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25 Nov 2005: Show that f and g havethe same splitting field if and only if b = cmar for some c K and r N with gcd(r, m) = 1. ... Deduce that every finite abelian groupis the Galois group of some Galois extension K/Q.

8 Dec 2005: Suppose F is an ultrafilter and A, Ac 6 F. Then we must have B Fwith B A = (for otherwise F ′ = {C N : C A B for some B F}is ... 26. Proof. The first statement says that inside (A, M )(ω) there is some (B, L)(ω).

17 Jun 2005: So we can assume that t (j/k, (j 1)/k) for some j. ... 1. Tlog. B(x,T ). nT (x,y) dy. Now assume that x = f(z) for some z K.

4 Apr 2005: X =s. j=1. P〈X,Pj〉j. for some well-defined 〈X,Pj〉 N. Proposition. Let X,Y be finitely generated projective kGmodules. ... Because is finite, the image of ρ iscontained in Aut(E) for some lattice E in V.

21 May 2005: χ(z w) χ(z) χ(w). Remark for readers of Ahlfors or Jones & Singerman: their σ and ζ are not quite the same asψ and χ here, but for some ... f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C.

12 Apr 2005: intersection of A with (x,y)A has expected size cδ2N2 for some absolute constant c > 0,lives inside the grid [N,N]2, and has the property that B = ... It is straightforwardto show that B has density at least η > 0 for some η that depends on δ only.

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... subgroup of Q/Q2 generated by a,b. Determine whether the field Q(.

26 Sep 2005: We now describe some applications of these results. 1.1. Infinitesimal spectral rigidity. ... References. [1] D.V. Anosov, Y.G. Sinai, Some smooth ergodic systems, Russ.

21 May 2005: χ(z w) χ(z) χ(w). Remark for readers of Ahlfors or Jones & Singerman: their σ and ζ are not quite the sameas ψ and χ, but for some constants ... 9. (i) Prove Schwartz lemma: if f : is holomorphic and f(0) = 0 then either|f(z)| < |z|, for every z ,

14 Mar 2005: intersection of A with (x,y)A has expected size cδ2N2 for some absolute constant c > 0,lives inside the grid [N,N]2, and has the property that B = ... It is straightforwardto show that B has density at least η > 0 for some η that depends on δ only.