Posted by martin on
Monday, 08 June 2009 at 13:32
The Part III exams took place in the last couple of weeks. It was quite strange for me to do exams where you are expected to answer all or most of the questions - the undergraduate maths exams have several times more questions to choose from than anyone could possibly do and in olympiad exams completing each question is an achievement.
The most common form of a Part III exam is questions which simply ask you to write out proofs from the course. Three of my five exams took this form, and it’s quite boring - except one of those, where the content was very hard so just recalling bits of the lectures is a challenge. Elliptic Curves livened things up a bit with some computation, although that is rather tedious in a different way. And my last exam, in Modular Forms, had a much better balance of bookwork and problems, although it was the hardest exam I did.
Tags partiii, tripos
Posted by martin on
Saturday, 16 May 2009 at 19:54
In my last post on the Yoneda lemma for groups, I ignored the naturality part of the lemma. I want to work in detail what this means once - it is a lot of fiddly composing of morphisms and I probably won’t do it again (at least in public). If you’re not in the mood for following such details, then there is little point in reading this, although you could skip to the last paragraph.
Tags categories, groups, maths, yoneda
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Posted by martin on
Sunday, 10 May 2009 at 16:07
When I wrote my first post on Cayley’s theorem, I noticed that Wikipedia claims that the Yoneda lemma is “a vast generalisation of Cayley’s theorem”. In this post I will try to understand why, and end up concluding that this is probably false.
Tags categories, groups, maths, yoneda
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Posted by martin on
Saturday, 02 May 2009 at 17:12
This post explains how we can consider groups as categories, along with treating the G-sets and G-homomorphisms I considered in my last post on group actions as category-theoretic objects. This is preparation for talking about the Yoneda lemma. Before reading this post, you will need to know the definitions of categories, functors and natural transformations.
Tags categories, groups, maths
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Posted by martin on
Monday, 27 April 2009 at 13:23
This continues my earlier "post on groups and actions":http://www.martinorr.name/blog/archives/2008/10/2/active_and_passive_groups. I want to think some more about Cayley's theorem, and describe how it provides an example of a universal property. (With regard to James's comment on that post, I think universality may be a better way than injectivity of describing my concept of "active group" but I'm not sure how to do that in full).
Tags categories, groups, maths
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