What’s with all the calculus?

I hate calculus. So, so much. I hate that the first type of mathematics students see at university (the only kind for many of them) is calculus. No one does research in trigonometry. Real analysis is a vibrant research area, but modern real analysis has little to do with the series of tricks we expect students to learn in calculus class. That’s actually what annoys me the most: the calculus curricula are not about understanding calculus, but rather memorizing a series of tricks to then regurgitate them during the 4 (!) midterms and the final.

If I had a say in the development of mathematics courses at university, this is what I’d do:

Scrap calculus. The general math requirement would be fulfilled by a general introduction to mathematics. We would study the unit circle, coordinate geometry, basic number theory, basic combinatorics, the idea of limits, some interesting probability theory, and some graph theory. This course would introduce the breadth of mathematics, and be a proof-based course.

For science majors, there would be a calculus sequence that prepared them for engineering, physics, and applied mathematics. Since only science majors would take this class, it could be taught considerably faster than current calculus classes. The first semester would cover advanced trigonometry, derivatives, and integrals, the second multivariable calculus, and the third differential equations. This could be well integrated with the science courses to make it even more relevant to the students that actually need to take the calculus sequence.

Math majors would not waste two years taking calculus and differential equations before getting to the real math. Instead, they would take introduction to mathematics, and the first course in the calculus sequence. During their second semester, they would be encouraged to take courses like real analysis, linear algebra, abstract algebra, combinatorics, probability theory and number theory in parallel with the second calculus course.

All courses would be proof-based, for both math majors and non-majors. Any class that consists of learning and regurgitating a series of tricks does not belong at university, and I’m a bit ashamed that I’m TAing one right now. We skip all of the beautiful theory underlying calculus, and simply test the students on how well they can regurgitate tricks in a test situation. I read something wonderful on one of my favourite blogs, Math with Bad Drawings, the other day: math tests should be like Turing tests. They should test if there is something intelligent on the other side; something more than what a computer can do. I feel that the same should hold for math classes in general. If they don’t teach you anything that a computer can’t do, the class should not exist. And that’s why I want to scrap calculus as the introductory math class.

Many students sadly don’t have the background or motivation to learn the interesting parts of calculus, so introductory calculus classes focus on just teaching tricks instead. Moreover, the calculus sequence leaves students with a warped idea of what maths and maths research are. It’s not about tricks or calculating values. It’s about understanding how it all fits together, and how to use small pieces that other people found to build bigger things. I wish that the introductory math classes introduced the feeling you get when you managed to get all the pieces to fit together, and prove something for the first time.

What do you think?

A theorem by Erdös

This is a quick application of the probabilistic method to additive number theory that I personally think is really cool. A set is called sum-free if it contains no such that .

Theorem (Erdös, 1965). Every set of nonzero integers contains a sum-free subset of cardinality at least .

Proof. Let be a prime such that and let . Clearly, is a sum-free subset of the group , and .

Let us choose at random an integer , according to a uniform distribution on and define by $d_i \equiv xb_1 \pmod p$, and let $latex 0\leq d_i

1/3$. Therefore, the expected number of elements such that is more than .

Consequently there is an and a subsequence of of cardinality at least such that for all . Since is sum-free, so is . Q.E.D.

Let mathematics be abstract

I’m going to rant for a little while about my favourite pet peeve in math education: that everything has to be concrete and connected to the real world. Students suck at algebra? Try to relate it to the real world. Students hate geometry? Have them use similar triangles to calculate the height of a flagpole. Students question why they need to study derivatives? Come up with some ridiculously contrived reason why everyone needs to use derivatives in their working life.

It’s ridiculous. Students are not stupid. They know that they will never need to calculate the height of a flagpole, or the terminal velocity of a stone dropped from a building while ignoring the effects of gravity in most jobs. I have never needed to calculate the volume of a room, and never in my entire life have I used trigonometry outside a mathematics classroom. And I’m working on my PhD in math.

I strongly believe that all these contrived explanations for why mathematics is important to everyone are only alienating students. They know, just like I did when I was in school, that they will use very little mathematics outside a math classroom. Trying to tell them differently with the kind of ridiculous explanations I mentioned above (all are things I have a actually heard from teachers) will only convince them that mathematics is completely unnecessary, since teacher clearly needs to lie to them to convince them to study it.

