- Emacs kill (cut) commands
- Emacs point (cursor) movement
- Getting started with Emacs on Windows
- Notes on Unicode in Emacs

See also the Twitter account UnixToolTip and blog posts tagged Emacs.

**Last week**: Miscellaneous math notes

**Next week**: R resources

It’s not too hard to create a table of sines at multiples of 3°. You can use the sum-angle formula for sines

sin(α+β) = sin α cos β + sin β cos α.

to bootstrap your way from known values to other values. Elementary geometry gives you the sines of 45° and 30°, and the sum-angle formula will then give you the sine of 75°. From Euclid’s construction of a 5-pointed star you can find the sine of 72°. Then you can use the sum-angle formula to find the sine of 3° from the sines of 75° and 72°. Ptolemy figured this out in the 2nd century AD.

But if you want a table of trig values at every degree, you need to find the sine of 1°. If you had that, you could bootstrap your way to every other integer number of degrees. Ptolemy had an approximate solution to this problem, but it wasn’t very accurate or elegant.

The Persian astronomer Jamshīd al-Kāshī had a remarkably clever solution to the problem of finding the sine of 1°. Using the sum-angle formula you can find that

sin 3θ = 3 sin θ – 4 sin^{3} θ.

Setting θ = 1° gives you a cubic equation for the unknown value of sin 1° involving the known value of sin 3°. However, the cubic formula wasn’t discovered until over a century after al-Kāshī. Instead, he used a numerical algorithm more widely useful than the cubic formula: finding a fixed point of an iteration!

Define *f*(*x*) = (sin 3° + 4*x*^{3})/3. Then sin 1° is a fixed point of *f*. Start with an approximate value for sin 1° — a natural choice would be (sin 3°)/3 — and iterate. Al-Kāshī used this procedure to compute sin 1° to 16 decimal places.

Here’s a little Python code to play with this algorithm.

from numpy import sin, deg2rad sin3deg = sin(deg2rad(3)) def f(x): return (sin3deg + 4*x**3)/3 x = sin3deg/3 for i in range(4): x = f(x) print(x)

This shows that after only three iterations the method has converged to floating point precision, which coincidentally is about 16 decimal places, the same as al-Kāshī’s calculation.

Source: Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

]]>This morning I ran across the etymology of the word:

In the late 1800s, the physicist Ludwig Boltzmann needed a word to express the idea that if you took an isolated system at constant energy and let it run, any one trajectory, continued long enough, would be representative of the system as a whole. Being a highly-educated nineteenth century German-speaker, Boltzmann knew far too much ancient Greek, so he called this the “ergodic property”, from

ergon“energy, work” andhodos“way, path.” The name stuck.

Found here, footnote on page 479.

Other etymological footnotes:

]]>

There’s not a good way to find these pages except through search. So I plan to categorize them and write a short post each Wednesday for the next few weeks listing some related pages. This post starts the series with math notes that didn’t fall into any other category.

- Big-O and related notation
- Notes on Spherical Trigonometry
- Solving quadratic congruences
- The difference between an unbiased estimator and a consistent estimator
- General binomial coefficients
- How to calculate binomial coefficients

See also posts tagged math.

Next week: Emacs resources

]]>… the famous

googol, 10^{100}(a 1 followed by 100 zeros), defined in 1929 by American mathematician Edward Kasner and named by his nine-year-old nephew, Milton Sirotta. Milton went even further and came up with thegoogolplex, now defined as 10^{googol}but initially defined by Milton as a 1, followed by writing zeros until you get tired.

**Related post:** There isn’t a googol of anything

The first from Princeton was The Best Writing on Mathematics 2014. My favorite chapters were *The Beauty of Bounded Gaps* by Jordan Ellenberg and *The Lesson of Grace in Teaching* by Francis Su. The former is a very high-level overview of recent results regarding gaps in prime numbers. The latter is taken from the Francis’ Haimo Teaching Award lecture. A recording of the lecture and a transcript are available here.

The second book from Princeton was a new edition of Andrew Hodges’ book Alan Turing: The Enigma. This edition has a new cover and the new subtitle “The Book That Inspired the Film ‘The Imitation Game.'” Unfortunately I’m not up to reading a 768-page biography right now.

