The most important work in the last century was done in foundations:

– Gödel’s Incompleteness Theorems;

– Computable functions in the sense of Turing, Church, Kleene and others; especially Turing’s work is the origin of computing and today’s computers.

Oh – and did any algebraic geometrist help win a war like Turing did?

]]>After 47 years, this 1968 roadmap’s discussion of the significance of Grothendieck’s work stands up pretty well:

Homological Algebra and Category TheoryModern mathematics is characterized by an ever-increasing range of applications of algebra to other mathematical subjects. A particularly striking example is topology, a branch of geometry concerned with qualitative rather than quantitative aspects of shapes of geometric figures. In the early 1920’s it was recognized, under the influence of Emmy Noether especially, that the methods used by topologists are basically algebraic. But just as everu science that uses mathematics not only exploits the existing mathematical theories but reshapes them to its own needs, so topologists developed algebraic tools suitable for their needs. The next step was a return to pure algebra. Algebraic methods created for the needs of topology have been analyzed, codified, and studied for their own sake. This led to two new subdivisions in algebra: category theory and homological algebra. They are perhaps the most abstract specialities in algebra. Categories provide a language for discussing \emph{all} algebric systems of a given type. The result is, as is so often the case in mathematics, a wide variety of applications to diverse mathematical fields, in this case from logic to such “applied” fields as the theory of automata.

Algebraic GeometryAn oustanding problem was whether, given $k-n$ independent algebraic equations in $k$ unknowns, it is possible to represent all solutions of this system by a smooth geometric figure. (More precisely, can one transform the variety defined by this system into a smooth figure by using so-called birational transformations?). […] Recently the general case (any $n$) was settled affirmatively by Hironaka.Another achievement, of a quite different nature, is the systematic rebuilding of the foundations of algebraic geometry now led by Grothendieck in France. His work, which also leads to solutions of important concrete problems, has influenced many young mathematicians, including those working in different fields.

Nowadays engineers regard Hilbert space as the “smooth figure” of the above observations, which arises as the algebraically birational (and computationally efficient) resolution of an underlying varietal state-space.

Quantum dynamical trajectories computationally “unravel” (in the sense of Howard Carmichael) upon these algebraic state-spaces, in service of innumerably many objectives of modern system engineering. Mark Murcko’s recent video essay **“Accelerating Drug Discovery“** (reference below) surveys the astounding pace of development of the resulting mathematical and computational capabilities capabilities that (as it seems to me) are grounded entirely in Grothendieck’s transformational vision.

Who authored this foresighted assessment from 1968? One candidate is lattice theorist Robert P. Dilworth (whose thesis advisor Eric Temple Bell and student Juris Hartmanis will be familiar to many students of mathematics and theoretical computer science). And it is impressive too (as it seems to me) that this early assessment of Grothendieck’s work could receive the public *imprimatur* of a committee so diverse as the collection of mathematicians listed below.

**Conclusion** Not every STEAM roadmap committee gets it right … this one did!

@book{NAS:1968, Title = {The mathematical

sciences; a report}, Year = {1968} Address =

{Washington}, Author = {{Committee on Support of

Research in the Mathematical Sciences of the

National Research Council for the Committee on

Science and Public Policy of the National Academy

of Sciences}}, Publisher = {National Academy of

Sciences}, Series = {Publication number 1681},

Annote = {Committee members: Lipman Bers, T. W.

Anderson, R. H. Bing, Hendrick W. Bode R. P.

Dilworth, George E. Forsythe, Mark Kac, C. C.

Lin, John W. Tukey, F. J. Weyl, Hassler Whitney,

C. N. Yang }}

```
```

`@inproceedings{Murcko:2014aa, Title =`

{Accelerating Drug Discovery: The Accurate

Prediction of Potency}, Author = {Mark Murcko},

Note = {URL:

\url{https://www.youtube.com/watch?v=wa59ZgflJD8}

, Booktitle = {Advances in Drug Discovery and

Development}, Month = {24 September},

Organization = {Chemical \&\ Engineering News

(Virtual Symposium)}, Year = {2014}}

“(for shapes more complicated that disks)”

Typo: “that” -> “than”

Also I must have missed the reference – what do you mean by a ring punishing Moby for his sins?

]]>He broke his proofs down into small steps so that each step is “trivial” if you remember all the new concepts and all the previous steps. But developing an intuition for all the new concepts takes real work. You have to rewire your brain.

]]>“Interestingly, if you start with a variety (over the complex numbers), there’s a standard way to associate a ring with it, and in that case Grothendieck’s construction doesn’t give you anything new. It’s for the other kinds of rings that you get something new.”

I’ve seen a bunch of these “schemes for the layman” things since Grothendieck died, and I think it’s *really* worth saying that this ring is just the ring of coordinate functions on the variety. The way it is now this just sounds like some arbitrary, inscrutable algebraic invariant, whereas in reality it’s something very simple. Most people will be able to make a mental picture of things like the latitude and longitude functions on a sphere. (Okay, technically one of those isn’t an actual coordinate function, but it at least gives a morally correct picture.)

Also it’s probably worth making the point that Grothendieck didn’t just say “Let’s associate geometric-ish objects to arbitrary rings and see what happens.” He knew that representable functors had better categorical properties than arbitrary ones, that certain functors “ought” to be representable, and that the prime spectrum of a ring was what controlled the associated hom-functor. I think it’s really more accurate to say that Grothendieck’s primary insight here was that instead of taking the objects you’re interested in and dealing with whatever category they form, you should build a good category including them and then understand the new objects.

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