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8 Nov 2006: 4O:=<CKI@asgD6m{¡MO<>MK?b |7| IU6 J8;:U47L :@JMnMOD:@J¢{0:[ADIPM<=8;:ib |j b |g Sh,£?b | b |g SG[ADIPMub |7|; VT:iU6 J8;:U47L ... B64u<CKC º 1 jsgD6m{MO<>Mub |7|; U6 J8;:U47L :@I@G}<JDWWD:UMO:U47[tJD:nKMOIkCK[tKM@» t4O:=<ChJAD[aV :4ubvea¿>b b b d

27 Oct 2006: 0|f| a normed space? 3. Which of the following subsets of R2 are (a) open, (b) closed? ... Show that there exist disjoint closed subsets A, B of R (or R2 if you prefer) withd(A, B) = 0.

27 Feb 2006: ngi{_bie|g?t}&X;phYXXq&ZY[X+]. u:/u&9qfYucwE9& :q&>j uc Efc;A :q&> uc 9&E U ... k&mnbi]n}ZY[X}&XlZ<X;h<{y_]nbi]pZZ<ggi¥&Zb?_]b?]b?eoZYXhY]nb?ZY_iXµt:gphY{eb?ZY_gi]vg?tZY[_wWt9bpEZA[nb?Z«}&gfXWqZY[_wWh<X+}&

31 Jan 2006: O. é{ê_êcë¥ìë¢íî_ïðKñzò_óôìë¢íî_ôOO WZnWYXZY}YwXr|.qRqXERq}fYasd eYfvWYX> ][w >qWYXmO KKI|.0X}d efpcv ][w nKRaWhX}uowuqRqusoaZnÝ 4e|Yd ef_o{uY4Rx[w RqdQWs dexXr >XuRqvr_b.

10 Feb 2006: bWam%a_WØ y ¡Ø ¡Ø Ü: | >b[É]i_5] amWYWO>f;_C]i_W;>_hMàFá(â%âµBãäC9â>ã. ... Ì w Ã>kâYi[aeW_]i>] _>f;_¥ghVdh_b y | >bi]i>] _%_:ªib_]i_W_ÖeY_biXZ_ p VbiYiX];Vdc_:e phxzy Ì#!

4 May 2006: Prove that X+ is a topological space and prove that it is compact (N.B. ... 18. A continuous surjective map p : E B is a covering if every point b B has an openneighbourhood V such that the connected components of p1(V ) are open in

10 Mar 2006: O. ò«óXóôõôöXøùKúûXüýõôöXýOO v xw>)b]ef 6S [ikQW[þ v |Cÿrw |ÉÿN|ÿr9¿ÿrÉ)ÊCÿr)Ê|ÊÿN O ]>z |Cÿµ}9Ï¿ KQ)Q.

10 Oct 2006: n}.How large can we make a family A of subsets of [n] subject to thecondition that for all A, A′ A, there is some B B with B A A′? ... n}.How large can we make a family A of subsets of [n] subject to thecondition that for all A, A′ A, there is some

12 Apr 2006: We nd σ(A) = sign(σ)A and σ(B) = B forall σ S3. ... The only non-trivial subgroups of 〈a,b〉, not containing I2, are the conjugatesof 〈b〉.

13 Nov 2006: ô-õ M £Æ¡= Ê 9¡ 9MòË 7µbKO-b f Û Úï!ÚÙ %>Û Rï i#Ml frx "%"%}l frx ri(! ... Õ-bpè Ð ZR0N? PO QC Þ¢Q> b# Øë Øì ÜÛ F Øì ëÍÚÙRÛ jì#ëÛcÚÜÉî(Ú#ïzì#ÛÑÜSRùÛë0% ñ Øí î=Ú RÙmì ëÛcÚÜ # Øë 4! «

16 May 2006: Write B for Bi(A). Then for the set-systems A and B, wehave |A| = |A||A+||A4A+| and |B| = |B||B+||B4B+|.Now, |B| |A| and |B+| |A+| (by the ... system B = Ak satisfies |B| = |A|, |B| |A| and Bi(B) = B forall i.

1 Jun 2006: A A A (1 A) and A (1 A) =. (ii) |A| = min{|B| : A B (1 B)}.(iii) |A| = max{|B| : B A,B (1 B) = }. Proof: (i) We may assume ... B : B Z/nZ, |B| = r 1,B (1 B) = }. In particular I(SecrΓn) is generated by a space of (r 1)-ics of dimen-sion β(r 1,n).

22 Oct 2006: Then there exist a A, b B and c C with ab = c.In particular, this is true if all of A, B and C have size greater than 2n8/9. ... Equivalently, there exist a A and b B suchthat ab = c, as claimed.

13 May 2006: Then thereis an isomorphism of Brauer-Severi diagrams defined over K,. C //. φ. Pn1. ψ. E|n.0|. // Pn1. with σ(φ)φ1 = τξσ. We lift ψ to a matrix B ... It is evident that yσ =σ(B)B1 is an element of ΘE projecting onto ξσ.

26 Apr 2006: 49:904[þ%Hü;490HS68P?ý?üxú40hø684 F úLP?ý?þ%?ýý70þ:AYý76a=b[Rc;dfe , }r6n.E.32g4h4? ... þ6.04ø[ÉÈLÊ$[,ûB2iË([%b [%b ¿ dfe.