Saunders Mac Lane, 1909--2005
April 22, 2005 7:31 AM Subscribe
Saunders Mac Lane, mathematician, has died, age 95. Winner of the National Medal of Science, Vice-President of the National Academy of Science, President of the American Mathematical Society, author of three of the canonical texts in algebra [reg. maybe req., here's a local copy], Mac Lane was also mathematical ancestor to over a thousand mathematicians, father of category theory and homological algebra, and expert in topology, topos theory, group cohomology, logic, and applied mathematics. He was one of the towering figures of postwar mathematics. Remembered by his students and all of us who were affected by his work and his life.
gleuschk: "I discovered last night that I've lost his letter."
That's a great story, and it's sad that the letter has been lost. It's always interesting to hear a "brush with fame" story that's a little more than a brush.
posted by Plutor at 8:19 AM on April 22, 2005
That's a great story, and it's sad that the letter has been lost. It's always interesting to hear a "brush with fame" story that's a little more than a brush.
posted by Plutor at 8:19 AM on April 22, 2005
Wow...that says a lot about the man's character gleuschk. That's really too bad about the letter.
posted by peacay at 8:27 AM on April 22, 2005
posted by peacay at 8:27 AM on April 22, 2005
I wondered whether I'd see this on the blue! I should've known it'd be by you, gleuschk. saw you in the hallway last weekend
I was rather disappointed by the NYTimes obit. It's not that hard to describe a category and give a couple of examples. I guess they can be forgiven for completely punting on defining an Eilenberg-Mac Lane space, but "It is used in the study of mathematical convergence and continuity" is a better description of "topology" than of "Eilenberg-Mac Lane space". (No mention of the Mac Lane coherence theorem at all.)
Heck, I'll take a stab at this.
One way to see that a sphere and a doughnut are different topologically -- and "different" here means that there's no continuous correspondence between points on one and points on the other -- is that any loop drawn on a sphere can be shrunk to a point, whereas on a doughnut that's not true. (If you had a correspondence, you could correspond the shrinking operation too.)
This trick doesn't work for comparing three-dimensional space vs. three-dimensional space with a point missing; in the latter space, every loop that misses the puncture can be shrunk in a way that also doesn't meet the puncture. However, if we go beyond loops to spheres (beyond x^2+y^2=1 to x^2+y^2+z^2=1), we notice that any sphere in 3-space can be shrunk, but a sphere in punctured 3-space that goes around the puncture can't. note: the loop is a "1-d sphere", and the next one is a "2-d sphere", because an ant wandering on them would believe itself to be on a line, or a plane, respectively.
Then that trick doesn't work in punctured 4-space, so we have to use the next bigger spheres (x^2+y^2+z^2+w^2=1), and so on. there's a 3-d sphere
There's a really important subtlety built into this idea of shrinking a sphere in X -- don't think of the sphere as embedded in X, but just mapping to X. The famous example is where the sphere is unit vectors in the 2-d complex vector space, and X is the "Riemann sphere" of complex numbers plus infinity, and the map is (x,y) goes to x/y. (Note that this ratio might be infinity, which is okay, and it can't be 0/0, which is usually better undefined.) This particular map, the "Hopf fibration", turns out to not be shrinkable. So the 2-d sphere has complicated 3-d "homotopy".
An Eilenberg-Mac Lane space is one where all spheres, of every dimension except for dimension n, can be shrunk. (So for example, the 2-d sphere isn't one; while loops (1-d spheres) can be shrunk in it, the whole thing (2-d) can't, and the Hopf fibration (3-d) also can't.) One example is a doughnut with g holes for g>0. But almost all other examples are infinite-dimensional, alas.
A great deal of topology has been done by a sort of factorization, where one starts with an arbitrary topological space and deals with its homotopy one dimension at a time -- each piece being an Eilenberg-Mac Lane space. So it's been a fantastic definition, one that will be used by mathematicians until the end of time.
posted by Aknaton at 8:49 AM on April 22, 2005
I was rather disappointed by the NYTimes obit. It's not that hard to describe a category and give a couple of examples. I guess they can be forgiven for completely punting on defining an Eilenberg-Mac Lane space, but "It is used in the study of mathematical convergence and continuity" is a better description of "topology" than of "Eilenberg-Mac Lane space". (No mention of the Mac Lane coherence theorem at all.)
Heck, I'll take a stab at this.
One way to see that a sphere and a doughnut are different topologically -- and "different" here means that there's no continuous correspondence between points on one and points on the other -- is that any loop drawn on a sphere can be shrunk to a point, whereas on a doughnut that's not true. (If you had a correspondence, you could correspond the shrinking operation too.)
This trick doesn't work for comparing three-dimensional space vs. three-dimensional space with a point missing; in the latter space, every loop that misses the puncture can be shrunk in a way that also doesn't meet the puncture. However, if we go beyond loops to spheres (beyond x^2+y^2=1 to x^2+y^2+z^2=1), we notice that any sphere in 3-space can be shrunk, but a sphere in punctured 3-space that goes around the puncture can't. note: the loop is a "1-d sphere", and the next one is a "2-d sphere", because an ant wandering on them would believe itself to be on a line, or a plane, respectively.
Then that trick doesn't work in punctured 4-space, so we have to use the next bigger spheres (x^2+y^2+z^2+w^2=1), and so on. there's a 3-d sphere
There's a really important subtlety built into this idea of shrinking a sphere in X -- don't think of the sphere as embedded in X, but just mapping to X. The famous example is where the sphere is unit vectors in the 2-d complex vector space, and X is the "Riemann sphere" of complex numbers plus infinity, and the map is (x,y) goes to x/y. (Note that this ratio might be infinity, which is okay, and it can't be 0/0, which is usually better undefined.) This particular map, the "Hopf fibration", turns out to not be shrinkable. So the 2-d sphere has complicated 3-d "homotopy".
An Eilenberg-Mac Lane space is one where all spheres, of every dimension except for dimension n, can be shrunk. (So for example, the 2-d sphere isn't one; while loops (1-d spheres) can be shrunk in it, the whole thing (2-d) can't, and the Hopf fibration (3-d) also can't.) One example is a doughnut with g holes for g>0. But almost all other examples are infinite-dimensional, alas.
A great deal of topology has been done by a sort of factorization, where one starts with an arbitrary topological space and deals with its homotopy one dimension at a time -- each piece being an Eilenberg-Mac Lane space. So it's been a fantastic definition, one that will be used by mathematicians until the end of time.
posted by Aknaton at 8:49 AM on April 22, 2005
Bonus link I just learned about: Mac Lane's autobiography will be published in the next few months.
Did I miss you flashing the secret hand signal, Aknaton?
posted by gleuschk at 9:39 AM on April 22, 2005
Did I miss you flashing the secret hand signal, Aknaton?
posted by gleuschk at 9:39 AM on April 22, 2005
The Springer GTM test made me laugh. Great post, gleuschk.
posted by weston at 10:58 AM on April 23, 2005
posted by weston at 10:58 AM on April 23, 2005
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I discovered last night that I've lost his letter.
posted by gleuschk at 7:32 AM on April 22, 2005 [1 favorite]