This handout is the first in a series aiming to teach calculus in a more rigorous manner to motivated middle/high school. It covers limits, which are fundamental to the subject of calculus.
By Kevin Xu
This article goes into the basics of random walks on groups, and may skip or skim over prerequisite information. It is advisable to look up the articles on those topics before reading this.
Recall the definition of a group:
Definition 1.1: A group is a set closed under a binary operation satisfying
- Identity: There exists such that for all
- Inverse: For every , there exists such that
- Association: For every , we have
Subsets of groups that are also groups are called subgroups. For example, the set containing just the identity is the trivial subgroup, and a group is a subgroup of itself. Because subgroups are closed under their operation (as a subset of the group), we can split a group into cosets of a subgroup , where each coset is for elements . Some of these cosets can overlap, though cosets are either mutually exclusive or are equivalent.
Exercise 1.2: For , prove that for any subgroup the cosets and are either the same set or contain no overlap.
Hint: What happens when we have for ?
In fact, is actually called a left coset, and would be a right coset. However, we usually talk about cosets of normal subgroups, which is when and are the same set. Note that this is equivalent to being closed under conjugation (i.e subgroup is normal if and only if for all ). It turns out that the set of cosets of a normal subgroup forms another group called the quotient group. One familiar quotient group is the remainders when dividing by , or , which is a quotient group of the integers under addition.
Now that we’ve introduced groups, let’s talk about probability. When we usually think of probability, we think of events such as rolling a die or flipping a coin, and associate a value from to to that event. We can do the same but with groups, by using a function to associate a probability to each element of a group.
Definition 1.3: A probability distribution is a function such that
Essentially, the probability of picking element is . Below are several examples of probability distributions:
Flipping a fair coin can be represented as a distribution on , or the remainders modulo , where both probabilities are . Rolling a fair die is a distribution on , again with equal probabilities for each element. When we have equal probabilities, or , we say the distribution is uniform. Here, is the order of , and is the number of elements in the group.
A game where you can move forward or backward at every step with equal probability is best represented by , the integers under addition, with distribution and otherwise. In these non-uniform distributions, the elements that have probability zero are not important as they can never occur, so in many cases we want to work with the support of , or the set of elements that have positive probability.
Now just like how we can construct a sequence of coin flips or dice rolls, we can construct a sequence by starting from the identity and then multiply by an element with probability at every step.
Definition 1.4: A random walk on a group along with its associated probability distribution is a sequence of elements starting from such that the probability of moving to is for .
A random walk is called irreducible if the probability of getting to from is positive for all , meaning we can always get from any element to any other element, and a random walk is called aperiodic if for time where is fixed the probability of reaching an element is positive, meaning we can reach any state at any time given that the time is large enough. When both conditions are satisfied, we say that the walk is ergodic. Below are a few examples of random walks on groups:
Example 1.5: A random walk on the plane moving one unit in any direction parallel to the axes can be represented as a random walk on defined on .
Example 1.6: Shuffling cards is a random walk on , the symmetric group. The probability distribution depends on what method is used to shuffle the cards.
One easy albeit inefficient way to shuffle a deck is by transposition, or switching the positions of two not necessarily distinct cards in the deck. This can be represented by the distribution
Exercise 1.7: Are Examples 1.5 and 1.6 ergodic random walks?
In standard probability, as long as we repeat for a large amount of times, our sequence will “converge” to some value. For example, a fair coin will always converge to a 50-50 distribution after a large number of tries no matter what the first few flips are.
In the same way, as long as we make a large amount of steps in any random walk, we also approach a “stationary distribution” consisting of the probabilities of landing on each element. But to find a stationary distribution, we need to find the probability of being at element after a long time , which is quite hard in practice. Thus we define a convolution recursively by
Basically, is the sum over all instances of first rolling element and then rolling , which results in being at element after two steps. The recursion then follows inductively.
The convolutions will converge to a unique stationary distribution when the random walk is ergodic (and in fact, will be uniform). However, ergodicity is also somewhat hard to establish.
Theorem 1.8: For a group and its probability distribution , the random walk on will be ergodic if and only if the support of cannot be contained by a proper subgroup or a coset of a proper normal subgroup of .
Here, proper subgroup means it is not equivalent to . Let the support of be . First suppose that the random walk is irreducible, so starting from we can arrive at any element . Because we can only multiply by elements in , in order to reach any element the support must contain a generating set for . Thus generates . But now, if can be contained by a proper subgroup of , then due to closure the set generated by , or , must be contained within the subgroup. This is a contradiction, so cannot be contained within a proper subgroup.