The first book from No Starch Press was a new edition of The Book of CSS3: A Developer’s Guide to the Future of Web Design by Peter Gasston. The book says from the beginning that it is intended for people who have a lot of experience with CSS, including some experience with CSS 3. I tend to ignore such warnings; many books are more accessible to beginners than they let on. But in this case I do think that someone with more CSS experience would get more out of the book. This looks like a good book, and I expect I’ll get more out of it later.

The final book was a new edition of How Linux Works: What Every Superuser Should Know by Brian Ward. I’ve skimmed through this book and would like to go back and read it carefully, a little at a time. Most Unix/Linux books I’ve seen either dwell on shell commands or dive into system APIs. This one, however, seems to live up to its title and give the reader an introduction to how Linux works.

]]>Front:

Back:

Designed by my friend Scott Bronstad. Scott also designed the new look of the web site. (If something doesn’t look quite right, that’s probably my doing.)

]]>The Pareto principle would say that importance is usually very unevenly distributed. The universe is essentially hydrogen and helium, with a few other elements sprinkled in. From an earthly perspective things aren’t quite so extreme, but still a handful of elements make up the large majority of the planet. The most common elements are orders of magnitude more abundant than the least.

The uniformitarian view is a sort of default, not often a view someone consciously chooses. It’s a lazy option. No need to think. Just trudge ahead with no particular priorities.

The uniformitarian view is common in academia. You’re given a list of things to learn, and they all count the same. For example, maybe you have 100 vocabulary words in your Spanish class. Each word contributes one point to your grade on a quiz. The quiz measures what portion of the *list* you’ve learned, not what portion of that *language* you’ve learned. A quiz designed to test the latter would weigh words according to their frequency.

It’s easy to slip into a uniformitarian mindset, or a milder version of the same, underestimating how unevenly things are distributed. I’ve often fallen into the latter. I expect things to be unevenly distributed, but then I’m surprised just how uneven they are once I look at some data.

**Related posts**:

Four reasons we don’t apply the 80-20 rule

Gerald Weinberg’s law of twins

The Lindy effect

]]>

1/7 = 0.142857142857… 2/7 = 0.285714285714… 3/7 = 0.428571428571… 4/7 = 0.571428571428… 5/7 = 0.714285714285… 6/7 = 0.857142857142…

We can make the pattern more clear by vertically aligning the sequences of digits:

1/7 = 0.142857142857… 2/7 = 0.2857142857… 3/7 = 0.42857142857… 4/7 = 0.57142857… 5/7 = 0.7142857… 6/7 = 0.857142857…

Are there more cyclic fractions like that? Indeed there are. Another example is 1/17. The following shows that 1/17 is cyclic:

1/17 = 0.05882352941176470588235294117647… 2/17 = 0.1176470588235294117647… 3/17 = 0.176470588235294117647… 4/17 = 0.2352941176470588235294117647… 5/17 = 0.2941176470588235294117647… 6/17 = 0.352941176470588235294117647… 7/17 = 0.41176470588235294117647… 8/17 = 0.470588235294117647… 9/17 = 0.52941176470588235294117647… 10/17 = 0.5882352941176470588235294117647… 11/17 = 0.6470588235294117647… 12/17 = 0.70588235294117647… 13/17 = 0.76470588235294117647… 14/17 = 0.82352941176470588235294117647… 15/17 = 0.882352941176470588235294117647… 16/17 = 0.941176470588235294117647…

The next denominator to exhibit this pattern is 19. After finding 17 and 19 by hand, I typed “7, 17, 19″ into the Online Encyclopedia of Integer Sequences found a list of denominators of cyclic fractions: OEIS A001913. These numbers are called “full reptend primes” and according to MathWorld “No general method is known for finding full reptend primes.”

]]>Hello-world programs are often intimidating. People think “I must be a dufus because I find hello-world hard. At this rate I’ll never get to anything interesting.”

The problem is that we confuse the *first* task with the *easiest* task. Hello-world programs are almost completely arbitrary. You can’t deduce what a compiler is named, where files must be located, how they must be formatted, etc. You have to be told. The amount of arbitrary material you need to learn is greatest up-front and slowly decreases.

When I started programming I thought I’d quickly get past the hello-world stage and only write substantial programs from then on. Instead, it seems I’ve spent a good chunk of my career writing hello-world programs with no end in sight.

***

No discussion of hello-world programs would be complete without mentioning possibly the most intimidating hello-world program: the first Windows program in Charles Petzold’s Programming Windows book. I was only able to find the program from the Windows 98 edition of his book. I don’t recall how it differs much from the program in his first edition, but I vaguely remember the original being worse.

