The tales of a mathematician

I think I have seen the most clear way to rigorously construct a point at “infinity” in in Kirwan. So, say we have the complex plane. Each point where can be expressed as in . Now, if we make an equivalence relation, ~, so that for any point , then ~ . So, we see that the natural map outlined above, ~, is an injective map, since ~, and for , , for all .

So, now the only points that are do not have any element in mapping to them via are of the form . So, if we add an extra element, , then we can make a new function, ‘~, where ‘. From here on, you can see that the end behavior makes ‘ continuous along with the inverse, and can check that it is indeed surjective, and clearly injective by construction. So, I have a homeomorphism between this under some equivalence relation, ~, and the complex plane with a point at infinity.

You can do this with higher dimensions, but “infinity” will increase in dimension as well. So, there’s a really need construction to think about for any topologist that likes to look at curves! Good night!

So, today I read Chapters 4-6 of my french book (and did all the non-translations and oral exercises), sections 1.2-1.3 of Ash’s Graduate Algebra (of UIUC) notes online, and sections 2.1-2.3 of Kirwan’s Complex Algebraic Curves book (of UOxford). Now, in high school, I wasn’t a geometer. I hated the stupidity of rigor here or there, but as I sit here reading this book, it keeps the rigor of algebra, but the wonderful conceptual and natural feel of topology working together. I feel like it really nails and fully explains projective planes as best as possible. This is a must read, as it is not a book that you really have to sit down all the exercises to get all the concepts. It’s really nice just to slowly read and digest without having to deal with the paper and pen that we all know and (sometimes) love. It keeps the feeling of a exposition to something much greater, while remaining steadfast in rigor.

I feel the ingenuity in algebraic curves is really quite astonishing. The machinery made is really well-thought out by the mathematicians of history to give us some well-crafted tools, that I will no doubt learn this year. I’ll keep you posted with some really cool things this year, since this blog actually has typesetting. Go wordpress!!  And thanks to Tim for telling me about the abilities of wordpress!

Today is part of the beginning of August, and that means that I am currently in the ending days of summer, which would usually mean that I would be going through the motions of hating having to go back to school, but I am very happy going back right now. I am happy with what I did this summer, but it has been a long journey to today, and I am done with that journey, I feel. I now can say that I have researched applied mathematics, and have learned Perl and all such lovely industrious things, but it doesn’t feel right. I cannot share my love for math with someone on the side of the street. That is probably why I want to be a professor so badly, to fully feel free to find out anything and tell anyone about what I figured out. There is great beauty, and deep magic based on the axioms of the world, and I am dedicating my life to not only finding them, but sharing them.

Deep magic. That’s all I really can say about how I feel about some things. Tietze’s Extension Theorem is a great example for deep magic. Mayer Vietoris is too. Stokes’ Theorem. All of these known and not-so-known theorems have beauty in their cold and austere appearance. It’s the art that makes me do it, not the application. I just hope that one day I can write a book that shows the beauty of math to others, people that are not of the math phenotype. Erik Demaine can do it with his applied mathematics. Pure mathematics can generalize that approach.