- What is Algebra (401)?
The importance of 401 and 405 to majors cannot be overstated. This course and 405 are the only two courses absolutely required for all majors. These two together provide an introduction to the ideas and the methods of modern abstract mathematics. In 401 and 405 you will learn algebra and analysis, but you will also learn how proofs are developed and used, and methods to construct your own proofs. This involves an important transition in mathematical maturity: Until this point you have focused on solving problems by applying methods, and the central unit of learning is the “new method.” Once you reach these more abstract courses, you will focus on uncovering information by applying theorems and their proofs. The central unit of learning will be the theorem. This is somehow more natural and more straightforward, because “theorem” is just another word for “fact.” Instead of learning to do things you will now learn facts, and infer how to do things. In the best case, this transition happens early, even in Calculus class. In any case, expect it to be complete after you do 401 and 405!Wondering what you’ll be getting into? Algebra I will teach you about groups, rings, and fields, but of course you don’t already know what groups, rings, and fields are. So what is algebra? Roughly speaking, our system of numbers has objects (3,0,e,pi), operations (+,-,*,/), and rules (a+b=b+a, a(b+c)=ab+ac, etc.). In algebra, we take away the objects, leaving operations, operating abstractly on mysterious unknowns, and rules, reassuring us of some remaining structure. We then ask “What do we still know, despite self-imposed ignorance of objects?” The answer, invariably, is “quite a lot,” which by the way means ultimately that most of our knowledge of numbers comes not directly from our great understanding of the numbers themselves, but from the operations and rules governing them. Eventually, we can “fill the void,” substituting other objects (e.g., matrices, polynomials, spatial rotations, remainders-by-seven, possible rearrangements of five chairs) in place of numbers, and produce pseudo-numerical knowledge about structures which are not at all numerical in nature.Groups, rings, and fields, are names for instantiations of these rules with various levels of requirements that make them more and more like the real numbers. Groups have the fewest requirements, and include lots of crazy structures: remainders-by-seven, the Rubik’s cube configuration space, the set of square matrices of nonzero determinant, the real number system, the set of spatial rotations, and elliptic curves. Fields are the most like the normal number system, but can still be strange: There is a field with 16 elements in which + and – are two names for the same operation!This course is proof-based (i.e., hard) and has very little to do with calculus.
- What is Algebra 2 (402)?
Obviously the successor to Algebra 1! There are many ways to satisfy the “and one more semester of algebra” requirement for the major, but this one is the most highly recommended. Because much of this course is Galois theory, we’ll attempt to give a brief overview of that theory for those who are taking or have taken Algebra 1. Although this is not a complete picture of the course, it should give you a feel for the material.The real numbers form a field. The complex numbers form a field extension. Notice that you get the complex numbers by adjoining a single root (i) of a polynomial p(x) = x2 + 1, and then including everything you have to include because you have to make a field (e.g., 1+i, 2i, -i, etc). Notice, further, that the complex numbers have a single automorphism, which takes i to -i and -i to i (flip around the real axis), “permuting the roots.”Galois theory tells us that in general (starting with any field and almost any polynomial p(x)), one may create an extension field in which p(x) has a root, and that the extension field will have automorphisms moving the roots of p(x) to each other, but fixing the smaller field. Moreover, intermediate fields may exist, and if they do, they will be the parts not moved by certain subgroups of the group of automorphisms of the larger field. A beautiful correspondence can be established, then, between all the intermediate subfields and all the subgroups of the group of automorphisms of the larger field. It’s “beautiful” because the subfields and subgroups match perfectly, with nothing left over on either side. Because we can start with the field of integers modulo a prime, these ideas have powerful applications to number theory.
- What is Analysis (405)?
The importance of 401 and 405 to majors cannot be overstated. This course and 405 are the only two courses absolutely required for all majors. These two together provide an introduction to the ideas and the methods of modern abstract mathematics. In 401 and 405 you will learn algebra and analysis, but you will also learn how proofs are developed and used, and methods to construct your own proofs. This involves an important transition in mathematical maturity: Until this point you have focused on solving problems by applying methods, and the central unit of learning is the “new method.” Once you reach these more abstract courses, you will focus on uncovering information by applying theorems and their proofs. The central unit of learning will be the theorem. This is somehow more natural and more straightforward, because “theorem” is just another word for “fact.” Instead of learning to do things you will now learn facts, and infer how to do things. In the best case, this transition happens early, even in calculus class. In any case, expect it to be complete after you do 401 and 405.What’s Analysis I? If calculus is learning how to climb the stairs, analysis is learning how to build the stairs. You will learn how mathematics puts a rigorous foundation underneath the ideas of calculus. Along the way, you’ll answer fundamental questions about the real number system (“Are you sure there are no holes?”, “Exactly which sequences converge?”, “Are there infinitesimal numbers?”), and discover the techniques of analytic proof. You can expect to become very familiar with epsilon-delta proofs, and you may find them easier the second time around! You’ll get a brand-new perspective on a familiar topic. There’s nothing wrong with taking 405 before 401—they are equally difficult.