/*------------------------------------------------------------ HELLOWIN.C -- Displays "Hello, Windows 98!" in client area (c) Charles Petzold, 1998 ------------------------------------------------------------*/ #include <windows.h> LRESULT CALLBACK WndProc (HWND, UINT, WPARAM, LPARAM) ; int WINAPI WinMain (HINSTANCE hInstance, HINSTANCE hPrevInstance, PSTR szCmdLine, int iCmdShow) { static TCHAR szAppName[] = TEXT ("HelloWin") ; HWND hwnd ; MSG msg ; WNDCLASS wndclass ; wndclass.style = CS_HREDRAW | CS_VREDRAW ; wndclass.lpfnWndProc = WndProc ; wndclass.cbClsExtra = 0 ; wndclass.cbWndExtra = 0 ; wndclass.hInstance = hInstance ; wndclass.hIcon = LoadIcon (NULL, IDI_APPLICATION) ; wndclass.hCursor = LoadCursor (NULL, IDC_ARROW) ; wndclass.hbrBackground = (HBRUSH) GetStockObject (WHITE_BRUSH) ; wndclass.lpszMenuName = NULL ; wndclass.lpszClassName = szAppName ; if (!RegisterClass (&wndclass)) { MessageBox (NULL, TEXT ("This program requires Windows NT!"), szAppName, MB_ICONERROR) ; return 0 ; } hwnd = CreateWindow (szAppName, // window class name TEXT ("The Hello Program"), // window caption WS_OVERLAPPEDWINDOW, // window style CW_USEDEFAULT, // initial x position CW_USEDEFAULT, // initial y position CW_USEDEFAULT, // initial x size CW_USEDEFAULT, // initial y size NULL, // parent window handle NULL, // window menu handle hInstance, // program instance handle NULL) ; // creation parameters ShowWindow (hwnd, iCmdShow) ; UpdateWindow (hwnd) ; while (GetMessage (&msg, NULL, 0, 0)) { TranslateMessage (&msg) ; DispatchMessage (&msg) ; } return msg.wParam ; } LRESULT CALLBACK WndProc (HWND hwnd, UINT message, WPARAM wParam, LPARAM lParam) { HDC hdc ; PAINTSTRUCT ps ; RECT rect ; switch (message) { case WM_CREATE: PlaySound (TEXT ("hellowin.wav"), NULL, SND_FILENAME | SND_ASYNC) ; return 0 ; case WM_PAINT: hdc = BeginPaint (hwnd, &ps) ; GetClientRect (hwnd, &rect) ; DrawText (hdc, TEXT ("Hello, Windows 98!"), -1, &rect, DT_SINGLELINE | DT_CENTER | DT_VCENTER) ; EndPaint (hwnd, &ps) ; return 0 ; case WM_DESTROY: PostQuitMessage (0) ; return 0 ; } return DefWindowProc (hwnd, message, wParam, lParam) ; }]]>

- CSS / responsive design
- WordPress customization
- Emacs customization
- Advanced LaTeX
- Data cleaning and visualization
- Python (miscellaneous automation scripts)

I don’t have an immediate project to outsource, but these tasks come up occasionally and I’d like to have someone to contact when they do. Mostly these would be small self-contained projects, though data cleaning and visualization could be larger.

]]>

Obviously the intended message is that scalpels are better than Swiss Army Knives. Certainly the scalpel looks simpler.

But most people would rather have a Swiss Army Knife than a scalpel. Many people, myself included, own a Swiss Army Knife but not a scalpel. (I also have a Letherman multi-tool that the folks at Snow gave me and I like it even better than my Swiss Army Knife.)

People like simplicity, at least a certain kind of simplicity, more in theory than in practice. Minimalist products that end up in the MoMA generally don’t fly off the shelves at Walmart.

The simplicity of a scalpel is superficial. The realistic alternative to a Swiss Army Knife, for ordinary use, is a knife, two kinds of screwdriver, a bottle opener, etc. The Swiss Army Knife is the simpler alternative in that context.

A surgeon would rightfully prefer a scalpel, but not just a scalpel. A surgeon would have a tray full of specialized instruments, collectively more complicated than a Swiss Army Knife.

