Introduction to Algebraic Geometry 1

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In a forthcoming series of posts, I plan to try and introduce Algebraic Geometry, a topic in Mathematics, to an audience interested in Mathematics, but unfamiliar with the topic at hand.

This can be seen as ambitious given that the topic has a reputation — even within the Mathematical community — for being abstract, perhaps even obtuse (the urban dictionary says the word ‘abtuse’ exists for such cases )

So you may ask why that reputation, and if the reputation is true, then why bother trying to get familiar the subject ?

The reputation of AG

Well, to deal with the reputation bit, the fact is that to understand the main ideas in Algebraic Geometry (AG for short), one needs to have background in:

1. some commutative algebra (aka ring theory)
2. some category theory
3. some rudiments of algebraic topology, including homology and cohomology
4. some sheaf theory
5. perhaps some basics of manifolds / Riemann surfaces

… and topics 2–5 are not normally taught at undergraduate level, so that a student of AG will first need to spend some time learning these in graduate courses before even beginning to talk about curves etc, let alone embark on AG proper.

Even on topic #1, abstract algebra courses will stick to groups/fields/rings (3 types of abstract algebra structures) and not deal with modules (another such structure).

So studying AG does come with pre-requisites, and one may feel discouraged in not having the pre-requisites for the pre-requisites, so to speak.

Furthermore, AG is a large topic with numerous sub-branches, almost all of them specialised, so that even the holder of a PhD in AG may not necessarily understand an paper posted to say, the AG section in ArXiv (a repository of scientific research papers popular with the AG community — https://arxiv.org/list/math.AG/recent)

Finally , the ‘Algebraic’ in ‘Algebraic Geometry’ means that one will normally only reach qualitative results rather than quantitative results — in short, nothing immediately practical even after many years of study.

There ARE practical applications of AG (in robotics, computer vision, control etc), but these require specialisation, and perhaps yet more study.

So why bother ?

Life is short, and there are lots of interesting, practical things to learn, so why bother spending years learning AG, or even labour through this forthcoming series of posts ?

In my subjective opinion :

• The methods involved in AG will introduce you to completely new ways of thinking, that you may apply to topics that do directly interest you
• if you are curious about what comparatively advanced, ‘pure’ mathematics looks like, then AG is very much ‘it’
• The subject is utterly fascinating (to me)

What now ?

I will keep this logical, so I will first try to go through the prerequisites and then embark on AG proper.

The posts are designed for a non-specialist audience, so this means a) I will not be rigourous [so this will not be mathematics at all, but something else] b) I will gloss over things [in other words, Iwill occasionally say incorrect things] c) I will wave my hands a lot and explain things away rather than explain them at all.

In the next post, we will begin by going though some of the abstract algebra involved — remember those pre-requisites !

Hopefully not everyone was put off , and some of you will find this series of some interest.

That’s it for now — see you in the next post (hopefully :) and do let me know in the comments just how hopeless that whole endeavour is !