Algebraic Topology

Algebraic Topology is a field of mathematics combining two distinct areas: Algebra and Topology. Its main goal is to find and develop certain properties of topological spaces (in two dimensions, these are essentially surfaces) that remain invariant under certain types of transformations to be able to classify surfaces. To understand what all this means and why algebra is needed, the fields of Topology and Algebra need to be elaborated on.

An important transformation between two topological surfaces is the homeomorphism. A homeomorphism f is a function from one topological space X to a topological space Y that satisfies certain properties: every point in X is mapped to one and only one point in Y, for every point in Y, a point in X can be found that is mapped to the point in Y, every open set (in two dimensions, can be thought of as the set of points taken up by a surface minus the points on the surface's boundary) in Y is mapped from an open set in X, and there exists some function f^(-1) such that the composition of f and f^(-1) is the identity function, and that for f^(-1), every open set in X is mapped from an open set in Y. The first condition says f is injective or one-to-one and the second condition says that f is surjective or onto. The third and fourth conditions state that f is continuous and has a continuous inverse. Homeomorphisms are studied because they are very special and behave very nicely-they preserve open sets, which also can be thought of as preserving neighborhoods-two points next to each other will always be mapped to two points next to each other. This is essentially what Topology is all about, because Topology deals not with distance as Geometry does but with "nearness". Topology can be thought of as a general form of Geometry, where distance is abstracted to "nearness". In Topology, if there exists a homeomorphism between two spaces, then the spaces can be thought of as equivalent. In this way, a donut and a coffee cup can be thought of as equivalent. Now the question is asked, how can one determine whether two spaces are homeomorphic to each other? This is where a diversion into Algebra is needed.

The Algebra that is referred to here is not high school algebra-it does not deal directly with such topics such as quadratic equations or conic sections. Rather, it is much more general, and is called "Abstract Algebra". There is a certain field in Abstract Algebra that can be applied to great effect in Topology; this topic is called "Group Theory" and deals with mathematical objects called groups. A group G is simply a set of elements endowed with some operation * with the following properties under the operation.

1.  For all x,y in G, x*y is an element in G
2. There exists an element e such that for all x in G such that x*e=e*x=x
3. For all x,y,z in G, (x*y)*z=x*(y*z)
4. For all x in G, there exists an element y in G such that x*y=y*x=e

  The first condition is G is closed under *, the second is the existence of an identity element, the third is transitivity, and the fourth is the existence of an inverse. Some examples of groups are the integers under addition, (Z under +), the reals minus 0 under multiplication ( R\{0} under x), and n dimensional space under vector addition ( R^n under +). In all these examples, the elements are numbers or points, but in general, the elements could be any element, even something as unspecific as the letter A. An interesting application of Group Theory is solving the Rubik's Cube, but now back to Topology; how can groups be applied to the study of Topology?
 

When viewed a particular way, surfaces can be assigned a group. Given two curves on a surface who have the same endpoints, a homotopy exists between the two if and only if there is a continuous function F(s,t) such that at t=0, F maps to one of the curves, while at t=1, F maps to the other curve. Essentially, two curves are homotopic if there exists  a continuous "deformation" from one curve to the other. It turns out that on surfaces, the set of sets of curves that are homotopic to one another but not to other sets of curves forms a group under the operation of path class multiplication, i.e. path concatenation. This is called the fundamental group of a surface. Surfaces with different fundamental groups are never homeomorphic to each other (the converse is not always true), and homeomorphisms between two surfaces preserve the surfaces' fundamental groups. Therefore the fundamental group of a surface is an invariant, and can be used to classify surfaces. Some examples of fundamental groups are that the circle has a fundamental group of Z (the integers), the two dimensional surface of the sphere has a trivial fundamental group (all curves are homotopic to each other, and therefore there is only one element in the group, which must be the identity element), and the torus (donut), which has a fundamental group of ZxZ (all ordered pairs of integers (a,b)).

The fundamental group of surfaces can be generalized to the fundamental group on manifolds (n dimensional surfaces). In n dimensions, the fundamental group would simply be the set of sets of (n-1) manifolds who are homotopic to each other but not to other sets of (n-1) manifolds. These groups are called higher homotopy groups.

By synthesizing two distinct fields of mathematics, mathematicians have successfully advanced the study of types and classes of surfaces and manifolds.

Math Contest #2

This blog will be about the answers to selected problems Westmont's second math contest.

One of the questions was asking for a five digit number that had the following property: when a 1 is added to the right of the number, the resulting six digit number is three times greater than if the 1 was added to the left of the number. This problem is easily solved by using the following equation:

let X be the 5 digit number we are looking for, then:

10X+1=3(10^5 +X)

which, when solved, gives 42857. Checking this result with the given properties shows that it indeed is the five digit number that is needed. The equation gave the correct number because it simply stated the conditions in mathematical terms: the left side of the equation essentially shifts all the digits of X one place holder to the left, and then adds a 1 to the right, while the right side keeps the digits of X in place, while adding a 1 to the right.