Instead, I propose we teach mathematics the way we teach poetry. Teach mathematics because it is beautiful, intriguing, and has fascinated the best minds in the world for thousands of years. Stop teaching kids that mathematics is just a tool. Teach them that mathematics is about the deep structures we can find in numbers: the basic building blocks of the universe. Stop forcing students to memorise endless tables of trig values and integrals, and focus on the deeper structures. Introduce number theory, graph theory, combinatorics, and logic. Show them the breadth of mathematics, not endless ways to calculate numbers.

Few students want to study math heavy subjects at university, because we have picked the most boring parts of math and show only those to young kids. Then, when they ask why they should care care, we give them contrived explanations that any 10 year-old can see through. Instead, show them why mathematicians do math: we want to understand the deepest of structures in the real world. Let them explore how numbers work instead of just giving them formulae to calculate numbers.

Allow math to be abstract, and deep, and beautiful, and try to get kids interested in that, rather than lie to them about how useful trigonometry will be in their future career.

Rant over.

Mindful practice

Oh impostor syndrome. I still feel like everyone else in my program is just gliding through the classes, and will ace the prelims in August, whereas I am struggling with most of it. And then I have a sudden breakthrough when I finally figure out a homework question on semidirect products that no one could figure out during the semester, and all the feelings of inadequacy go away. For a few hours, or maybe a few days, only to come back again. So I’ve been reading a good book on what connects successful people over all disciplines: math, music, sports, business, and anything else you can think of. It’s called “Talent is overrated” by Geoff Colvin. From the title, I imagine you have some idea what it might say. The author goes through the research on successful people from various disciplines, and discusses some very interesting findings. For example, there is no evidence that what we call “talent” even exists. Mozart started preforming at a very young age, but he had already been trained by his father, one of the top music teachers in Vienna, for years before he started. Those of his works that are still preformed today were composed when he was around 20, after 17 years of training.

The same pattern is found for all other successful people. Tiger Woods trained from a very young age, much harder than anyone else, and thus became a golf prodigy. Simply because he had more practice than anyone else that young. Not because he had magic golf playing genes. Simple practice. Benjamin Franklin spent years practicing how to write, and emulating famous writers, before he wrote any of the things he is now famous for.

This is where you object “but write 3000 word for my blog every week, and I’m not a professional writer”. That objection get at the heart of the problem: what sets successful people apart from average performers is what is called “conscious practice”. Playing the same piece over and over again on the piano will not help you improve, no matter how many hours you spend with it. Sitting down for an hour or two every day with a specific goal in mind: “today I will practice the transition between chord X and chord Y” will help you improve. Spending hour every day mindlessly playing chess will not make you a chess grad master. Spending hours every day analyzing games played by masters and your own games for advantages and weaknesses will help you get better. This is the reason ballerinas spend hours every day at the barre, practicing the same developpe they have been doing for 20 years.

The idea is that being mindful and practicing the basic, underlying skills of your trade will make you better at it: much, much better than simply mindlessly trying to dance that pas de deux, or playing chess over and over again. What separates the experts from average performers is their absolute mastery of the basic skills of their trade.

I sure hope that’s conscious practice (curtesy of SMBC).

So, how does this translate to math research? The basic skills needed are a deep knowledge of your particular area of interest, as well as any closely related fields, and a knowledge of the current research. I’m a graph theorist, so for the basic knowledge of my field, I have a few books to work through: currently “Graph theory” by Reinhard Diestel. Every day, I spend at least 30 minutes working through the book. I don’t focus on exercises: I focus on parts where I can get immediate feedback. I make sure I understand the text, try to predict what would be the next natural development, and try my hand at all of the proofs. I read the theorem or lemma, try to prove it, and then compare how my (sometimes only partial) proof compares to the one in the book. Which one is more elegant? easy to understand? This gives me immediate feedback on how I’m improving as a mathematician, and how well I’m internalizing the key methods used in my field. To supplement this, I also keep up to date with current research. Every week, I pick a newly submitted paper from ArXiV to read. I spend a half hour every night on this paper, with special focus on the proof technique(s) they used. After all, proof techniques are the tool kit of a mathematician.

I’ve only been adhering to the principle of mindful practice for a couple of months, but I already feel more confident in my weekly meetings with my adviser, and during the department graph theory seminar, because I have so much more background knowledge. As for how it will work out over the long term, I will keep you updated!

Can we teach kids real math with computers?

