The great reformulation of algebraic geometry

“Tate helped shape the great reformulation of arithmetic and geometry which has taken place since the 1950’s.” — Andrew Wiles

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 H0, H1, and H2.

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.

Writing down an unwritten language

In this post I interview Greg Greenlaw, a friend of mine who served as a missionary to the Nakui tribe in Papua New Guinea and developed their writing system. (Nakui is pronounced like “knock we.”)

JC: When you went to PNG to learn Nakui was there any writing system?

GG: No, they had no way of writing words or numbers. They had names for only seven numbers — that was the extent of their counting system — but they could coordinate meetings more than a few days future by tying an equal number of knots in two vines. Each party would take a vine with them and loosen a knot each morning until they counted down to the appointed time — like and advent calendar, but without numbers!

Continue reading

Interview with Sacha Chua

I spoke with with Sacha Chua last week. We talked about entrepreneurship, Emacs, having eclectic interests, delegation, and more.

Navigation cons from Sacha's blog

J: I ran into you by searching on Emacs topics. When I look at your blog, I see that you do a lot of interesting things, but it’s a little hard to get a handle on exactly what you do.

S: Oh, the dreaded networking quirky question. What exactly do you do?

J: Yeah, people have said the same thing to me. Not to put you in a box, but I was curious. I see from your site that you do graphic art — sketching and such — and it doesn’t create the impression that you’re someone who would spend a lot of time in front of Emacs. So I’m curious how these things fit together, how you got started using Emacs and how you use it now.

Image by Sacha Chua

S: So my background is actually fairly technical. I’ve been doing computer programming for ages and ages. In high school I came across a book Unix Power Tools, which is how I got interested in Emacs. And because I was interested in programming, in open source, a little bit of wearable computing as well, I got to know Emacs and all these different modules it had. For example, Emacspeak is amazing! It’s been around since the 1990s and it’s a great way to use the computer while you’re walking around. Because I love programming and because I wanted to find a way to help out, I ended up maintaining PlannerMode and later EmacsWiki mode as well.

When I went to university, I took up computer science. After I finished, I taught. Then I took my masters in Toronto, where I am now. Emacs was super helpful — being able to do everything in one place. After I finished my masters, I did a lot of software consulting with IBM. I did business consulting as well. Then in 2012, after saving up, I decided to go on pretty much the same adventure you’re on. I’m completely unhirable for the next five years! Most businesses struggle for the first five years, so I saved up enough to not worry too much about my expenses for the next five years. I’m one year in, four years to go, and that’s where I am.

At networking events, I like to shake people up a bit by telling them I’m semi-retired. I’m in this five-year experiment to see how awesome life can be and what I can do to make things better. I’ve done technical consulting, business consulting, sketching, illustration, writing, all sorts of things. Basically, my job description is context-dependent.

J: I understand that.

S: I use Emacs across all the things I do. When I’m doing technical and business consulting, I use Emacs to edit code, to draft documents, even to outline comic strips. And when I’m doing illustration, Emacs — especially Org Mode — helps me keep track of clients and deliverables, things to do, agenda, calendar, deadlines.

J: I’m basically running my life through Org Mode right now. When you say you use Emacs to draft documents, are you using LaTeX?

S: I used LaTeX when I was working on my master’s thesis and other papers, I think. Now I mostly use org mode and export from there.

J: Are you using Emacs for email?

S: I used to. But I’m stuck on Windows to use drawing programs like Sketchbook Pro on my Tablet PC. So it’s harder to set up my email like I had it set up when I used Ubuntu. Back when I used Ubuntu, I was very happy with Gnus.

J: Do you work entirely on Windows, or do you go back and forth between operating systems?

S: I have a private server that runs Linux. On Windows I run Cygwin, but I miss some of the conveniences I had when I had a nicely set-up Linux installation.

J: When you’re running Emacs on Windows, I’m sure you run into things that don’t quite work. What do you do about that?

S: Most things work OK if they’re just Emacs Lisp, but some things call a shell command or use some library that hasn’t been ported over yet. Then I basically wail and gnash my teeth. Sometimes I get things working by using Cygwin, but sometimes it’s a bit of a mess. I don’t use Emacs under Cygwin because I prefer how it works natively. I don’t run into much that doesn’t work.

J: So what programming languages do you use when you’re writing code?