- What is Analysis 2 (406)?
Obviously the successor to Analysis 1! There are many ways to satisfy the “and one more semester of analysis” requirement for the major, but this one is the most highly recommended.In this course, the tools of linear algebra start to be applied in a realm where vectors (finite sequences of numbers) are replaced by continuous functions. This may seem very odd, at first, but it makes sense if you think of a continuous function as an infinite sequence of numbers, indexed not by positions (1,2,3), but by the real numbers themselves. One coordinate per real number!Much as vectors can be decomposed in various ways as the sum of simpler functions, (remember Gram-Schmidt?), functions can be decomposed and written as a sum of “parts” in several very interesting ways. First, a (continuous periodic) function can be written as a sum of simple sinusoidal functions, giving the Fourier decomposition. If the original function is played on good quality speaker device, the simple sinusoidal parts will be the pitches distinguished by the ear, and their contribution to the sum will correspond to their relative perceived volumes.Functions can also be decomposed as the sum of polynomial functions (The Stone-Weierstrass theorem). This idea is strongly related to the notion of the Taylor series.
In order to prove these theorems, you’ll have to learn more sophisticated integration theory. In particular, you’ll learn to replace the Riemann integral with the more general Lebesgue integral. You will also learn about measure theory, which is the rather difficult answer to a simple question: “Given any arbitrary subset of the real numbers, how do we assign it a length?” If this question seems simplistic, let me convince you otherwise: How much of the total length of the integral [0,2] should be attributed to the rational numbers in that range, and how much of the length belongs to the irrationals?
- What is Calculus (106–109)?
Calculus is the study of change and accumulation. It forms the foundation and context for most forms of modern scientific thought. Differential Calculus introduces a general tool—the derivative—for understanding and quantifying change as a measurable thing. For position, there is velocity, a quantifiable (i.e., with units), measure of how position changes. For anything else (volume?), there is a derivative, which is to it as velocity is to position. Differential Calculus allows you to calculate these quantities and relate them to one another. Typically, if high-school math can solve a problem in which things are constant or change in simple proportions, then calculus can solve the same problem when things begin to change in complicated, curvy, nonlinear ways. A typical problem of differential calculus might look like this:A hemispherical bowl with radius 3 inches is half full, with water constantly entering at 3 cubic inches per second. How fast is the water level rising?Note the preoccupation with rates and such.Integral calculus is the study of how things accumulate over time or distance. A typical example of integral calculus might be this question:
If sunlight falls on my solar collector with an intensity of f(t) = 1/(1+t^2) [or whatever function], how much solar power will I collect in one day?
(Note: Accumulation!) or this instead:
What is the volume of a bowl manufactured by rotating the parabola y=x^2 around the y-axis?
What is the circumference of an ellipse with major radius 6 and minor radius 5?
This last is a much harder question than it looks!
Differential and Integral Calculus study rates of change and accumulations, respectively. They are taught in the same course because of one (two?) simple theorem(s), namely the Fundamental Theorem of Calculus. You’ll have to take the course to learn about that.
- What is Calculus III (202)?
Differentiate in six dimensions! Integrate in three dimensions! Impress your friends and dazzle your acquaintances!Think of temperature, which is a function of four variables (time, space, space and space). What does it mean to say “the derivative of temperature,” if it’s genuinely a function of all four things? What does it mean to say “the integral of temperature?” You’ll find out!Stokes’ Theorem, Green’s Theorem, and the like are absolutely essential for much of our modern physics, which needs calculus to study change, and multiple dimensions to study our big big world. Stokes’ theorem, in a nutshell, says: The fluid flow around the outside of a coffee cup is equal to the total vorticity within the cup. (Or perhaps I’ve paraphrased it beyond repair). The Divergence theorem, in a nutshell, says: The fluid flow out of the top of a (full) coffee cup is the flow into the cup plus total fluid expansion within the interior. Note: Rates of change + multiple dimensions = Calc 3.Many of the examples of Calculus 1 and 2 were artificial because the objects studied had to move in a line and change only in one dimension. With calculus 3, you can honestly study a satellite hurtling through space, affected by the gravities of many objects, and explain the nearby dustclouds as density functions. All this can be done with derivatives and integrals for added understanding.
- What is Complex Analysis (311)?
