Algebra and Algebraic Topology
A brief Introduction
My mathematical interests lie in the areas of algebra and algebraic topology. Broadly speaking, algebra studies the structure of mathematical objects, and topology might be described as a kind of qualitative geometry. Algebraic topology might thus be loosely described as a qualitative study of the structure of geometric objects.
What kinds of structure do algebraic topologists look for in geometric objects? The number of "holes" is one such critical piece of structural information. For example, an apple fritter is different from a doughnut topologically because the apple fritter doesn't have any holes in the middle of it, whereas the doughnut has one hole.
There's an old joke that an algebraic topologist is someone who can't tell the difference between a coffee cup and a doughnut. Because each has only one hole (think of taking the clay in the cup before it's fired in the kiln and smooshing it back into the handle), the coffee cup and the doughtnut are similar topological spaces.
Algebraic topology seeks to develop theories that associate algebraic structures to topological spaces in such a way that topologically similar spaces (such as the doughnut and the coffee cup) give rise to the same algebraic structure, while spaces that are topologically different (such as the apple fritter and the doughtnut) yield different algebraic structures. Each such theory involves some algebraic machinery designed to detect one of the qualitative properties of the space (such as the number of holes), and the vast array of theories that have been developed allow algebraic topologists to characterize and classify topological spaces according to the various properties that describe the space qualitatively.
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