Conversely, suppose cannot be contained within a proper subgroup . The set generated by must be closed and contain inverses, so it must be a subgroup. However, it cannot be a proper subgroup, so then it must be . Thus generates . Then any element can be written as a product of elements in , so we can always get from element to by multiplying the corresponding representation for . Therefore is irreducible.
Now suppose that is contained by some coset of a proper normal subgroup . Then the first steps can be expressed as multiplying by in order. Thus we arrive at their product
after steps. Now note that
because of association and normality. By carrying out this process multiple times, we see that the product becomes . Closure tells us that must lie in , and therefore the product lies in the coset . Since is a proper normal subgroup, we can take an element of the nonempty set , and observe that we cannot arrive at this element in steps. Because was chosen arbitrarily, the random walk cannot be aperiodic.
Conversely, suppose the random walk is not aperiodic. Since the walk is irreducible, this means that we will arrive at any element periodically with interval . We can express this formally as the common factor
This is because two sequences will arrive at the same element at different steps only if we multiply by the representation of the identity in . Now define
We prove that these sets do not overlap with each other. First note that if , then is the product of elements in and thus lies in , which further implies . Now suppose we have overlap between two sets, i.e. . Rearranging, this is
By the definition of this product must lie in , so we have , which implies . Therefore and have no common elements unless are equivalent. Lastly, note that is a normal subgroup of . If , then and . For some element , we have . Therefore since is a subset of , it is contained in a coset of .
Besides wanting to know what a distribution will converge, we also want to find out how many steps we need to get a close approximation of the stationary distribution.
Definition 1.9: Total variation measures how much a distribution deviates from its stationary distribution after a certain amount of steps, and is calculated by
When the total variation is small, commonly measured with or other small number, we say that the random walk is sufficiently mixed. The mixing time, is the minimum time for which a random walk is mixed.
It is always nice to find the mixing time or find better bounds on it in a random walk in order to have more efficient processes. For example, Persi Diaconis and Mehrdad Shahshahani proved in 1980 that it takes transpositions on a 52-card deck to ensure the total variation was less than . In comparison, the popular riffle shuffle takes about shuffles to ensure the same “randomness.” Many real-life events can be expressed as random walks on groups, and I encourage you to explore them yourself and find their stationary distributions/mixing times!
- [Dia87] Persi Diaconis. Random Walks on Groups: Characters and Geometry. Stanford University, 1987.
- [Dia12] Persi Diaconis. Group Representations in Probability and Statistics. Harvard University, 2012.
- [DS80] Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with random transpositions. Stanford University, 1980.
- [Out15] Nolan Outlaw. Markov Chains, Random Walks, and Card Shuffling. Illinois Institute of Technology, 2015.
What is Pure Mathematics?
Pure mathematics is the general branch of mathematics that deals with mathematical concepts without any relation to the real world. In contrast with applied mathematics, its concepts are not usually applied to other fields outside of math. For example, while applied math might seek to study mathematics for the sake of applying it in physics or engineering, pure math focuses on results obtainable within mathematics.
There is immense motivation to study both pure and applied math. Pure math has many different branches, such as abstract algebra and analysis, but subjects can range to include things like string and knot theory as well. Here are some of our resources to get you started:
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What is Set Theory?
Set theory studies sets, which are essentially unordered collections of objects. These objects could be anything from functions to integers to imaginary numbers. Set theory has quite a foundational role in mathematics, as it is helpful background for many different fields in higher mathematics, including group theory, ring theory, and topology. Sets are also very commonly used to signify functions, such as a function which maps from some input set to some output set .
Basic operations and properties of sets
An example of a set is , where the numbers 1, 2, and 4 are referred to elements of the set. denotes the empty set, or the set with no elements, and sets can also be elements of sets. For example, is a set, where its elements are and .
Set theory introduces the idea of logical operators, such as AND, OR, and NOT. Some helpful concepts to know for set theory are:
- Union: The union of two sets and , written as denotes all the elements in , , or both. This corresponds to the AND operation.
- Intersection: denotes all the elements in both sets and .
- Complement: , or the complement of in , denotes all the elements in both sets which are not in .
Prerequisites/What I need to know
There are really no prerequisites for set theory, which is what makes it a great place to start in pure math. Some basic background knowledge such as the rational and real numbers as well as of functions is helpful, but not required to learn basic set theory.
Set theory is overall a great foundational subject, but also can be studied as its own subject.
- A thorough introduction to set theory from Stanford
- Paper introducing set theory (more technical than the one above)
- Quicker, more high-level article from GeeksForGeeks
- Video introducing set theory
- Another video introducing set theory
Skimming these resources should be enough to give a basic background in set theory which can be applied to more advanced endeavours.
