A quick proof of the classification of surfaces

... The classification of surfaces is one of the cornerstones of low-dimensional topology. The goal of this brief note is note is to explain an amazingly eﬃcient proof of this result that is due to Zeeman [Z]. Most textbook sources for the classification are aimed at beginning students and include so mu ...

... The classification of surfaces is one of the cornerstones of low-dimensional topology. The goal of this brief note is note is to explain an amazingly eﬃcient proof of this result that is due to Zeeman [Z]. Most textbook sources for the classification are aimed at beginning students and include so mu ...

14.2 Flat Mirrors!

... Flat (Plane) Mirrors Simplest of all mirrors Light rays bounce off objects in front of the mirror and reflect from the mirror’s surface. An object's reflection is said to be located behind the mirror (not literally) The object distance (do) is equal to the image distance (di) ...

... Flat (Plane) Mirrors Simplest of all mirrors Light rays bounce off objects in front of the mirror and reflect from the mirror’s surface. An object's reflection is said to be located behind the mirror (not literally) The object distance (do) is equal to the image distance (di) ...

Do every problem. For full credit, be sure to show all your work. The

... Instructions: Do every problem. For full credit, be sure to show all your work. The point is to show me that you know HOW to do the problems, not that you can get the right answer, possibly by accident. ...

... Instructions: Do every problem. For full credit, be sure to show all your work. The point is to show me that you know HOW to do the problems, not that you can get the right answer, possibly by accident. ...

GEOMETRY OF SURFACES b3 course 2004 Nigel Hitchin

... equivalence relation. For example, in constructing the torus from the square we define (x, 0) ∼ (x, 1) and (0, y) ∼ (1, y) and every other equivalence is an equality. The torus is the set of equivalence classes and we give this a topology as follows: Definition 3 Let ∼ be an equivalence relation on ...

... equivalence relation. For example, in constructing the torus from the square we define (x, 0) ∼ (x, 1) and (0, y) ∼ (1, y) and every other equivalence is an equality. The torus is the set of equivalence classes and we give this a topology as follows: Definition 3 Let ∼ be an equivalence relation on ...

1.5 Introduction to Surface Electronics

... These terms are not specific to surfaces: they are also used for atoms and molecules generally, as the energy level that a) the next electron goes into, and b) the last electron comes from. c) Surface States and related ideas A surface state is a state localized at the surface, which decays exponent ...

... These terms are not specific to surfaces: they are also used for atoms and molecules generally, as the energy level that a) the next electron goes into, and b) the last electron comes from. c) Surface States and related ideas A surface state is a state localized at the surface, which decays exponent ...

Pizzas, Bagels, Pretzels, and Euler`s Magical χ

... What is topology? Given a set X , a topology on X is a collection T of subsets of X, satisfying the following axioms: 1. The empty set and X are in T. 2. T is closed under arbitrary union. 3. T is closed under finite intersection. ...

... What is topology? Given a set X , a topology on X is a collection T of subsets of X, satisfying the following axioms: 1. The empty set and X are in T. 2. T is closed under arbitrary union. 3. T is closed under finite intersection. ...

First Page

... denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for ≈ 1.81 < m ≤ 2, the corresponding solutions to the original geometric problem are not singlevalued as functions of x, where x varies along the unperturbed exterior surface of the ...

... denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for ≈ 1.81 < m ≤ 2, the corresponding solutions to the original geometric problem are not singlevalued as functions of x, where x varies along the unperturbed exterior surface of the ...

THE GEOMETRY OF SURFACES AND 3

... surface, not on how the surface is embedded in R3 ! If you bend a surface without stretching it, the geodesics don’t change! For example, a circular cylinder can be squashed into a cylinder over an ellipse, or a cylinder with a square cross section, and when you do that, the geodesics bend along wit ...

... surface, not on how the surface is embedded in R3 ! If you bend a surface without stretching it, the geodesics don’t change! For example, a circular cylinder can be squashed into a cylinder over an ellipse, or a cylinder with a square cross section, and when you do that, the geodesics bend along wit ...

WORKSHEET ON EULER CHARACTERISTIC FOR SURFACES

... In topology all polyhedra are just spheres, and we can think of edges and vertices as a graph on the sphere. Hope: maybe the above number does not depend on the graph you put on a sphere! Problem 2 Show that the above hope is too much to ask for. Find (simple) graphs on the sphere for which the abov ...

... In topology all polyhedra are just spheres, and we can think of edges and vertices as a graph on the sphere. Hope: maybe the above number does not depend on the graph you put on a sphere! Problem 2 Show that the above hope is too much to ask for. Find (simple) graphs on the sphere for which the abov ...