I basically agree with the Unix philosophy that tools should do one thing well, but even Unix doesn’t follow this principle strictly in practice. One reason is that “thing” and “well” depend on context. The “thing” that a toolmaker has in mind may not exactly be the “thing” the user has in mind, and the user may have a different idea of when a tool has served well enough.

]]>In particular, the URL http://johndcook.com/blog may take you to the new home page rather than the latest blog post, at least temporarily.

If you subscribe via email or RSS posts will come to you as usual; you shouldn’t notice any changes.

]]>

Perhaps in reaction to knee-jerk antipathy toward Bayesian methods, some statisticians have adopted knee-jerk enthusiasm for Bayesian methods. Everything’s better with Bayesian analysis on it. Bayes makes it better, like a little dab of margarine on a dry piece of bread.

There’s much that I prefer about the Bayesian approach to statistics. Sometimes it’s the only way to go. But Bayes-for-the-sake-of-Bayes can expend a great deal of effort, by human and computer, to arrive at a conclusion that could have been reached far more easily by other means.

**Related**: Bayes isn’t magic

Image via Gallery of Graphic Design

]]>This is a variation on a problem I’ve blogged about before. As I pointed out there, we can assume without loss of generality that the samples come from the unit interval. Then the sample range has a beta(*n* – 1, 2) distribution. So the probability that the sample range is greater than a value *c* is

Setting *c* = 0.9, here’s a plot of the probability that the sample range contains at least 90% of the population range, as a function of sample size.

The answer to the question at the top of the post is 16 or 17. These two values of *n* yield probabilities 0.485 and 0.518 respectively. This means that a fairly small sample is likely to give you a fairly good estimate of the range.

Since the range of integration is symmetric around zero, you might think to see whether the integrand is an odd function, in which case the integral would be zero. (More on such symmetry tricks here.) Unfortunately, the integrand is not odd, so that trick doesn’t work directly. However, it does help indirectly.

You can split any function *f*(*x*) into its even and odd parts.

The integral of a function over a symmetric interval is the integral of its even part because its odd part integrates to zero. The even part of the integrand above works out to be simply cos(*x*)/2 and so the integral evaluates to sin(1).

Strictly speaking, a professional in some area is simply someone who is paid to do it. But informally, we think of a professional as someone who not only is paid for their services, they’re also good at what they do. The two ideas are not far apart. People who are paid to do something are usually good at it, and the fact that they are paid is evidence that they know what they’re doing.

Experts, however, are not always so pleasant to work with.

Anyone can call himself an expert, and there’s no objective way to test this claim. But it’s usually obvious whether someone is a professional. When you walk into a barber shop, for example, it’s safe to assume the people standing behind the chairs are professional barbers.

Often the categories of “professional” and “expert” overlap. But it is suspicious when someone is an expert and not a professional. It implies that their knowledge is theoretical and untested. If someone says she is an expert in the stock market but not an investor, I wouldn’t ask her to invest my money. When I need my house painted, I don’t want to hire an expert on paint, I want a professional painter.

Sometimes experts appear to be professionals though they are not. Their expertise is in one area but their profession is something else. Political pundits are not politicians but journalists and entertainers. Heads of scientific agencies are not scientists but administrators. University presidents are not educators or researchers but fundraisers. In each case they may have once been practitioners in their perceived areas of expertise, though not necessarily.

**Related posts**:

Three reasons expert predictions are often wrong

Applied is in the eye of the client

]]>

**You can’t divide 3 by 4** (inside the ring of integers, but you can inside the rational numbers).

**You can’t take the square root of a negative number** (in the real numbers, but in the complex numbers you can, once you pick a branch of the square root function).

**You can’t divide by zero** (in the field of real numbers, but you may be able to do something that could informally be referred to as dividing by zero, depending on the context, by reformulating your statement, often in terms of limits).

When people say a thing cannot be done, they may mean it cannot be done in some assumed context. They may mean that the thing is difficult, and assume that the listener is sophisticated enough to interpret their answer as hyperbole. Maybe they mean that they don’t know how to do it and presume it can’t be done.

When you hear that something can’t be done, it’s worth pressing to find out in what sense it can’t be done.

**Related post**: How to differentiate a non-differentiable function

“What will happen when you’re done with this project?”

“I don’t know. Maybe not much. Maybe great things.”

“How great? What’s the best outcome you could reasonably expect?”

“Hmm … Not that great. Maybe I should be doing something else.”