Construction of the Integers
You may have heard of the integers before. Namely, it’s the set of numbers
stretching on infinitely. But what fundamental properties does this amazing set satisfy? How much can we derive based on these axioms? Are the theorems we see and take for granted really trivial?
The integers, commonly denoted as , are governed by two operations: addition () and multiplication (). Specifically, we have additive and multiplicative closure, which means
This means that if we take two integers and , their sum and their product are integers as well. We also quickly see that the integers form a ring, as for all we have the following properties:
With the integers defined, we now look at one of its special subsets: the positive integers. How do we define positivity? We cannot simply say an integer is positive when it’s greater than 0, as we have not defined the notion of order in the integers, and thus cannot say when integers are greater or less than each other.
Exercise: Is there a way to define “greater than” or “less than” with our above axioms?
Let be a subset of the integers that is closed under addition and multiplication. Remember that this means if we have , then and must also be in . But this is not enough to construct the positive integers; the integers is a subset of the integers satisfying additive and multiplicative closure. Thus, we stipulate further that the identity cannot be in , and that for any integer , only one of and lie in (here, refers to the additive inverse guaranteed by Negativity).
Exercise: Does the positive integers satisfy these two properties?
Exercise: How do we know that are not both in ?
can either be the positive integers or the negative integers. To simplify things, we call the positive integers and the negative integers . Now we can define the symbols and . Let be integers where . We say if , and if .
If you’ve ever tied your shoes or made a bracelet, you probably have heard of knots. You start out with a line and then fold it over and under itself until it forms a shape for you to use. However, this is not quite correct in the mathematical world. In mathematics, a knot is an embedding of a circle in the 3D space. You may be confused about this definition, but the gist of it is that a knot is a connected strand that goes over and under itself. Here is the simplest kind of knot:
This knot is called the trivial knot or the unknot. Here is the unknot, but in a slightly different form:
A little harder to see, right? We consider this to be the unknot because it can be twisted and turned in 3D space to get the unknot. These twists and turns form equivalence classes for knots. When twisting and turning, there are three moves called Reidemeister moves that we can make:
It turns out that all possible transformations of twisting and turning a knot can be expressed in these three moves. But how does this have to do with proving two knots are equivalent? Notice that if we had some characteristic of a knot that was constant under these three moves, then every knot in a knot’s equivalent class shares the same characteristic, called an invariant.
We can easily find some simple knot invariants. For a knot, we define an intersection to be a place where the knot goes over or under itself. One invariant would be the minimum number of intersections in all knots equivalent to it. Of course this characteristic would stay constant for all knots in its equivalence class; they all point to the same knot!
There are many complex and invariants in knot theory, for example tricolorability, or even knot polynomials such as the Jones and Alexander polynomials. Knot theory is a fascinating subject that bears a lot of similarity to braid theory, and thus has applications in both physics and cryptography.
Ring theory is a branch of mathematics that is closely interlinked with group theory. To create a ring, we take a group with its “addition” operation, and add “multiplication” to the mix! We say that this group is a ring if we can apply the distribution law like standard addition and multiplication. If the nonzero elements of the ring form a group under multiplication, then we call the ring a field. It is important to include this “nonzero” part, because 0, or the identity of the group, does not have an inverse under multiplication. Instead, the identity of the multiplicative group is another element, commonly referred to as 1.
The integers, rationals, and reals are all examples of rings, using standard addition and multiplication. Another example is the set of n by n matrices under matrix addition and multiplication. Note that the rationals and reals are also fields, as we can take the multiplicative inverse of every nonzero element.
Given some field , we can actually construct another field by considering the polynomials in , denoted as . An element of this field has the form
where are elements of F.
Exercise: Check that is indeed a field.
If is a field, then we can substitute for any number, and get a field as well. These fields are called field extensions of .
Fields and their extensions are widely used in many fields relating to algebra and number theory, including abstract algebra and Galois theory. They are one of the most fundamental mathematical objects, and are a way to describe the overall structure of sets that (integers modulo ), the rationals, and others share.
Prerequisites/What you need to know
Rings do not take a lot of background in mathematics to understand. However, ring theory can be thought of as an expansion to group theory that adds on an additional binary operation to the set.
What is Topology?
Topology deals with “the properties that are preserved through deformations, twistings, and stretchings of objects”. Without delving into too many specifics: it deals with manifolds, which you can stretch and distort in any way, as long as you do not make any rips or glue any parts of the manifold together. Specifically, surfaces are a type of manifold. They are made of 2-dimensional shapes but exist in 3-d. You can think of this like creating a paper cube – it’s made out of something 2-dimensional (and is thus “hollow”) but it is still 3-dimensional.