This is one way to continue studying calculus after Calc 3. Calc 1 and 2 can be called “the theory of functions of one real variable.” This course allows that variable to take complex values, by admitting the square root of -1 as a bona fide number. One might expect this theory to be basically identical, with minor adjustments here and there for “i”, but in fact it differs in very significant ways, and often feels like a completely different branch of math. The reason for this change can be roughly explained: In order for a function f from the complex numbers (x+iy) to the complex numbers (u+iv), to be differentiable, of course four partial derivatives (du/dx, du/dy, dv/dx, dv/dy) must exist. But multiplication by i rotates the u-coordinate into the v-coordinate, and the v-coordinate into the -u-coordinate (and similarly for x and y). Because multiplication by i (or any other complex number) must commute (more or less) with the process of taking derivatives at a point, these four partials have to “match up when rotated”.This statement amounts to the two equations called the Cauchy-Riemann equations. These two equations together imply that complex-differentiable functions are very special. In particular, they preserve angles (which makes them useful in modeling fluid flow), and have harmonic real parts (which makes them great models for temperature fields).One difficulty of reasoning about functions from the complex plane to the complex plane is that their graphs should be drawn on a four-dimensional space (two for the domain, and two for the range), but that’s impossible.Modern physics gives us strong empirical reasons to believe that imaginary and complex numbers do indeed exist, whatever that means. The tools developed in this course have very “real” applications, especially in quantum physics, despite our tendency to call them “imaginary”!
- What is Differential Equations (302)?
We need to learn about differential equations because there are many processes in the real world that are easy to describe, qualitatively and exactly, in the “right now,” but which are nevertheless very difficult to predict exactly.For example, an electron might travel through a container. It is affected by gravity, by various charges, and by magnetic fields. Gravitational and electrical affects depend on its position, and magnetic affects depend on its velocity. All the affects feed back into its acceleration, which immediately influences its velocity and position. This complicated feedback loop gives us an equation involving a(t), v(t), p(t), and t. Because the equation involves lots of derivatives, we call it a differential equation. To solve the equation is to find a formula for p(t) in terms of t, which is an exact prediction of the electron’s future path.Here’s another example: When a chain hangs, the tension on any point on the chain is determined by the overall shape of the whole chain, because points supporting more weight (i.e., higher points) have greater tension. But the curvature of the cable at a point must be inversely proportional to the tension. So the second derivative (curvature), which is of course determined by the function, also determines it via the tension equation. This feedback loop creates a different sort of differential equation. Solving the equation gives the exact shape in which the chain will hang. It turns out to be a catenary.In each example, the differential equation feels like the exact description of the process itself. Often it comes directly from a general law of physics. Solving the equation tells the implications of the laws, and describes the expected consequences of the rules of the particular system.
Many, many different techniques exist to solve differential equations, because many different kinds of differential equations exist. You will learn all the standard techniques, and along the way you will make good use of all of the prerequisites for the course!
- What is Linear Algebra (201)?
Vectors and matrices and more vectors and more matrices. This course (or often Calc 3) is usually the first in which you learn to truly do mathematics in multiple dimensions—two, three, or more! But the mathematics of six dimensions is very complicated, and here you simplify by studying the higher-dimensional analogues of linear functions (mx+b) only. So forget parabolas, trig functions, exponentials, and all the other functions which made matters complicated, and focus on functions that do not curve. There is a great deal to be said even about such a simple class of functions when you say it in 12 dimensions!When passing from one dimension (mx+b) to several, the numbers x get replaced with vectors. Matrices are the replacement for m. The simple multiplication of m*x becomes matrix*vector multiplication, which you’ll have to learn how to do!
- What is Number Theory (204)?
This is a gentle introduction to abstract mathematics, at a level majors and non-majors alike can enjoy. It’s not required by the major (although it does fill the important second algebra course requirement), and doesn’t serve as a prerequisite for lots of things to follow, but it’s an interesting survey of a branch of mathematics very, very different from the standard (calculus) fare. By Number, read Integer. Number theory is the study of integers only. So forget pi, e, and 1/2—you won’t need them. This doesn’t make number theory easy. On the contrary, it makes it rather hard!Here are some interesting questions which properly belong to number theory, just to give you a sense of the field:
- Why are 3^7-3, 4^7-4, and 5^7-5 all divisible by 7?
- What integers a,b,c can be Pythagorean triples? (i.e., a^2+b^2=c^2)
- The factors of 28 (excluding 28 itself) are 1,2,4,7, and 14, which sum to 28. 6 also does this. Are there any odd
- numbers which do this?
- Is 2^127-1 prime? (Note that’s a very large number. Trial division is not an option!)
- Is every even number the sum of two prime numbers?
- What is Putnam (225)?
The Putnam Exam is a highly competitive yearly national undergraduate math competition, taken by students very interested in math. You will be given six hours to solve 12 problems, ranging in difficulty from hard to incredibly hard. This is a fun course for majors and for others who like intensity and competition. Future math classes may be easier because of Putnam experience, because the most difficult skill trained by the Putnam class is proving theorems.