A surface is a two-dimensional space where, if you zoom in close enough at any point, it looks like a flat plane. Common examples include the surface of a sphere, a torus (donut shape), or even a Möbius strip—each one locally flat, but globally shaped in interesting ways. In this tutorial lecture, we will learn about the mathematical formalization behind surfaces and their topological invariants. Along the way, we will explore mathematical cut-and-paste, gluing diagrams, Euler characteristics, surface classification theorems with various computations and applications of these notions for fundamental examples.
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