It’s a little paradoxical to think that asking an optimistic question — What’s the best thing that could happen? — could discourage us from continuing to work on a project, but it’s not too hard to see why this is so. As long as the outcome is unexamined, we can implicitly exaggerate the upside potential. When we look closer, reality may come shining through.

** Related posts**:

Obsession

How much does typing speed matter?

Wouldn’t trade places

I bought Marshall Goldsmith’s book by that title shortly after it came out in 2007. As much as I liked the title, I was disappointed by the content and didn’t finish it. I don’t remember much about it, only that it wasn’t what I expected. Maybe it’s a good book — I’ve heard people say they like it — but it wasn’t a good book for me at the time.

***

I’ve written before about The Medici Effect, a promising title that didn’t live up to expectations.

***

“Standardized Minds” is a great book title. I haven’t read the book; I just caught a glimpse of the cover somewhere. Maybe it lives up to its title, but the title says so much.

There is a book by Peter Sacks Standardized Minds: The High Price Of America’s Testing Culture And What We Can Do To Change It. Maybe that’s the book I saw, though it’s possible that someone else wrote a book by the same title. I can’t say whether I recommend the book or not since I haven’t read it, but I like the title.

***

I started to look for more examples of books that didn’t live up to their titles by browsing my bookshelves. But I quickly gave up on that when I realized these are exactly the kinds of books I get rid of.

What are some books with great titles but disappointing content?

]]>Nothing can be discussed rationally. Even narrow scientific questions lead to emotionally-charged political arguments. Those who have a different opinion must be maligned.

The big question is whether the Ebola virus can spread by air. Experts say “probably not” but some are cautious. For example, Ebola researcher C. J. Peters says “We just don’t have the data to exclude it.” But people who know absolutely nothing about virology are firmly convinced one way or the other.

]]>

I haven’t read more than the introduction yet — a review copy arrived just yesterday — but I imagine it’s good judging by who wrote it. Havil’s book Gamma is my favorite popular math book. (Maybe I should say “semi-popular.” Havil’s books have more mathematical substance than most popular books, but they’re still aimed at a wide audience. I think he strikes a nice balance.) His latest book is a scientific biography, a biography with an unusual number of equations and diagrams.

Napier is best known for his discovery of logarithms. (People debate endlessly whether mathematics is discovered or invented. Logarithms are so natural — pardon the pun — that I say they were discovered. I might describe other mathematical objects, such as Grothendieck’s schemes, as inventions.) He is also known for his work with spherical trigonometry, such as Napier’s mnemonic. Maybe Napier should be known for other things I won’t know about until I finish reading Havil’s book.

]]>Many think this is stupid. They say that Microsoft should call the next version Windows 9, and if somebody’s dumb code breaks, it’s their own fault.

People who think that way aren’t billionaires. Microsoft got where it is, in part, because they have enough business savvy to take responsibility for problems that are not their fault but that would be *perceived* as being their fault.

I thought about my short academic career [1]. If I had been wildly successful, the most I could hope for would be to be one of these laureates. And yet I wouldn’t trade places with any of them. I’d rather do what I’m doing now than have an endowed chair at some university. Consulting suits me very well. I could see teaching again someday, maybe in semi-retirement, but I hope to never see another grant proposal.

***

[1] I either left academia once or twice, depending on whether you count my stint at MD Anderson as academic. I’d call my position there, and even the institution as a whole, quasi-academic. I did research and some teaching there, but I also did software development and project management. The institution is a hospital, a university, a business, and a state agency; it can be confusing to navigate.

]]>At the Heidelberg Laureate Forum I has a chance to interview John Tate. In his remarks below, Tate briefly comments on his early work on number theory and cohomology. Most of the post consists of his comments on the work of Alexander Grothendieck.

***

**JT**: My first significant work after my thesis was to determine the cohomology groups of class field theory. The creators of the theory, including my thesis advisor Emil Artin, didn’t think in terms of cohomology, but their work could be interpreted as finding the cohomology groups H_{0}, H_{1}, and H_{2}.

I was invited to give a series of three talks at MIT on class field theory. I’d been at a party, and I came home and thought about what I’d talk about. And I got this great idea: I realized I could say what all the higher groups are. In a sense it was a disappointing answer, though it didn’t occur to me then, that there’s nothing new in them; they were determined by the great work that had already been done. For that I got the Cole prize in number theory.

Later when I gave a talk on this work people would say “This is number theory?!” because it was all about cohomology groups.

