complex ... Hint: is there a point in the plane fixed by all theelements in the group?

11 Nov 2008: Let G be the set of all 3 3 matrices of the form 1 x y0 1 z. ... If we think of the plane as C you may assumethat all isometries have the form z 7 az b or z 7 az̄ b, where a and b are complex

2 Nov 2008: 0, 1]. )denote the vector space of all integrable functions on [0, 1]. ... 9. (a) Show that the space is complete. Show also that c0 = {x : xn 0}, the vector subspace of consisting of all sequences converging to 0, is complete.(b) Define a

3 Nov 2008: 6. Solve (i.e., find all solutions of) the following congruences:(a) 7x 77 (mod 40).(b) 12y 30 (mod 54).(c) 3z 2 (mod 17) and 4z 3 (mod 19). ... 13. Recall the Fibonacci numbers Fn from question 5.(a) Prove that (Fn,Fn1) = 1 for all n.(b) Let m 3.

3 Mar 2008: B′), Hn(A) Hn(A′), and Hn(B) Hn(B. ′) are isomorphismsfor all n, then the homomorphisms Hn(K) Hn(K. ′) ... 9) (a) Compute the Euler characteristic of all closed surfaces.(b) Show that the Klein bottle is homeomorphic to the con-.

12 Nov 2008: 5. Let Q(2) denote the set of all real numbers of the form a b. ... described is indeed an equivalence relation.(b) Define addition and multiplication of rational functions, and verify that they satisfythe usual laws of arithmetic, so that the set of all

10 Mar 2008: b) Find all C2(R2) functions with |Du| bounded and u constant in R2. ... Prove that if Γ B,then M B. (Hint: use the maximum principle of problem (4) above.).

21 Feb 2008: Deduce that traces of all words in A and B andtheir inverses (i.e. ... of all elements of the group generated by A and B) aredetermined by the three numbers {tr(A), tr(B), tr(AB)}.

8 Feb 2008: Instead we send K photons at a time all with the same polarization. ... i.e. T(xy) = 0 for all y K implies x = 0).(b) Show that the sequence xn = T(α.

14 Oct 2008: for all w with |wf(a)| < δ,there is a unique g(w) such that f(g(w)) = w. ... 5) (i) Show that a bounded holomorphic function on D(0,1) extendsholomorphically to all of D(0,1).

19 Mar 2008: 0 for all x, otherwise d(x, y) = 3n if x y = 3na/b where a, b are prime. ... φ(x, t) φ(y, t)| M |x y| , for all x, y R and t [a, b],.

14 Nov 2008: Show that all functions f from Rwith metric d′ to R with metric d are continuous. ... Show that f has a fixed point. (b) Suppose instead we assume only that for all x, y X we have d(f (x), f (y)) Kd(x, y), ord(f

14 Oct 2008: But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting. ... n 1}.]. 10. A binary operation , defined on the positive integers, satisfies the conditions(a) 1 n = n 1 for all n,(b) m 1 = (m 1) 2 for all

23 Feb 2008: all vertices, 1-simplices, and 2-simplices. For a vertex a, showthat π1(K2,a) π1(K,a) is an isomorphism. ... Suppose that the maps A A′, B B′, D D′, and E E′.

27 Feb 2008: More precisely, the Cartan map Ki(B) Gi(B) is an isomorphism for all i > 0. ... a b) (c d) = ac bd for all a,c A,b,d H.