As so many blog posts and articles with a question in the headline, this one follows Betteridge’s law of headlines. I disagree with the sentiments put forth in this following TED talk by Conrad Wolfram, the creator of Wolfram Alpha, and best friend of calculus students everywhere. Before you read on, you can watch the talk here:

Like I said above, I strongly disagree with the conclusion of Wolfram’s talk. The idea he promotes is that students shouldn’t have to bother with mechanical calculations, because there are now computers that can do mechanical calculations faster and more accurately than any human. Instead, mathematics should focus on teaching kids how to apply their knowledge in the real world, and how to make the computer do the difficult, numerical parts for us. We can do all these really cool things with math – math underlies Google, rockets, the financial market, and countless other things. However, since students mostly do mechanical calculations and occasionally see a very simplified model of a real world event, they never see how cool mathematics is and lose interest. I agree completely with this problem statement. However, his solution is entirely off-base.

Wolfram is making the same mistake countless elementary school teachers do by dumbing down mathematics to be strictly about applications. I personally realized early, like Wolfram points out in his talk, that the applications we get to see in school are silly. There has never been a time in my life where I needed to estimate the height of a building by knowing the length of my shadow and its shadow and applying similar triangles to it. This far, Wolfram has a point. But after that, once he starts talking about how we therefore should skip mechanical calculations and move on to realistic computer simulations of the real world, he loses me.

He believes that we are teaching math for three reasons: to prepare kids for technical jobs, to better navigate our modern world (mortgages, statistics and those kinds of things) and to teach kids to think logically. If this were really the reasons, then maybe Wolfram would have a point. However, there is no other subject that has to justify its existence in this way. When will you ever dissect a frog or need to recite Shakespeare? Probably never, for the overwhelming majority of you. Why does then math have to be all about what it’s good for, when no one questions why kids should have to read Shakespeare?

Consider what would happen if we were told in English class that we need to be able to read to interpret road signs, recognize the letters for a vision test and occasionally write a memo at work. And then we spend the next 12 years practicing vision tests, reading road signs and writing memos. How many authors, poets, and songwriters would we have if we were taught in school that English is strictly about such mundane applications? But that’s not what we do in English class. Instead we read Shakespeare, Austen, and WH Auden, and try our hand at writing poetry, and fiction, and plays. And today everyone and their cat has a blog, or is working on a novel, or wants to be a screenwriter.

We don’t learn math in school because math will be useful in the rest of our life. For the vast majority of you, you will not use derivatives any more than you use that close-reading of Romeo and Juliet. We learn math because it is a major area of study, and hence one that students should be familiar with to be able to decide what they want to do with their life. Don’t show kids more realistic applications of math in school, most people will still never use it: show them more math instead.

Teach kids graph theory, combinatorics, number theory, cryptography, probability theory, non-euclidean geometry, or propositional logic instead. Bring in some of the wonderful areas of mathematics that can be extremely accessible to kids if presented in the right way. Show them the breadth of mathematics. At the risk of sounding like a cliché: show them the beauty of mathematics; its power and adaptability. See how many kids will still want to study math after learning math, not applications of math like they do today.

Virtually all of the applications Wolfram talks about are applications of calculus. This is one of the main problems I see with mathematics education: most kids even at university see nothing but trigonometry and calculus. When I tell people I’m a mathematician, they have no idea what kind of research there even is to do in math. Do I just sit around all day and take derivatives and integrals of things? Figure out new ways to calculate areas of triangles? We would never accept an English curriculum that only teaches kids how to read technical texts. Why do we accept a math curriculum that only teaches mechanical calculations?

I agree with Wolfram that the focus on mechanical calculations of derivatives and integrals is too strong and probably scares off quite a few students. But don’t substitute it with more calculus using computers as tools: substitute in more math. Show students the whole breadth and beauty of mathematics that is out there, and that only very few of us ever get to come near. Show students that, and see what happens to the number of kids who still like math in high school and onwards.

Ribbon graphs of link diagrams

This has been a weird week. Grad school seems to go in waves: first I feel really stupid and like I understand nothing, and then suddenly I’m on top of everything. Up until now, this rollercoaster has been out of sync in all of my classes, but this week I’ve been on top of everything. I wonder how long it will last. I even understand what we’re doing in analysis (intro to functional analysis), though that will probably change when we start with abstract measure theory next week…

Update: It did not change when we did abstract measure theory. Shocking, I know. I found the abstract stuff so much easier to do than the concrete Lebesgue measure. When we did abstract measure theory, I didn’t get confused by all the things I do know about the real numbers and I could focus on just understanding measures on sets, rather than mixing in all the weird things that happen with real numbers. Much more fun.

I have a vague recollection of promising a post on the topic of my conference presentation, so here it goes. First of all, you need to know what a knot is. A normal knot, that you tie your shoes with, is not mathematically interesting. It can be tied, so it can (mathematically at least) be untied. Instead, to define a mathematical knot we tie a knot in a piece of string, and connect the ends. That’s the intuitive definition. There are some tweaks to make it behave nicely, but you get the general idea. A link is a knot with several components.