Another question asked to find the number who is the sum of consecutive integers who's squares differ by 2011. Again, a simple set of equations gives the answer:

x=y+(y+1)=2y+1

(y+1)^-y^2=2y+1=2011

x=2y+1=2011

This equation also worked because it simply restated the given conditions mathematically. The number that needed to be found, x, can be expressed as y+(y+1), which simplifies to 2y+!, but this turns out the be exactly 2011. Therefore x=2011.

The first problem asked for the maximum value of the following expression:

(x^2-2x+1)^3 + (1-2x-x^2)^3

This question is made trivial after simplifying:

(x^2-2x+1)^3 + (1-2x-x^2)^3

= (x^2-2x+1)^3 + ((-1)^3)(x^2-2x+1)^3

= (x^2-2x+1)^3 - (x^2-2x+1)^3 = 0

Since the expression is 0 for all values of x, the maximum value of this expression must simply be 0.

The last problem that will be covered is the only problem whose solution is in doubt. It asked for the minimum amount of numbers required to express the number 1 as the sum of positive decimal expansions with only 0's or 8's. It is 95% likely that the solution is 7; here is one way to obtain it:

1 = .8 + .08 + .08 + .008 + .008 + .008 + .008 + .008

There is probably a way to prove mathematically that the answer is 7, or some other number, but given the thirty minute time period and the five other problems, no mathematical proof was found.

Metric Spaces

The topic of today's blog is the concept of a Metric Space. A metric space is a set of elements(in most cases numbers) that are endowed with a "metric" that satisfies certain conditions. The "metric" is simply a function of two variables on the set of elements. It will be denoted d(x,y). For metric spaces, the metric d must satisfy the following properties:

1. d(x,y)>0 for all inequal x,y
2. d(x,y)=0 if and only if x=y
3. d(x,y)=d(y,x)
4. d(x,z)< or =d(x,y)+d(y,z)

In words, the conditions state that the distance function is positively defined(it is never negative)and symmetric (d(x,y)=d(y,x)). Also, d satisfies the triangle inequality. One elementary example of a metric space is the set of real numbers R endowed with the absolute value function d(x,y)=absolute value of (x-y). In all high school and lower math classes, and some college classes it is subconsciously assumed that when dealing with real numbers, the absolute value function is the metric that is given for R.However, this function is not the only possible metric on R, take for example the function:

d(x,y)=0 when x=y,

d(x,y)=1 when x is inequal to y

This function satisfies properties 1-4, because it is never negative, only 0 when x=y, symmetric, and not in violation of the triangle inequality. This function is called the discrete metric, and is a metric for any set of elements(the absolute value function, on the other hand, is not, because there is no definition for the absolute value of the difference of two functions). In addition the absolute value metric and the discrete metric, there are an infinite number of metrics on R, as well as any set of elements.

Why study metric spaces? Metric spaces are essential to the further study of calculus. One big part of calculus deals with entire functions as points in a space, instead of singular numbers as points, and the functions on these so called "function spaces". Taking limits, derivatives, and integrals rigorously all require the use of metric spaces. The most general definitions of limits, continuity, open and closed sets, and compactness, completeness, and convergence are all formulated in the language of metric spaces. The terms listed are all central to calculus, and in fact, calculus can be thought of as based on four of the terms: continuity, compactness, completeness, and convergence. Without metric spaces, calculus would be nowhere as sophisticated as it is today.

Brouwer Fixed Point Theorem

This blog post will be on a very interesting theorem called the Brouwer Fixed Point Theorem, focusing on the case for the 2 dimensional disk, otherwise known as the circle (with all points inside it included). The Brouwer Fixed Point Theorem for the 2 dimensional disk states that every continuous mapping f from the disk to itself has a fixed point, e.g. a point x such that f(x)=x. One can visualize this theorem by stirring a cup of coffee; the point that does not move when the cup is stirred is the center point. Similarly, the eye of a hurricane is the one place where no winds are swirling.

First of all, the definition of a function, continuous function and a function from the disk to itself should be precisely defined. A function is a relation that given an element, assigns one and only one element. Elements are commonly thought of as points, but they could also be entire functions, or other, more general concepts. A continuous function is a function that over its entire domain (all possible x values), has the following property: as x approaches a, f(x) approaches f(a). Intuitively speaking, the function is "smooth" and has no "jumps" in it-values of x that are very close together are assigned values of f(x) that are also very close together.Graphs of continuous functions can be drawn without ever lifting the writing instrument. There are other, even more precise definitions of continuity, but this one is sufficient. Finally, a function that maps the disk to itself is simply a function whose domain is the disk and whose range (all assigned values) does not include any points outside of the disk.

