The Beauty I See in Algebra: Margot Gerritsen at TEDxStanford (Full Transcript)

Margot Gerritsen – TRANSCRIPT

I’m going to take you back to class, to school, to algebra, and that’s probably the last thing that you wanted to do in a TED talk today. Maybe you remember in algebra class that you learned how to solve small systems of equations in variables X, Y, Z, and maybe they didn’t tell you exactly what these X, Y, Z were, and you were left with the impression that these equations were not particularly interesting nor particularly beautiful.

But actually equations like these are the very core of science and engineering, and all the algebra around them. And because each of these equations defines a relationship between these variables X, Y, and Z, and relationships and connections are all around us.

Now, for that reason, we love them, and I want to share some of that love with you today. Maybe you remember how to solve these equations, you express most of these variables in terms of one or another, here Y and after some manipulations, you get a final equation for Y that you can then solve that gives you back X and then Z. That works really well for these small systems of equations.

But perhaps you had these nightmares of going into class one day, and your teacher writing down on the board this test of equations and saying, “Go ahead and solve them.” Now the equations that I work with, have thousands if not millions of variables, and obviously we would really need a very big piece of paper, and a lot of patience to solve in the way you were taught. So, of course, we don’t. Instead we use computer programs.

Now, before we can use these computer programs though, we need to reorder these equations a little bit.

So we’re going to write them really really neatly under each other and where variables are missing, for example, Z in that first equation we add it but we multiply it with a zero. And then we explicitly write all the coefficients in front of it. Write everything in terms of addition like this. It seems a little bit silly, why would you do that?

But now you see, every equation looks the same. Something times X plus something times Y plus something times Z is something else. And so we don’t have to write X, Y and Z all the time, we just remember the order in which they occur, right? And we store these X, Y and Z in a little skinny table that we call a vector, like this, and then we store these coefficients the ones and the zeros and here also a minus one in a separate table, like – I will show you in a minute, here.

So now the system of equations that I have is really just a table of coefficients like this and then these little vectors with all these unknowns. Now this table of coefficients we call the matrix. And the matrix is so famous, they even made a few movies after it. And like Morpheus says in this movie, “The matrix is everywhere. It’s all around us.” Even now in this very room. So I want to give you a few examples of where matrices occur. And as first example I am going to take you to the San Francisco Bay.

So here in the San Francisco Bay, some of my colleagues designed a really nice computer simulation of the tidal flows going in and out the bay. And this is a really interesting simulation that can show you for example, salinity gradients in the bay or maybe surface velocities that are good and useful for the America’s Cup which is exactly what they did last year.

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Now the flow is much too complex to understand and computer velocity in every single point. So instead we want to be computing the velocities in a set of points distributed throughout the domain and here the points are the vertices of these triangles. Now through the laws of physics, we can relay the velocity in each of these vertices to velocities in neighboring points. There is a relationship between them. If there is a relationship, there must be an equation, and if I have a whole bunch of these equations, what do I get? The matrix.

Now, if I write down this matrix it will be extremely large, and it will have a lot of numbers in it, so I don’t do that, instead for every non-zero in this matrix I put a little blue dot, right? And then what I get is a matrix like this that we can actually look at from afar and then you can see structure in this. Now we use computer programs to create matrices like this or visuals of matrices like this and these are called spy plots, and sometimes these computer programs they have little Easter eggs in them. And so, one of these programs when you type the command, spy you see this. Now we can do the same thing in your body I don’t know if you realize this, but your body has matrices in it.

And now I’m going to take you to a simulation of my colleagues in Med USC in San Diego of blood flow through the aorta. Now this blood flow model was created using CT scanning of the aorta itself. And besides the blood flow it will also give you really interesting information about wall shear stresses that are important for blood clotting that you want to know about for bypass operations. And again, the velocities and the wall shear stress are computed in points distributed like this. And the matrix that comes out has a really interesting structure, too.

Now, search engines use matrices that tell them which words occur in which website. And I want to take four randomly chosen words: Stanford, beating, California, and the word, egg, right? And five websites that have one or more of these words in them. And so, there is a website over cooking, over TEDxStanford, the Berkeley website. Now what the search engine does it creates a beautiful table, where each row of the table corresponds to a word, and each column of the table corresponds to a web page. And if a word occurs in the web page then you put a little one and if it doesn’t, then you put a zero. And what do you get out of this? A matrix.

