The topic of Topology revolves around studying mathematical shapes and abstracting concepts in Geometry. There are three major areas:
Algebraic Topology: Given a geometric inquiry, it has proven fruitful to try and cast that into an algebraic problem, and then translate the results back into geometric notions. Three basic tools are:
Smooth Manifolds: A manifold is a mathematical abstraction of a surface. The theory of smooth manifolds begins with a number of technicalities and language around making sure that there is an atlas that completely describes the manifold, via local patches. These patches allow one to use the analytic theory in $\mathbb{R}^n$, in this setting. One then works towards the definitions of tangent spaces , vector fields , and flows . In what follows, $M$ is an $n$-dimensional manifold, $T_pM$ the tangent space of $M$ at $p$, and $f: M \rightarrow \mathbb{R}$ a smooth function on $M$.
Morse Theory: In an introduction to Calculus, one learns maximization and minimization methods, as an application of the first and second derivatives. From this, we know that interest in when the first derivative vanishes, and the sign of the second derivative are significant. How can one take these techniques into higher dimesions, and more abstract settings?
Here's a first step. A critical point of a smooth function $f: M \rightarrow \mathbb{R}$ is a point $p \in M$ such that the differential $df_p: T_pM \rightarrow T_{f(p)}\mathbb{R}$ vanishes. This is the equivalent of the first derivative being zero.
Here's a second step. The Hessian $H_p(f)$, of a smooth function $f: M \rightarrow \mathbb{R}$, at a critical point $p$, is the symmetric bilinear mapping $H_p: T_pM \times T_pM \rightarrow \mathbb{R}$ defined by mapping $(v,w) \mapsto v_p (w' \cdot f)$, where $w'$ is an extension of $w$ to an open neighborhood of $p$. The Hessian fulfills the role of the second derivative. Here are a couple of definitions:
The first example of a Morse function is usually that of the height function on a torus standing up. There are four critical points. If one considers dumping water and letting it "flow" over the torus, the water will flow from the maximum to the minimum. The only "stability" is at the critical point and any perturbation is "unstable" and will want to flow to more stable point, where it can come to rest. If one considers a similar scenario, with the "relative" maximum "inside" the torus, then it will flow down to the "relative" minimum. One can do a local decomposition into stable and unstable manifolds.
The point, of all of this, is that Morse theory allows one to study the manifold via the smooth functions on $M$. Further, the Morse functions are dense in the set of smooth functions. Hence, given some smooth function, that is not a Morse function, we can approximate it via Morse functions, through small perturbations. The Morse Lemma states that there is a canonical way to write a function around a nondegenerate critical point.
This is but the beginning of an extrodinarily interesting story. On a Riemmanian manifold $\langle M, g \rangle$, one can consider the negative gradient flow. Smooth functions decrease along gradient flow lines. It can then be shown that a flow line, of a smooth function $f$, begins and ends at a critical point. So, we again have a connection between what's happening analytically and the geometry of the manifold.
A deep question: What if we are working on an infinite-dimensional manifold? As always, moving from the finite-dimensional situation to an infinite-dimesional situation leaves one with great difficulties. The ideas behind Floer Homology were developed by Andreas Floer, in the 1980's. He developed these ideas in pursuit of a proof to the Arnold conjectures. The Arnold conjectures are statements that take place in the setting of symplectic manifolds . It turns out that a symplectic manifold is, in some sense, very rigid and has to be even-dimensional. The various constructions of Floer homology led to major developments in answering open questions, in differential geometry.
Remark: If symplectic manifolds have to be even-dimensional, you may very well be asking yourself: What about odd-dimensional structures? The analog, if you will, is that of a contact manifold
My personal experience, with the subject of Topology, was a difficult one. The first course usually delves deeply into the structure of point-sets. This involves a great deal of technicality, an understanding of set theory, and learning new language. The presentation of this material was done by regurgitation of the book onto an instructor's notes, and regurgitation of those notes onto a blackboard. This was immensely frustrating, as it ignored the threads that bind the subject together and provided little motivation.
As I delved further into geometry, from an algebraic point-of-view, I became more curious about what the analytic picture looked like. This lead me to take an introduction to smooth manifolds. A course that once again contained multiple sequences of consume / regurgitate. My first course in Algebraic Topology was much the same way. They did serve a purpose, as an introduction to the language and definitions.
My last semester of graduate school I did an independent study in Morse Theory. Through this study, the topological concepts I had been so frustrated by, in the past, finally came together. Having a context that interested me, provided motivations for the techniques learned, and allowed me to delve deeply. Morse theory was as compelling as my studies in representation theory. Sometimes patience is required, until a topic comes along that provides the proper context, and a mentor that's good for you.
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