S: I do a lot of quick-and-dirty things in Emacs Lisp. When I need to do some XML parsing or web development, I’ll use Ruby because a lot of people can read it and there are a lot of useful gems. Sometimes I’ll do some miscellaneous things in Perl.

I love doing programming and putting together tools. And I quite enjoy drawing, helping people with presentation and design. So this is left brain plus right brain.

It does boggle people that you can have more than one passion, but others are, like, “Yeah, I know, I’m like that too.”

J: I think having an interest in multiple things is a healthier lifestyle, but it’s a little harder to market.

S: Actually, no. I finally figured out a name for my company, ExperiVis, after a year of playing with it. People reach out to me and we figure out whether it’s a good fit. I don’t need to necessarily guide people to just this aspect or another of my work. I like the fact that people bump into these different things.

J: When we scheduled this call, I went through your virtual assistant. How do you use a virtual assistant?

S: One of the things I don’t like to do is scheduling. I used to get stressed out about scheduling when I did it myself. I’ve always been interested in delegating and taking advantage of what other people enjoy and are good at. I work with an assistant — Criselda. She lives in the Philippines. I found her on oDesk. She works one to four hours a week, more or less, and keeps track of her time.

J: What else might you ask a VA to do?

S: I’ve asked people to do web research. I’ve had someone do a little bit of illustration for me. I’ve had someone do a little bit of programming for me because I want to learn how to delegate technical tasks. He does some Rails prototyping for me. I have someone doing data entry and transcription. It’s fascinating to see how you can swap money for time, especially for things that stress me out, or bore me, or things I can’t do.

Every week I go over my task list with my VA to see which of the tasks I should have delegated. Still working on it!


Later on in the conversation Sacha asked about my new career and had this gem of advice:

Treating this as a grand experiment makes it much easier for me to try different approaches and not be so scared, to not treat it as a personal rejection if something doesn’t work.

Related posts:

People I’ve interviewed and people who have interviewed me

Fifteen interviews

Seven Nine people I have interviewed:

  1. Rick Richter, CIO of Food for the Hungry
  2. Eric Floehr, owner of ForecastWatch
  3. Frederick Brooks, computer pioneer and author
  4. Robert Ghrist, applied topologist
  5. Cliff Pickover, mathematician and author
  6. Dan Bricklin, software developer and author
  7. Carl Franklin, musician and software developer
  8. Michael Hammer, GIMP developer and author (new)
  9. Sacha Chua, sketch artist, consultant, and Emacs fan

Two Four people who have interviewed me:

  1. Vincent Tan in his online magazine Singularity
  2. Samuel Hansen in his podcast Strongly Connected Components
  3. Erik Meijer and Channel 9 (new)
  4. Romy Misra on Data Science and Analytics (new)

Six interviews I have blogged about:

  1. Henry Petroski, engineer
  2. Arnold Kling, economist
  3. David Spiegelhalter, statistician
  4. Billy Hollis, software developer
  5. Eugenia Chang, mathematician
  6. Charles Bamforth, beer scholar

Why Food for the Hungry runs Ubuntu

Rick Richter is CIO of Food for the Hungry. In this interview Rick explains why his organization is moving all of its computers to Ubuntu.

Ethiopian farmer Ato Admasu

Ethiopian farmer Ato Admasu. Photo credit Food for the Hungry.

John: Tell me a little about Food for the Hungry and what you do there.

Rick: Food for the Hungry is a Christian relief and development organization. We go in to relief situations — maybe there has been a natural disaster or war — and provide life-sustaining needs: food, shelter, whatever the need may be.  For example, the recent earthquake in Haiti. But the other part of what we do is the sustained, long-term development on the community level. The idea is to work with leaders and churches to better take care of themselves rather than relying on outside organizations for support.

I’m the CIO. I’m in charge of the information and technology for the organization. We’re in 25 countries. I have staff all over the world, about 25 people. There are about 12 who work directly for global IT, mostly in Phoenix, and the rest in various countries.  There are also people who work directly for local offices, for example in Kenya, that coordinate with global IT. We’re responsible for about 900 computers.

John: You and I were talking the other day about your organization’s project to move all its computers over to Ubuntu.

Rick: We started an informal process to convert to Ubuntu two and a half years ago. It started when my son went to Bangladesh. He spent the summer there and converted some of their computers to Ubuntu. At first we didn’t have full management support for the process. They don’t really understand it and it scares them.