**JC**: Can you explain what the great reformulation was that Andrew Wiles spoke of? Was it this greater emphasis on cohomology?

**JT**: Well, in the class field theory situation it would have been. And there I played a relatively minor part. The **big **reformulation of algebraic geometry was done by Grothendieck, the theory of schemes. That was really such a great thing, that unified number theory and algebraic geometry. Before Grothendieck, going between characteristic 0, finite characteristic 2, 3, etc. was a mess.

Grothendieck’s system just gave the right framework. We now speak of *arithmetic* algebraic geometry, which means studying problems in number theory by using your geometric intuition. The perfect background for that is the theory of schemes. ….

Grothendieck ideas [about sheaves] were so simple. People had looked at such things in particular cases: Dedekind rings, Noetherian rings, Krull rings, …. Grothendieck said take *any* ring. … He just had an *instinct* for the right degree of generality. Some people make things too general, and they’re not of any use. But he just had an instinct to put whatever theory he thought about in the most general setting that was still useful. Not generalization for generalization’s sake but the *right* generalization. He was unbelievable.

He started schemes about the time I got serious about algebraic geometry, as opposed to number theory. But the algebraic geometers classically had affine varieties, projective varieties, … It seemed kinda weird to me. But with schemes you had a *category*, and that immediately appealed to me. In the classical algebraic geometry there are all these *birational* maps, or *rational maps*, and they’re not defined everywhere because they have singularities. All of that was cleared up immediately from the outset with schemes. ….

There’s a classical algebraic geometer at Harvard, Joe Harris, who works mostly over the complex numbers. I asked him whether Grothedieck made much of a difference in the classical case — I knew for number theorists he had made a tremendous difference — and Joe Harris said yes indeed. It was a revolution for classical algebraic geometry too.

]]>When I mentioned that some people had reacted to the original post saying the scheme was too hard, Blum said that he has taught the scheme to a couple children, 6 and 9 years old, who can use it.

He also said that many people have asked for his slide summarizing the method and asked if I could post it. You can save the image below to get the full-sized slide.

This slide and my blog post both use a 3-digit password for illustration, though obviously a 3-digit password would be easy to guess by brute force. I asked Blum how long a password using his scheme would need to be so that no one with a laptop would be able to break it. He said that 12 digits should be enough. Note that this assumes the attacker has access to many of your passwords created using the scheme, which would be highly unlikely.

]]>

Mathematics and art restoration

Studying algorithms to study problems

]]>It’s well known that software needs to be maintained; most of the work on a program occurs after it is “finished.” Proof maintenance is common as well, but it is usually very informal.

Proofs of any significant length have an implicit hierarchical structure of sub-proofs and sub-sub-proofs etc. Sub-proofs may be labeled as lemmas, but that’s usually the extent of the organization. Also, the requirements of a lemma may not be precisely stated, and the propositions used to prove the lemma may not be explicitly referenced. Lamport recommends making the hierarchical structure more formal and fine-grained, extending the sub-divisions of the proof down to propositions that take only two or three lines to prove. See his paper How to write a 21st century proof.

When proofs have this structure, you can see which parts of a proof need to be modified in order to produce a proof of a new related theorem. Software could help you identify these parts, just as software tools can show you the impact of changing one part of a large program.

]]>**JC**: Orthogonal polynomials are kind of a lost art, a topic that was common knowledge among mathematicians maybe 50 or 100 years ago and now they’re obscure.

**DS**: The first course I taught I spent a few lectures on orthogonal polynomials because they kept coming up as the solutions to problems in different areas that I cared about. Chebyshev polynomials come up in understanding solving systems of linear equations, such as if you want to understand how the conjugate gradient method behaves. The analysis of error correcting codes and sphere packing has a lot of orthogonal polynomials in it. They came up in a course in multi-linear algebra I had in grad school. And they come up in matching polynomials of graphs, which is something people don’t study much anymore. … They’re coming back. They come up a lot in random matrix theory. … There are certain things that come up again and again and again so you got to know what they are.

***

More from my interview with Daniel Spielman:

What is smoothed analysis?

Studying algorithms to study problems

]]>

I believe there is, in mathematics, in contrast to the experimental disciplines, a character which is nearer to that of free creative art.

There is evidence that the relation of artistic beauty and mathematical beauty is more than an analogy. Michael Atiyah recently published a paper with Semir Zeki *et al* that suggests the same part of the brain responds to both.