The title of this post mentions link diagrams. We can find a link diagram of a link by simply projecting the link “nicely” onto the plane. I’m fudging a lot of the details about knots, because they are not the main topic here. I might do a post later on knots where I explain all the details properly. Here “nicely” should be taken to mean that one can recover the original knot from the projection.

Now that we have a vague idea what a link diagram is, we define a ribbon graph. A ribbon graph is a collection of discs called vertices, and a collection of discs called edges. The edge discs intersect exactly two vertex discs along an arc on each vertex, and no edges intersect. An easier way to think about a ribbon graph is as a graph embedded in a surface, and then we cut out a thin neighbourhood around the graph to obtain a ribbon graph.

Next, I’m going to tell you how ribbon graphs are connected to link diagrams. We take a link diagram, and eliminate each crossing just like when constructing a knot polynomial, choosing a negative or a positive splicing at each crossing. We can choose whichever of the two ways to eliminate each splicing we like. When all the crossings have been eliminated, we are left with a number of disjoint circles. These will be the vertices of our ribbon graph. Connect the vertices where there were crossings with edge discs, and you will end up with a ribbon graph. Since we could choose any splicing we for each crossing, there are up to possible ribbon graphs associated with each link diagram.

The natural questions to ask is then: how are these connected? What can we say about the ribbon graphs of a given link diagram? If two link diagrams have the same set of ribbon graphs associated with them, how are the link diagrams related?

My undergraduate advisor already answered both of those questions for links in . It turns out that the ribbon graphs arising from the same link diagram are partial duals of each other. For the second question, it turns out that the link diagrams are related by a very simple “flip” move. In my undergraduate thesis, I proved that the same results holds for checkerboard-colourable link diagrams of links in . This is roughly what I presented at the conference a few weeks ago, to a room full of grad students. Here is a link to the paper on ArXiV.

It was a very nice conference, by grad students, for grad students. The organizers did a really good job putting it together, and there were a lot of interesting people there. It was all very friendly, in part I think because we are all so early in our careers. There were none of the “let me interrupt to talk about why my research is superior disguised as a question for 10 minutes”, which was nice. I also saw quite a few interesting talks, in particular one on the Erdös-Ko-Rado conjecture that I might try to turn into a blog post on a later date. No promises though, since apparently things keep popping up that keep me from blogging for weeks at a time. It should be easier for me to write regularly now that term is over, but no promises on the topics.

Unusual application of the Brouwer Fixed Point Theorem

Sometimes strange things happen in topology class. Some days we spend 20 minutes arguing about what to conjugate with what to show that something is in the normaliser of a group, other days no one knows what they are talking about, and yet other days we see some really surprising results. This particular one made me stop and go “What!” loudly the first time I read the problem ,and took several days to figure out. We had just proved the Brouwer Fixed Point Theorem, and were generally proving things about fundamental groups when suddenly this question comes up:

Prove that a matrix M with positive real entries has an eigenvector with a positive eigenvalue. (Hint: consider

So, the hint wants us to look at a triangle in , and use that to prove this strange result from linear algebra. It turns out that the solution is amazingly short and neat. Here we go.

Consider the transformation . This is a new, scaled version of , so a function . Say that the transformation takes a point to a point . Now, this transformation has a fixed point: for some , . So, for some , , where . Hence is our eigenvector and our positive eigenvalue.

This is such a cool result: don’t you think? Do you have a favourite surprising application of one branch of mathematics in another?

Graph embeddings – The Map Colour Theorem

Happy 2015 everyone! Last year was a good one. I graduated with a good undergraduate degree, and passed my first semester in grad school. I’ve spent some time thinking about what area I want to specialise in during the next four and a half years. I’m leaning towards graph theory, and in particular graph embeddings as my specialisation in grad school. This will be the first post of many on graph embeddings, why they are interesting and different useful techniques. I’m sure most of you have heard of the map colour theorem before: it states that given a simple planar graph, the vertices can be coloured using at most four colours such that no two adjacent vertices have the same colour.