There were individual country directors interested in the project and I talked it up. There’s some independence in the organization to make those kind of decisions. Now, for the first time, we have full support of management for the conversion on a wide scale. I’m going to Cambodia next week. Right now they’re all running Windows but before I leave they’ll be running Ubuntu. In Asia we probably have about 80% of our computers on Ubuntu. We don’t have big offices in Asia. Our bigger offices are in Africa and they’re a little slower to adopt. Until now, a lot of it depended on whether the local country director was ready to change.

We found it was important for a number of reasons. One is security. Linux is not as vulnerable to viruses. We have so many places where entire computer systems have been totally crippled because of viruses. A lot of networks are very primitive, so the network is basically a thumb drive between offices in a country. A thumb drive is the best way to transmit viruses you can find.

We’ve also found in the last few years anti-virus software has become less and less effective. Three or four years ago, if you had up-to-date anti-virus software you wouldn’t get a virus. These days, you still get them. Some of our staff have other jobs within FH besides their IT responsibilities and may not have a lot of IT experience. As a result, staff often do not have the time to pro-actively manage IT.

Another issue is maintainability. Windows computers don’t run as well over time. With Ubuntu, when we come back to a computer two years later it’s in as good a shape as we left it.

Linux requires much less hardware to run than Windows. We have eight- or nine-year-old computers at a lot of our sites that will no longer run or barely run Windows.

John: So saving money on software licenses is a benefit, but not the main consideration.

Rick: Saving money on licenses is important, but it’s not the driving force. We’re a non-profit and we have a contract with Microsoft where we get pretty good prices.

Another reason for moving to Ubuntu is that in some countries it is very difficult to legally obtain licenses. Sometimes it’s next to impossible. You can’t buy legal Microsoft licenses in some places, or if you can, the price is outrageous. So many legalities and so many weird hoops you have to jump through.

As a Christian organization we need to set a good example and make sure all our licenses are legal. We want to be clear and up-front about our software. Ubuntu eliminates that problem.

John: What experience have you had retraining your IT people to support Linux?

Rick: We have IT professionals and we have people who are much less skilled. Most of the IT people who do the support have really bought into it. They’re excited about it and they’re pushing it. Those who do support in the field who have had less exposure, some of them have bought into it, some have not as much. It requires time. It requires dedication. It also required commitment from their management.

Related posts:

Robust, scalable, and the keyboard works
Free Ubuntu Linux book
Geek fatigue
New spin on The Cathedral and the Bazaar

Applied topology and Dante: an interview with Robert Ghrist

Robert Ghrist A few weeks ago I discovered Robert Ghrist via his web site. Robert is a professor of mathematics and electrical engineering. He describes his research as applied topology, something I’d never heard of. (Topology has countless applications to other areas of mathematics, but I’d not heard of much work directly applying topology to practical physical problems.) In addition to his work in applied topology, I was intrigued by Robert’s interest in old books.

The following is a lightly-edited transcript of a phone conversation Robert and I had September 9, 2010.


JC: When I ran across your web site one thing that grabbed my attention was your research in applied topology. I’ve studied applied math and I’ve studied topology, but the two are very separate in my mind. I was intrigued to hear you combine them.

RG: Those two are separate in a lot of people’s minds, but not for long. It’s one of those things that the time has come and it’s clear that the tools that were developed for very abstract, esoteric problems have really concrete value with respect to modern challenges in data, or systems analysis.

JC: Could you give some examples of that?

RG: Certainly. One of the first groups of people who do full-scale applied algebraic topology were Gunnar Carlsson’s group at Stanford doing applications to data analysis. The setup is you have a collection of data points in a space, a point cloud, that is a discrete representation of some interesting structure. For example, you might want to know how many connected components does this data set have. That might correspond to different features. For example, this might come from customer surveys that a corporation has out. It’s trying to cluster these customers. Or if it is medical data, say they are trying to discern different types of cancers. Then you might look at what statisticians call clustering, grouping data sets into connected components.

Well, topologists know that’s just the first step in a larger program of finding global structure. Besides having connectivity properties, spaces can have holes in them of various types. There are formal algebraic methods for finding and classifying those holes. That’s homology, cohomology, homotopy theory. And applying that to data turns out to give some really revolutionary techniques that don’t rely on projecting the data set down to a two-dimensional picture and trying to visualize what’s happening. It’s automatic.

