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I have continued my work on Belton http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf. I had done Tychonov’s Theroem for finite collections – here I saw the proof for the general case. I looked at the Banach-Alaoglu Theorem (the closed unit ball in the dual space of a normed vector space is compact in the weak* toplology). I saw that every normed vector space is isometrically isomorphic to for some compact, Hausdorff space . I looked at topological vector spaces and convexity. Finally I looked at the Krien-Milman Theorem (in a locally convex topological space , every non-empty, compact, convex subset of is the closed, convex hull of it’s extreme points: ). I have done the first two questions 5.1, 5.2.

Having completed my MSc on Random Walks on Finite Groups, it’s time to go quantum! Quantum here refers primarily to non-commutative geometry. In studying a group (or some other geometry), often a study of the algebra of complex functions on the group, (or some similar object depending on the class of group), can tell us everything about the group – we can reconstruct the underlying structure from the algebra of functions on it. In particular, the algebra of functions is a commutative algebra that can encodes the group axioms in a certain way.

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The Cut-Off Phenomenon in Random Walks on Finite Groups