As mentioned on the syllabus,

I expect groups of two or three per project. The deadline for submission is the scheduled time of the final exam. This will constitute 20% or your total grade. Your final project must be typeset; I encourage you to consult with me through the semester in terms of how it looks and its contents.

What I expect is a paper where you explain the topic, and present its history and a few results on it with complete proofs. Work out a few examples. If relevant, do some numerical simulations. List all the references you consult. (Of course, do not plagiarize.) Some of the topics may end up being too ambitious, and if that occurs, let me know. In that case, it would be better to restrict your presentation (to some aspects of the topic at hand) rather than trying to be comprehensive.

I’ll give you a list of references you may find useful once you pick your topic, but of course if you find additional references, use those instead.

Suggested topics:

1. The Banach-Tarski paradox. (Chosen by K. Williams)
2. Quaternions and Octonions.
3. The 17 plane symmetry groups.
4. The word problem.
5. The Gordon game (See 5.5.2 on the book.)
6. The Redfield–Pólya enumeration theorem.
7. Character theory.
8. Conway polynomials.

0. During lecture I have sometimes skipped some arguments or not given as much detail as you may have wanted. If there was a result that in particular required of you some effort to complete in detail, please state it here and show me how you filled in the gaps left in lecture. Also, if there is a result for which you do not see how to fill in the details, let me know as well, as I may have overlooked something and it may be worth going back over it in class.

1. Give an example of a bounded set for which

does not exist.

2. Compute .

3. From the book, solve exercises 1.1.3, 1.1.5, 1.1.6, and 1.1.15.

[To get you started on 1.1.3: First verify in that assigns value 0 to any point. For this, use monotonicity and translation invariance, arguing first that . Then find that in terms of , and use this to find for any box with rational coordinates. Use this to compute for any box, and conclude by analyzing arbitrary elementary sets.

Note we essentially solved 1.1.15 in class, but under the assumption that 1.1.6 holds.]

4. From the book, solve Exercises 1.1.7-10. Make sure to explain in 1.1.9 why Tao’s definition of compact convex polytopes coincides with what should be our intuitive definition. Please also verify that convex polytopes are indeed convex.

(For a nice argument verifying that indeed , at least for even values of , see the paper “On the volumes of balls” by Blass and Schanuel, available here.)

5. From the book, solve exercise 1.1.11.

(If you are not comfortable with linear algebra beyond size , at least argue in the plane and in .)

6. From the book, solve exercise 1.1.13.

7. From the book, solve exercise 1.1.17.

1. Find the general form of a matrix (with real entries) satisfying the equation where and , or explain why no such exists. Note that the method explained in class does not work here, since is not invertible.

2. With the same and , find the general form of (or explain why it does not exist) if now we require that .

3. Consider the set of matrices whose entries are elements of , and that have the form . How many matrices are there in the set ?

Show that if and are in , then .

Show that if and , then exists and also belongs to .

Show that one of these matrices, “,” satisfies .

Solve the equation with , and check that the solution you obtain coincides with the solution described by the quadratic formula, that in this case looks like where of course by we mean a matrix (in ) whose square is .

4. We examine here the solutions of the cubic , following what is essentially Tartaglia’s method.

Show that if we define by , then the equation now takes the form .

Show that, no matter what and are, we can always find and such that and .

Suppose that are such that , , and . Show that is a root of . [Hint: Note that .]

Let be a cubic root of 1, . (Remember that this means that .) Check that and are also roots of , and that are all the roots.

Use this method to solve . Now note that are the roots of this polynomial. Reconcile this with the expressions you have found.

Use this method to solve in .

5. Here we solve the cubic using trigonometry. First, prove that .

Start with a cubic equation . As before, we can turn it into one of the form by means of a simple translation. Now, if happens to be 0, show how to find the roots. Suppose then that . Find a value of such that , and show that (for this ) if , then the equation becomes for some .

Comparing the results from the previous two paragraphs, we see that if is such that , then is a root of the cubic in . Use this to solve .

[This method seems more limited than the previous one, because we are used to thinking of as a real number, in which case must be a number between and . However, if we allow to be complex, then can take any value, and the formulas we obtain by this method actually coincide with the ones found in the previous problem.]

6. This is the method discussed in class. Suppose we are given the cubic equation and it has roots . [Note the main coefficient is 1 and I'm writing instead of .]

The point here was to find a polynomial in 3 variables with the property that some power of would take only two values (rather than six) as we permute the variables. The goal was (using a quadratic) to find these values when are used as the variables,  and then use these values to find themselves.