Think about a circle. Topologically, since we are allowed to stretch and distort it, it is the same as an oval (think about grabbing it with your hands and pulling them apart) or even a square (pinching 4 points and pulling apart to make a square). A circle with a hole in it is even the same as a cylinder (pull the hole up to the top and pinch the outer edge of the circle down)
We can get lots of interesting results when we think about surfaces topologically. One other curious structure is the Mobius strip, which you can get if you glue the ends of a paper strip together with one twist. This results in a surface that has only one side! Try drawing a line on the front side of a Mobius strip until it reaches itself – this should never reach the back side. But you’ll find that the back side is the front side – this is one of the curious properties of the mobius strip.
Point-Set and Algebraic Topology
Topology comes in two different flavors, point-set and algebraic topology. Typically, one starts out with point-set topology and moves on to algebraic
Topology is a challenging undergraduate course. Point-set topology does not really have prereqs other than “mathematical maturity” and being comfortable with proofs. Basic set theory is also helpful but not necessarily required.
What is a group?
Group theory is a subject associated with abstract algebra, a core course in undergraduate math. It deals with structures called groups, which are a collection of objects that interact with each other using an operation. Examples of groups include the set of integers, the set of real numbers, and the set of possible rotations and reflections of a square. Groups obey the following characteristics:
- Closure: Performing the operation on two elements in a group results in an element that is in the group.
- Identity: The group has an identity, i.e. performing the operation on the identity and an element in the group results in the same element.
- Inverse: For every element in the group, there exists an element called the inverse such that the operation on an element and its inverse results in the identity.
Example of a group
One great example of a group is the integers under addition (a way of saying that the group operation is addition). The identity element would just be 0 (as 0 + any number is just that number). The inverse of a number n is just -n (since n + (-n) is 0, the identity). This group has infinite elements, but other groups can have a finite number of elements. One especially interesting group is a Rubik’s Cube – there are only a set number of twists you can make on a Rubik’s Cube, and their operation is composition.
Prerequisite/What I need to know
For group theory, there are really no prerequisites other than what could be called “mathematical maturity”. This means being comfortable with reading, understanding, and writing proofs. Some set theory is also helpful, but not necessary to learn the subject. Beware: group theory is possibly one of the hardest math classes you can take. It’s not impossible to learn, but it strikes people as difficult because most people are not used to thinking about mathematics in this way. It’s very general and abstract, which makes it somewhat difficult to visualize, but the best way to become comfortable with it is to do practice problems that apply the concepts covered in these resources.
Group Theory Resources
- Socratic – Abstract Algebra (A video series that helps to explain some of the core concepts in abstract algebra – first 23 videos are on group theory)
- Contemporary Abstract Algebra by Joseph Gallian (an introductory college-level text to group theory)
- A book of Abstract Algebra by Charles Pinter (another textbook)
- Even Chen’s Napkin project (Chapters 1, 3, 10, 11)
- John Conway – Monster Group (Numberphile) (an interesting supplemental video)
- Rubik’s cube group (MIT) (A paper expanding on the idea of a rubik’s cube being a group)
- Ted-Ed intro to group theory (A fascinating video that touches on the Rubik’s cube group concept and what got me into group theory)
What is Calculus?
Calculus is the study of “continuous change”, dealing with tangent lines to a curve, or function, and the area under a curve. It deals with change at a specific point, with the instantaneous rate of change at a point being known as the derivative, and the area under a curve (between the function and the x-axis) being known as the integral. It is one of the first courses of undergraduate higher-level mathematics, and it is split into two different categories, integral and differential calculus.
Differential and Integral Calculus
Differential calculus is the typical first semester, or half, of an introductory calculus course. This deals with the derivative and introduces the concept of limits, which approximate the value as a function approaches a certain point or infinity. This naturally ties in with the concept of finding a tangent line at a point. This concept of the derivative has applications in many other areas and fields of mathematics.
Integrals are the second half of calculus, the complement of the derivative. Integrals represent the area under a curve, which also has many useful applications in fields like engineering, optimization in computer science, and machine learning.
Calculus is known as the gateway to higher mathematics, and it has applications in a wide variety of fields. Calculus is used in every branch of the sciences, including computer science, statistics, and engineering. One particularly interesting application is in data science or machine learning. The fundamental concept of optimization, or how a machine learning model learns, is through calculus. It uses calculus to minimize the errors made by the computer until an almost-optimal model is reached. It also has many applications in engineering as well as physics, particularly anything that involves motion, speed, and acceleration, which can be represented in terms of functions and their derivatives.
Prerequisites/What you need to know
To understand calculus, you will need to go through the standard high school math curriculum. This is typically the sequence Algebra I -> Algebra 2 -> Trignometry -> Precalculus -> Single-variable Calculus -> Multivariable Calculus