The proof of the theorem is fairly complicated, and will be omitted. Rather, applications of the theorem will be discussed. John Nash used a version of this fixed point theorem to prove the existence of the Nash Equilibrium, which is when in a game of two or more parties, each party individually makes the best decision it can make taking into account the decisions that the other parties can make. This finding won Nash the Nobel Prize in Economics. The Fixed Point Theorem is also used to prove the existence of other equilibria in economics, such as market equilibrium.

The Brouwer Fixed Point Theorem applies also for any line segment, and can be generalized to the n dimensional sphere. The theorem also applies to any surface, or in higher dimensions, "manifold" that is similar enough to the sphere. The meaning of "similar enough" is rigorously established in a field of mathematics called "Topology," which deals with the distinct shapes and figures of various manifolds.

Some Properties of Infinity

How big is infinity? There is a surprising answer to this question: there are actually multiple sizes of infinity. Actually, there are infinity sizes of infinity. Let us look at a few examples to illustrate these points. First of all, how many numbers are in the intervals [0,1], and [0,2]? Obviously, there is an infinite number of numbers, but in some sense, the two intervals contain the exact same number of numbers. This may seem counter-intuitive, because the length of the second segment is two times longer than the length of the first segment. However, one simple bit of reasoning will show the truth of the statement. Given any number in the first interval, if we double that number, it will be a unique number in the second interval. Similarly, given any number in the second interval, if we halve it, it will be a unique number in the first interval. What we have just done is establish a one-to-one correspondence between the two intervals. Now if the two intervals did not have an equal number of numbers, then we could not establish a one-to-one correspondence between the two, because there will always be at least one number in one of the intervals that is left over and does not correspond to any number in the other interval. This example illustrates the fact that the size of infinity does not depend on any finite quantity, such as length. Another similar example is that if we take every positive even integer and halved it, it would be the integers, and if we took every positive integer and doubled it, it would be the even integers. This shows that the sets {1,2,3...} and {2,4,6...} have the same number of numbers.

Now onto different sizes of infinity. It seems pretty obvious that there are more real numbers (numbers that can be expressed as infinite decimals) than integers, but the result takes a little bit of subtle reasoning to prove. Say there were as many integers as reals. We could then list out all the real numbers starting from the first real number, to the second real number, to the third, and so on. All the real numbers can be expressed as infinite decimals. We could then choose a digit different from the first decimal place of the first real number, a digit different from the second decimal place of the second real number, and so on. We then use these digits to construct a real number with the first decimal place as the first number chosen, second decimal place as the second number chosen, and so on. This real number would then differ in at least one spot from all of the listed real numbers, which means that no matter how we try to list in order out all the real numbers, there will always be a real number not listed. This implies that the number of real numbers is greater than the number of integers.

One last, and more counter-intuitive example is the fact that there are just as many integers as there are  rational numbers (numbers that can be expressed as p/q, where p and q are integers). The proof of such as result is very hard to explain without a picture:

Every single rational number is touched by the line; the line essentially amounts to an ordered list of rational numbers. Therefore the number of integers is actually just as great as the number of rationals. This, and the preceding example, establish the concept of different sizes of infinity.

The Geometry of Space

I've been reading this novel lately written by the Chair of the Harvard Mathematics department, Shing-Tung Yau, titled The Shape of Inner Space. So far, it is primarily on Geometry and how it relates to a particular area of physics called String Theory. String Theory says that space is governed by some sort of objects called strings that wiggle and twist and determine the laws of the universe. These strings can be described using an area of Geometry called Differential Geometry. The Geometry learned in high schools is called Euclidean Geometry, which deals with straight lines and measurements on space that is flat. Differential Geometry deals with measurements on space that is curved, such as the Earth. It turns out that Differential Geometry is essential to the study of Physics, because at the beginning of the 19th century Albert Einstein showed that space, or rather spacetime, is itself curved. Differential Geometry has been applied many times to the study of Physics, and is currently being applied right now to study String Theory. One of the consequences of String Theory, if it is true, is that spacetime is actually ten dimensional, instead of the four dimensional point of view proposed by Einstein. The ten dimensions consist of the four dimensions of spacetime, plus an additional six dimensions so tiny that we will most likely never see them. These six dimensions are stored in shapes known as Calabi-Yau manifolds, named after their discoverers, one of which is the author of this book. Essentially what a manifold is a shape that in tiny little regions, looks "flat". The surface of the Earth is roughly a manifold (not considering mountains or steep valleys), because to a person standing at the surface, it simply looks like a plane. Similarly, manifolds can be extended to higher dimensions, where locally they look like "flat" space, but globally, they could look entirely different.

Sadly, since these Calabi-Yau manifolds are said to be so miniscule, we will probably never be able to perform experiments to determine if String Theory holds or not. However, because of String Theory, we have been able to study Geometry from completely different points of view, which has helped advance the current field of Geometry as we know it.