As mentioned in class, if we let be a cubic root of 1, , then the polynomial works because only takes 2 values, namely and .

When are used in place of , we get the expressions and Explain how to find if we know these two expressions.

Show that these two numbers are the roots of the quadratic

.

Use this to solve the cubic .

Extra credit problems:

(Extra credit problems can be turned in by February 8 at the latest.)

7. To solve the quartic equation, one would use the same procedure: Start with a quartic, find a polynomial in 4 variables a power of which takes at most 3 values (rather than ), Evaluate these powers when the roots of the quartic are used as the variables, and use a cubic to find these three values. Then use these three values to find the actual roots.

Suppose the quartic is . Under the substitution this becomes . Find in terms of .

Let the roots of be . Check that .

Consider and prove that takes only 3 values as we permute . When evaluated at , these 3 values are

, , and .

Prove that the cubic has roots .

Check that we get

,

,

,

and

.

Use this to solve .

What Galois proved can be phrased in these terms as saying that if we want to solve in general the equation of degree , then we need to be able to find a polynomial in variables, a power of which takes fewer than values as the order of the variables is permuted, and using these values we should be able to recover the roots of the original polynomial of degree . He proved that this is not possible for .

8. The Arabian mathematicians of the middle ages where able to solve quadratic equations but not cubics, and could not understand equations of degree 4 or higher. This is because they understood the equations geometrically, so squares represented areas and cubes volumes. In addition, they only understood positive numbers. So, an equation such as necessarily had to be presented as , while something like would be written as and thought of as being meaningless. Their main other drawback was that their arguments were rhetorical, meaning that they never used variables, which greatly complicated their exposition. For example, instead of asking to solve , they would say

“If three times an unknown added to 5 is equal to the square of that unknown, what is the value of the unknown?”

Investigate how they used geometric diagrams to solve quadratic equations, and write a (short) exposition of their techniques. Since numbers are non-negative, there are at least three cases, that need slightly different techniques: For equations of the form , of the form , and of the form . If we insist that quantities cannot be zero, then there are additional cases , , etc.

3. THE EQUALITY OF THE VOLUMES OF TWO TETRAHEDRA OF EQUAL BASES AND EQUAL ALTITUDES.

In two letters to Gerling, Gauss expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i.e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.

Hilbert’s student Max Dehn solved the problem in 1901 with the introduction of what we now call Dehn invariants:

Theorem. If two polyhedra in are equidecomposable into polyhedra, then they have the same volume and the same Dehn invariants.

In 1965, J.-P. Sydler proved the converse of Dehn’s result:

Theorem. Two polyhedra in with the same volume and the same Dehn invariants are equidecomposable into polyhedra.

A couple of years ago, Richard Schwartz, from Brown university, wrote a couple of very nice notes explaining both Dehn’s and Sydler’s theorems. He also developed a Java applet illustrating Sydler’s argument (for his “Fundamental lemma”). They can be downloaded here.

(The nicest presentation of the Bolyai-Gerwein result that I’ve found is in Howard Eves’ “A Survey of geometry“. The text of Hilbert’s original lecture delivered before the International Congress of Mathematicians at Paris in 1900 was expanded to a paper, “Mathematical problems”, Bull. Amer. Math. Soc. 8 (1902), 437–479. It has been recently (I’m old) reprinted, in Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436 and can be downloaded here.)

Ars Magna, “The Great Art”, by Gerolamo Cardano.

Ars Magna, whose first edition dates back to 1545, is an Algebra book, and generally considered one of the key scientific texts of the XVI century. It contains the earliest recorded formulas for solving cubic equations. It is also the subject of one of the earliest priority disputes in the history of mathematics.

Cardano attributes the method leading to the formulas to  Scipione del Ferro (1465–1526), but the general result is due to Niccolò Fontana, called Tartaglia, “the stutterer.”

However, Cardano does not recognize the formulas he uses as originating with Tartaglia (1499/1500–1557), who wanted to keep his method private, and had apparently disclosed it to Cardano under sworn secrecy.

Cardano acknowledges help by his student Ludovico Ferrari (1522–1565), who also discovered the formula for solving quartic polynomials.
Tartaglia’s description of the exchange that led to his disclosure of the method is highly entertaining, and can be found here. I quote the relevant part. For a slightly different translation of some of this exchange, see “Abstract Algebra“, by Robert Redfield.

