• 1962 IMO, Problem 2: an inequality with a twist

    Updated: 2011-09-28 14:15:28
    1962 IMO, Problem 2: an inequality with a twist

  • 254A, Notes 3: Haar measure and the Peter-Weyl theorem

    Updated: 2011-09-28 00:29:51
    In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of Hilbert’s fifth problem. Now, we will work on Hilbert’s fifth problem from the other end, starting with [...]

  • 1=2 via Continued Fractions

    Updated: 2011-09-27 09:48:05
    1=2 via Continued Fractions

  • Finite Sums of Terms 2^(n-i) i^2

    Updated: 2011-09-20 19:15:19
    Finite Sums of Terms 2^(n-i) i^2

  • The Brunn-Minkowski inequality for nilpotent groups

    Updated: 2011-09-16 22:24:58
    One of the fundamental inequalities in convex geometry is the Brunn-Minkowski inequality, which asserts that if are two non-empty bounded open subsets of , then where is the sumset of and , and denotes Lebesgue measure. The estimate is sharp, as can be seen by considering the case when are convex bodies that are dilates [...]

  • 254A, Notes 2: Building Lie structure from representations and metrics

    Updated: 2011-09-08 23:10:32
    Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group to ensure that it is actually a Lie group. In the previous set of notes, we saw that one could reduce the regularity hypothesis imposed on to a “” condition, namely that there was an open neighbourhood of that was [...]

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