Now in reality these matrices are billions of web pages long and millions of words thick. And you need to be a pretty good mathematician in order to build a good search engine. So, you know now that this is a good field to be in to earn a little bit of money. So here are those matrices.

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Now a very variation on this theme is shown in the next slide of a matrix and here every column, every row corresponds to a text document, and there is a little blue dot if these two documents have a lot of words in common. So this is a connection matrix for 20 000 documents from the Classics. We can do the same with web pages, we can put web pages both along the rows and the columns, and put a little blue dot if there is a hyperlink from one web page to another.

And what you are looking at here now is the Stanford and Berkeley web domains. With Stanford nicely clustered, Berkeley nicely clustered, but there is communication between these two, because there are some blue dots in the other side as well, which is actually kind of surprising since we keep beating them. But they still want to communicate with us.

Now, matrices are also in your brain. Here you are looking at white matter fibers that connect gray matter regions in your brain. And looking at this, I could create a matrix that has gray matter regions in your brain along the rows and the columns and show connectivity between these regions. So, here you can stare this matrix for a long time and understand how your brain is all connected up. So matrices are everywhere, they model systems of equations, they’re in your brain, they’re in the search engine, and as mathematicians, we work for them daily, and we really love them. And these matrices, they have personalities for us.

So, when I prepared for this talk, I asked my students, what are your favorite matrices? And here are a couple that I wanted to share with you. So we have some matrices that are sparse with a lot of zeros, and then in the top right, the matrices that are symmetric, and symmetry, as we know, is always signifying beauty, so we love those. And then the matrices that are symmetric and sparse, and sort of bended like this so they are really fantastic. And then there are other matrices which I really like, a sort of blockage structures. But the very winner of this competition was the matrix we called the Toeplitz matrix, and that comes up a lot in signal processing. And it may not look like much to you, every diagonal in this matrix has a constant number and it makes it much easier to work with.

Now what about the nastiest matrices and here is the one that we hate the most. There are two elements that are very different in size and makes it really hard to work with. We call them ill-conditioned. But the worst matrices in the whole world are very very large matrices that are ill-conditioned, and to be able to work with those, we need to write specialized computer programs, and so there are millions of lines of codes everywhere.

Now the people wrote them in order to manipulate these very large matrices. For example, the matrix you heard about this morning for climate models, it is a very nasty matrix indeed. And these lines of code, they are behind of a lot of the simulation tools that you will see out there and that people use in science and engineering. Now, I said, I was going to talk about the beauty of math, and maybe so far, you’re not super excited to say, we’ve seen some blue dots from the screen that’s not very beautiful.

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So let me show you how we put beauty into mathematics and algebra. Here is a matrix and I’m going to associate a number with each row and column, one through four, right? And every number is now going to be related to a note or a little ball. There are going to be four of them, one, two, three, four. And now whenever I have a non-zero in the matrix, for example, at positions in the first row, positions one and three, it means there is a connection between one and one and one and three. Now one and one doesn’t show up, but one and three does. So the next row gives me a connection between two and four, the next row gives me an additional connection between three and four. And now I draw a connection, the last one, between one and four.

Now we have a little graph that comes out, and that looks much more pleasing than just a table with numbers. But you can just imagine that if I have a really big matrix with lots of ones and zeros what a mess that would be. All these little balls and all these little links between them So here is the trick. Whenever there is a link between these two balls we imagine there is a spring that pulls these balls together. But at the same time, we give the balls also repelling charges, so that when they’re not connected by a link they repel each other, push each other away, and then we just let things go, and sort themselves out. And what looked to be a mess, just a simple matrix that was messy is turning into a beautiful structure. It just finds the minimum energy, and the outcome is this fantastic geometrical figure.

And when we study them, we can actually see some of the properties of the physics, even coming out in these pictures. Now we can apply those to lots of different matrices. And I want to share some of the prettiest ones. Here is people and the web pages they like. Modelling of a lung, the matrix corresponding to that. Financial portfolio analysis. Shallow water models, estuary flows and so on. The Stanford web. MRI modelling. Analog circuits… makes your hair stand up. Tidal flow models. And my very favorite one, is called the galaxy that tells you how the catalogs and sub catalogs in the Library Congress are all connected together.

So, that math that you learned at high school that these equations that you didn’t like so much, that’s behind everything. It is beautiful, it’s omnipresent, It’s everywhere and who knew you could make such pretty pictures. Thank you very much.

The Beauty I See in Algebra: Margot Gerritsen at TEDxStanford (Full Transcript)

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