Cardano: And I also wrote to you that if you were not content that I should publish them, I would keep them secret.

Tartaglia: Enough that on that head I was not willing to believe you.

Cardano: I swear to you by the sacred Gospel, and on the faith of a gentleman, not only never to publish your discoveries, if you will tell them to me, but also I promise and pledge my faith as a true Christian to put them down in cipher, so that after my death nobody shall be able to understand them. If you will believe me, do; if not, let us have done.

Tartaglia: If I could not put faith in so many oaths I should certainly deserve to be regarded as a man with no faith in him; but since I have made up my mind now to ride to Vigevano to the lord marquis, because I have been here already three days, and am tired of awaiting him so long, when I am returned I promise to show you the whole.

Cardano: Since you have made up your mind at any rate to ride at once to Vigevano to the lord marquis, I will give you a letter to take to his excellency, in order that he may know who you are; but before you go I should wish you to show me the rule for those cases of your, as you have promised.

Tartaglia: I am willing …

[Tartaglia then gave Cardano his rule in a poem he had written.]

When the cube and things together
Are equal to some discreet number,
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing
In the second of these acts,
When the cube remains alone,
You will observe these other agreements:
You will at once divide the number into two parts
So that the one times the other produces clearly
The cube of the third of the things exactly.
Then of these two parts, as a habitual rule,
You will take the cube roots added together,
And this sum will be your thought.
The third of these calculations of ours
Is solved with the second if you take good care,
As in their nature they are almost matched.
These things I found, and not with sluggish steps,
In the year one thousand five hundred, four and thirty.
With foundations strong and sturdy
In the city girdled by the sea.

This verse speaks so clearly that, without any other example, I believe that your Excellency will understand everything.

Cardano: How well I understand it, and I have almost understood it at the present. Go if you wish, and when you have returned, I will show you then if I have understood it.

Tartaglia’s approach is slightly different from the one we presented in class, as the idea of studying symmetries of the set of roots would not appear until much later, with the work of Galois and Abel.

His method, and a second approach, based on trigonometric identities, are discussed in the first homework set.

Ferrari’s method for the quartic led to much work trying to generalize it to quintic equations and beyond. Galois work shows that this is impossible in general, and led to the isolation of two key notions: Groups and, specifically, solvable groups.

We will discuss them in due time.

I have been unable to find a portrait of Ferrari, and will be thankful if you let me know of one.

The proof that are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass -test applies.

The proof that is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of can be seen in page 49 of Thim’s master thesis.

As usual, the function is given as a series where the functions are continuous, and we can find bounds with and . By the Weierstrass -test, is continuous. Also, the strategy for nowhere differentiability is typical: We associate with each point a pair of sequences and with strictly decreasing to and strictly increasing to . The key lemma (shown, for example, in Johan Thim’s Master thesis available here) is that, if a continuous function is differentiable at , then we have

In the case of the Faber functions, the functions add `peaks’ in the neighborhood of any point, and the locations of these peaks can be used as the points and ; moreover, the slopes accumulate at these peaks, and so the limit on the right hand side of the displayed equation above does not converge and, in fact, diverges to or .

Faber’s original paper, Einfaches Beispiel einer stetigen nirgends differentiierbaren Funktion, Jahresbericht der Deutschen Mathematiker-Vereinigung, (1892) 538-540, is nice to read as well. It is available through the wonderful GDZ, the Göttinger Digitalisierungszentrum.

The presentation is nice: As usual with these functions, this one is defined as the limit of an iterative process, but the presentation makes it very clear the function is a uniform limit of continuous (piecewise linear) functions, and also provides us with a clear strategy to establish nowhere differentiability.

Actually, the function is presented in a similar spirit to many fractal constructions, where we start with a compact set and some continuous transformations . This provides us with a sequence of compact sets, where we set and . Under reasonable conditions, there are several natural ways of making sense of the limit of this sequence, which is again a compact set, call it , and satisfies , i.e., is a fixed point of a natural “continuous” operation on compact sets.

This same idea is used here, to define the Katsuura function, and its fractal-like properties can then be seen as the reason why it is nowhere differentiable.

Syllabus for Math 515: Advanced calculus AKA Analysis II.

Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 9:40-10:30 am.
Place: MG 124.
Office Hours: MF 11-12.

Text: “An introduction to measure theory“, by Terence Tao. AMS, Graduate studies in mathematics, vol 126, 2011. ISBN-10: 0-8218-6919-1. ISBN-13: 978-0-8218-6919-2.