This result was first conjectured in the 19th century by a Francis Guthrie. There were many attempted proofs, but no correct ones until over 100 years later. The final proof was done using a computer. It is based on the idea that if the conjecture is false, then there exists a smallest graph which requires five colours to be coloured. The proof showed that no such counterexample can exist. They managed to show that most graphs can be reduced to smaller ones with the property that if the smaller map can be 4-couloured, so can the original. Using this idea, they reduced the infinitely many planar simple graphs to 1936 cases that were checked by computer. The complete proof was announced in 1976 by Kenneth Appel and Wolfgang Haken at University of Illinois. It was considered quite controversial at the time, and still is to some, since it was the first proof of a major theorem that was done in large part by computers. Several people have claimed to have found major errors in the proof, but none have stood up. The proof has also been made more efficient by several computer scientists, minimising the chance that there might be a mistake in it. In 2005, Benjamin Werner and Georges Gonthier formalised the proof in the Coq proof assistant (possibly a future post). This mans that we don’t have to trust that Appel  and Haken’s algorithm’s are correct any more, we just have to trust Coq.

One of the first things that was proven related to the four colour theorem was the five colour theorem. It was proven by Percy John Heawood in 1890, and is based on the false proof of the four colour theorem by Kempe in 1879. The proof is by contradiction as follows:

We assume that is the smallest graph which cannot be coloured using only five colours. Using the Euler formula we can show that either every vertex is of degree at most 4, in which case the graph is trivially five-colourable, or it has a vertex, say , of degree 5. Now consider the graph , obtained by removing vertex . It had five adjacent vertices. Since is smaller than , it can be five-coloured, so these vertices, (in cyclic order around ) are coloured with at most five different coulours. If they are coloured with less than five coulours, we can use one of the unused ones for to obtain a five-colouring of . Hence they all have different colours: . Consider the subgraph containing only the vertices coloured by or . Either and are in different components, or they are in the same component. If they are in different components, then the colouring of say the component containing can be reversed, freeing up colour for the vertex . Hence they are in the same component, so there is a path joining and , using only colours 1 and 3. Now consider the subgraph . By a similar argument, vertices and must be connected by a path coloured only with colours 2 and 4. However, since the vertices were labeled cyclically, this path must intersect the one between and , which is a contradiction. Hence can be five-coloured.

A few last words: the four-colour theorem can be generalised to graphs on any surface, and this is where graph embeddings come in. Heawood conjectured that the maximum number of colours needed for any graph is given by the formula . This was proven true by Ringel and Young in 1968, with one exception. I’ll talk more about this in my next math post.

Interesting property of matrices

A few weeks ago, Gilbert Strang from MIT gave my department’s colloquium on the topic of banded matrices. A diagonal matrix is a 1-banded matrix, one where the semi-diagonals above and below the diagonal are non-zero as well is a 3-banded matrix, and so on. An example of a 3-banded matrix is .

Now, the interesting property of matrices is as follows: take a -matrix, say . Notice that the ratio on the diagonal is  . Now square the matrix: . The ratio on the diagonal is still . If we cube we get . Now the ratio on the diagonal is . It actually turn out that thus ratio along the diagonal is preserved for all powers of a -matrix. There are several ways to prove that it holds: but the most obvious is by induction on the powers of . This proof is fairly mechanical, so I will leave it as an exercise for the reader 🙂 (I’ve always wanted to write that).

The first question we ask as mathematicians is “how general is that property”. For these matrices, it turns out that it doesn’t generalise to general -matrices. However, it does generalise to 3-banded matrices. All the ratios along the diagonal of a 3-banded matrix are preserved under powers of the matrix. This is once again easy to prove by induction on the power of .

I think this is a really neat little fact, and I’ve never seen it before. I don’t know if it shows up in any applications, so if you know of any please leave a comment! I’ll be back after Christmas with some graph theory 🙂 Merry Christmas!

Algebra Basics – Algebras and Groups

Today we’ll continue on with short but dense post on some more basic algebra. We are going to define what an algebra is, and one basic algebraic structure: a group.

First, some notation. We will denote by the set of all maps from a set to a set . A type is a collection of natural numbers indexed by the set . An algebra of type is an ordered pair where is a non-empty set called the universe set of and a family of maps called the fundamental operations of .

The fundamental operation is called a -ary operation. Specifically, we talk about nullary (0), unary (1), binary (2) and ternary (3) operations. Now we can start looking at different algebraic structures.

1. A groupoid is an algebra with a single binary operation. A semigroup is a groupoid which satisfies the associative law.
2. A monoid is an algebra with a single binary operation such that is a semigroup and a nullary operation such that .
3. A group is a monoid in which every element has an inverse. It can also be thought of as an algebra , with an additional, unary operation

This is a fairly dense text if you’re not familiar with the terminology used, so I’m going to leave it at this and post later in the week about more group theory: permutations, isomorphism theorems and Lagrange’s theorem.