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.
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.
Posted by martin on Monday, 27 April 2009 at 13:23
This continues my earlier post on groups and actions. 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).
Posted by martin on Sunday, 15 March 2009 at 22:28
Another term has just finished. This term I studied:
I have also done some work on an essay (worth the same credit as a lecture course) on Complex Multiplication (which is about a special type of elliptic curves). I gave a seminar on this last Friday but beyond the content of that I don’t know much about it yet - I shall have to work on it over the holiday.
Other big news is that I have a place at the Université Paris Sud in Orsay for the second year of a French masters next year (because getting funding to go straight to a PhD in Paris would be a problem), and this week I heard I have got the Rouse Ball Travelling Studentship in Mathematics from Trinity to pay for it.
Posted by martin on Sunday, 07 December 2008 at 20:07
Term has just finished and it’s been hard work so I have had no time to post all the way through. At the end of term, I gave a talk on Kummer theory (a bit of Galois theory - you can read the abstract) as part of the Part III seminar series. My talk was fairly easy, intended to fill in some stuff that was assumed in the Part III Local Fields lectures but not included in Part II Galois Theory. None of the number theory talks provoked many questions afterwards.
Out of the seminars I attended (although these were only on one day out of 2.5) the number theory seminars had the largest audience, twenty-something; this included a significant proportion of PhD students and academic staff, while the others (category theory and differential geometry) were mostly attended by Part III students.