Contents: From the Course Description on the Department’s site:

Introduction to the fundamental elements of real analysis and a foundation for writing graduate level proofs. Topics may include: Banach spaces, Lebesgue measure and integration, metric and topological spaces.

We will emphasize measure theory, paying particular attention to the Lebesgue integral. Additional topics, depending on time, may include the Banach-Tarski paradox, and an introduction to Functional Analysis.

Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.

There will be no exams in this course. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term.

I will use this website to post additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

Section 1.
Instructor: Andres Caicedo.
Time: MWF 8:40-9:30 am. (Sorry.)
Place: MG 120.
Office Hours: MF 11-12. See here for contact information.

Text: “Adventures in Group Theory. Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys”, 2nd edn. By David Joyner. The Johns Hopkins University Press (2002). ISBN-10:0801869471. ISBN-13: 978-0801869471. Errata.

I will provide additional handouts and references as needed.

It may be a good idea to get a Rubik’s cube, as many examples we will see may be easier to understand with a cube in front of you. There are several online cube solvers (I particularly like this one), and they may be used as well, but I still recommend you get a physical copy.

The book presents many examples using the mathematics software SAGE. SAGE, developed by William Stein, is open source and may be freely downloaded. Consider installing it in your own computers so you can practice on your own. SAGE is very powerful and you will probably find it useful not just for this course.

(It was recently proved that Rubik’s cube can be solved in 20 moves or less, and 19 moves do not suffice in general.)

Contents: The usual syllabus for this course lists

Introduction to abstract algebraic systems – their motivation, definitions, and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups), followed by a brief survey of rings, integral domains, and fields.

Joyner’s textbook emphasizes group theory through permutation representations. The theory is illustrated by several permutation games. Other natural examples of groups come from geometric considerations. We will see many additional examples.

(An interesting example of groups arising from geometric considerations are the plane symmetry groups, which one can see nicely illustrated in La Alhambra. I visited Granada in 2005 and have uploaded to Google+ some pictures from the trip, where you can see further examples.)

Prerequisites: 187 (Discrete and foundational mathematics). Knowledge of 301 (Linear algebra) will be useful, though I will review the matrix theory we will need.

Grading: There will not be exams. Instead, the grade will be determined based on homework.

I will frequently assign problems (many will come directly from the book) and provide deadlines. Some of these problems are routine, others are more challenging, a few may give you extra credit points due to their difficulty. Although collaboration is allowed, each student should write their own solutions. If a group of students collaborate in a problem, they should indicate so at the beginning of their solutions. Also, if additional references are consulted, they should be listed as well. It may happen that while reading a different book you see a solution for a homework problem. This is fine, as long as it is not done intentionally, and I trust your honesty in this regard. For some problems, I may specify that no collaboration is allowed.

No problems will be accepted past their deadline, and deadlines are non-negotiable.

I will pay particular attention not only to the correctness of the arguments, but also to how the arguments are presented. Your final grade will be determined based on the total score you accumulate through the term.

It may be that you do not see how to completely solve a problem, but you see how to solve it, if you could prove an intermediate result. If so, indicate this clearly, as it may result in partial credit. On the other hand, the fact that you write something does not mean you will get partial credit.

In addition, you will be assigned a project (to work in groups of two or at most three), to be turned in at the latest by the scheduled time of the final exam. This will constitute 20% or your total grade.

Attendance to lecture is not required but highly recommended.

As the term progresses, I will be getting pickier on how you write your solutions. Introduce and describe all your notation. Use words as necessary; strings of equations and implications do not suffice. You may lose points even if you have found the correct answer to a problem but it is not written appropriately. Do not turn in your scratch work, I expect to see the final product. I am not requiring that you typeset (or LaTeX) your solutions, but I expect to be able to read them without any difficulty. Additional remarks are encouraged; for example, if a problem asks you to prove a result and you find a proof of a stronger statement, this may result in additional extra credit points.Your final project must be typeset; I encourage you to consult with me through the semester in terms of how it looks and its contents.

Once your total score is determined, I will then grade on a linear scale:

• If your final score is 90% or higher, you receive an A.
• If it is between 80 and 89%, you receive a B.
• If it is between 70 and 79%, you receive a C.
• If it is between 60 and 69% you receive a D.
• If it is 59% or lower, you receive an F.
• There may be a small curve up if the distribution of scores warrants this. Plus and minus grades might be used for grades near the top or bottom of a grade range.

I will use this website to post any additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.