There are similarly themed application in the work I do in engineering systems where you take data that comes from, say, a network of sensors or a communications network or a networked connection on computers or a networked collection of autonomous robots. And you try and take all that local information, say in the context of sensor networks, you take collections of local data and try to patch it together to give you global understanding of an environment. That kind of local-to-global transition is what the techniques of topology were built to do. And they are surprisingly efficacious in these very applied problems.

JC: I’m familiar with Van Kampen’s theorem in homotopy. Is that the sort of thing you’re talking about?

RG: Yeah, homotopy tends to be less computable than homology. Homology is much more natural in these contexts. The corresponding principle is the Mayer-Vietoris principle, the homological analog of Van Kampen. And the Mayer-Vietoris principle is really telling you something about integration, how you stitch together local bits of data, how you integrate it into a global understanding of your network. That is a very deep idea that is very important to transitioning from local to global data.

JC: One mental block I have when I’m thinking about these sort of things is that in my mind topology seems sorta fragile in the sense that one random connection can change something. Connect just one pair of dots and suddenly a disconnected space becomes a connected space. It seems that would be a problem in these settings where you make some sort of topological statement, but any missing data or erroneous data invalidates your conclusions. But I remember you said something that was sort of the opposite on your web site, that topological methods can be more robust. I could see that, but I’m having trouble resolving these two views.

RG: There are different types of robustness that are critical in different types of applications. Because the constructs of algebraic topology are invariant under homotopy or deformation, it turns out to be very robust with respect to, for example, coordinate changes. That is extremely useful when you’re dealing with data that has some anchor to the physical world.

Let’s say data that’s being collected by our cell phones. Put a couple sensors on a cell phone and we have lots and lots of interesting data streams. And that data is tied to physical locations. But you might not know exactly where it is. GPS doesn’t work so well when you’re inside a building, for example. In contexts like that, you want the kind of robustness that doesn’t depend on having a carefully laid-out coordinate system.

Now robustness with respect to noise, especially robustness with respect to errors, is a much more difficult problem to solve in general. But even in that case there are some topological tools that in specific examples can be deployed. This gets into slightly more technical stuff involving persistent homology and topological properties that persist over a range of samplings.

JC: Could you give an example of how knowing the homology of a data set might tell you something about a physical phenomena?

RG: One example that I’ve worked with a lot has to do with coverage problems in sensor networks. Let’s say we’re talking about cell phone coverage, because everyone’s familiar with that. If you get into a hole in the cell phone network where you’ve dropped coverage, that’s frustrating. You’d like to know whether you have full coverage over an area or not, whether you have holes.

This gets much more critical when you’re talking about a security setting. You have video cameras or satellites that cover a region and you want whether you’ve covered everything, or whether there are holes where you are missing information. One of the things I’ve used homology theory for is to give criteria that guarantee coverage, that guarantee no holes in your sensor network based exclusive on coordinate-free data. So even though the sensors may not know where they are, and only know the identities of the sensors near by, it’s still possible to verify coverage based on homological criteria.

JC: Interesting. I suppose especially if you’re looking at higher-dimensional data you can’t just draw a lot of circles on a map and see whether they overlap. You have to do something more computational.

RG: Exactly. Especially if those circles are in motion and you want to know what’s happening over time. Especially if there are no coordinates and you don’t know where to draw the circles to see how things overlap.

One thing I want to get across when I’m talking with people is that I view a mathematics library the same way an archaeologist views a prime digging site. There are all these wonderful treasures that are buried there and hidden from the rest of the world. If you pick up a typical book on sheaf theory, for example, it’s unreadable. But it’s full of stuff that is very, very important to solving really difficult problems. And I have this vision of digging through the obscure text and finding these gems and exporting them over to the engineering college and other domains where these tools can find utility.

Now, a lot of esoteric mathematics has already crossed the fence. No one will claim any more that number theory is useless. But topology is the place where you have the most-useful, least-used tools. So that’s my vision for what I want to see happen in mathematics and what I’m trying to accomplish.

JC: Switching gears a little bit, another thing on your site that caught my eye was your quote about old books.

Reading anything less than 50 years old is like drinking new wine: permissible once or twice a year and usually followed by regret and a headache.

