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This is my semi active blog, each article is a by-product of me attempting to understand some mathematics. Most articles, hence, are just rehashes of existing literature and most original contributions are either through mistakes which I apologise for in advance, or through non-traditional reordering of topics to suit my taste. It can however provide a starting point to some of the content, or an alternate perspective if you're already familiar with the material.
Splitting on the line
June 7, 2023 (geometry)
Grothendieck proved any algebraic vector bundle on \(\mathbb{P}^1\) splits into a direct sum of line bundles, and consequently any coherent sheaf on the variety is a direct sum of line bundles and torsion sheaves. This behaviour, and its ramifications on the derived category, exhibit how \(\mathbb{P}^1\) is very close to a principal ideal domain. The results above can be proven with as much or as little machinery as one desires– indeed one can take the point of view that this classification of coherent sheaves is a mere consequence of existence of Jordan normal forms of matrices. In this article however we look at a more cohomological proof, showing how Serre duality shines in these situations.
Among the first ideas one encounters in the study of homotopy groups is to consider covering spaces– things that locally look like products with some discrete topological space \(\Delta\) but can have vastly different global properties than a simple product space. In fact the number of non-trivial covering spaces is a measure of the topological complexity of the space, in a sense made precise by the Galois correspondence of fundamental groups. It is then natural to ask what happens if one replaces \(\Delta\) with more a complicated topological space \(F\): what results is a richer analogue of a covering space called a fiber bundle, or a vector bundle if \(F\) is a vector space.
We define the signature of a first-order language as a collection of symbols for functions, predicates and constants. Does removing any of them make the language less expressive? Here we look at an example that shows constant-symbols (or some equivalent) are necessary to express certain properties.
Alonzo Church’s Lambda Calculus is what makes type-free functional programming work. A powerful model of computation, at its heart is a very simple abstract rewrite system– a directed graph. This short article touches upon why Lambda-calculus might be a natural thing to think about, and presents an elegant proof of the Church-Rosser theorem.
The summer of 2020 found me working on set theoretic questions with Dr Thomas Forster– one of the first tasks he set me was to read about Conway’s Sylver Coinage. This simple question opens up a discussion of the theory of Gale-Stewart games, where two players take turns to build infinite sequences of natural numbers. The winner is decided based on whether or not the resulting sequence has some property– and knowing things as simple as the cardinality of all winning configurations can reveal a lot about whether the game has a winning strategy, i.e. is rigged in favour of one of the players.