I thought about C. S. Lewis’ exhortation to mix in old books with your reading of new books because each age has its own blind spots. Old authors have their own blind spots, but maybe they’re complementary to the ones we have.

RG: Exactly. I definitely have followed that dictum. Maybe a little too much so, in that I rarely read anything modern at all. When it comes to books. I don’t follow that rule when it comes to music or movies or blogs. But on the level of books, there is so much good stuff out there that has stood the test of time, I don’t run out of interesting things to read. I had the wonderful experience as a college student to take a great books-type course that involved a lot of reading, a lot of discussion. Really changed my outlook and got me loving the classics and really living inside a lot of those books.

JC: Were you a math major when you took this great books class?

RG: No, I was an engineering major. My undergraduate degree was in engineering. I came to math a little late in life.

JC: You said you were particularly fond of Dante.

RG: That’s correct. Yeah, I’ve lived in that book [The Divine Comedy] a long time and I still find new and very engaging ideas in it every time I crack it open. Most people don’t get very far past the Inferno. The Inferno is the exciting, action-movie part of the story. But the later parts of the story — purgatory, paradise — those are really nice places to live.

JC: Have you had to do a lot of historical research to be able to read Dante?

RG: If you get a good translation with a good set of notes, that makes it much easier. I find the translation by Dorothy Sayers has excellent notes. She got turned on to Dante late in life. There’s an interesting story. While they were having bombings in the early 1940’s, she was going down to the bomb shelter to spend a few hours, she decided to grab a book off the shelf on the way down. She saw a copy of Dante. She said “You know, I’ve never really read Dante.” Pulled it off the shelf. For the next two days she didn’t eat or sleep. She was engrossed with the story and how masterful it was. And she wound up devoting the rest of her life to mastering the Italian and producing her own translation.

Other interviews:

Fred Brooks
Cliff Pickover
Carl Franklin
Dan Bricklin

A few questions with Frederick Brooks

The shelf life of software development books is typically two or three years, maybe five or ten years for a “classic.” Frederick Brooks, however, wrote a book on software development in 1975 that remains a best-seller: The Mythical Man-Month. His book has remained popular because he wrote about human nature as applied to software development, not the hottest APIs and development fads from the 1970’s.

Frederick Brooks has written a new book that should also enjoy exceptional shelf life, The Design of Design: Essays from a Computer Scientist. In this book, Brooks looks back over a long successful career in computing and makes insightful observations about design.

Frederick Brooks, Jr.

The following interview comes from an email exchange with Frederick Brooks.

JC: You did a PhD in math with Howard Aiken in 1956. Did you study numerical analysis? Did you intend at the time to have a career in computing?

FB: Oh, yes indeed. That’s why I went to Aiken’s lab for graduate work.

JC: I was struck by your comments on conceptual integrity, particularly this quote: “Most great works have been made by one mind. The exceptions have been made by two minds.” Could you say more about that?

FB: I give lots of examples in the book of the one mind case, and a few of the two-mind case.

JC: You said in your book that your best management decision was sending E. J. Codd to get his PhD. That certainly paid off well. Could you share similar examples of successful long-term investments?

FB: Well, IBM’s decision to gamble everything on the System/360, terminating six successful product lines to do so, is a great example. DARPA’s funding of the development of the ARPAnet, ancestor to the Internet, is another great example.

JC: What are some technologies from decades ago that are being rediscovered?

FB: I find it useful to write first drafts of serious things, such as scientific papers and books, by hand with a felt-tip pen. I can type faster than I can think, so composing on a keyboard yields unnecessary wordiness. Writing by hand matches my thinking and writing speeds, and the result is leaner and cleaner.

JC: What are some that have not become popular again but that you think should be reconsidered?

FB: Probably the previous example answers this question better than it does the previous question.

JC: Apart from technological changes, how have you seen the workplace change over your career?

FB: I’ve been in academia for the past 46 years. The biggest change in academia is a consequence of personal computers and networks: faculty members don’t use secretaries as such, they write their own letters, and make their own phone calls. Our assistants are indeed administrative assistants, rather than secretaries.

JC: What change would you like to see happen as a result of people reading your new book?

FB: Even more recognition of the role of a chief designer, separate from a project manager, in making a new artifact, and more attention to choosing and growing such.

Related links:

The Design of Design
The Mythical Man-Month
Many hands make more work

Other interviews:

Cliff Pickover
Carl Franklin
Dan Bricklin