{\huge Alexander Grothendieck 1928--2014}

Creating vast beautiful mansions from the becoming of nothing \vspace{.2in}

Alexander Grothendieck, who signed his works in French ``Alexandre'' but otherwise kept the spelling of his German-Jewish heritage, passed away Thursday in southwestern France.

Today we mourn his passing, and try to describe some of his vision.

Part of the story of this amazing mathematician is that in 1970 he renounced his central position at the Institut des Hautes \'Etudes Scientifiques (IHES) in Paris, and made himself so remote shortly after formally retiring from the University of Montpellier in 1988 that not even family and friends could track him. He declined a Fields Medal in 1966 and the Crafoord Prize in 1988.

As captured by this obituary, he had left to seek a society kinder and more just than the ones that killed his father at Auschwitz and convicted him in 1977 of violating a French law dating to 1945 against feeding and sheltering an unregistered alien. More will be told of this story as his voluminous writings from the hinterland are being read. But between 1945 and 1970 he published mathematics of unparalleled sweep and power, conveying escalations of abstraction to the solution of concrete problems, and this is the part we wish to appreciate.

The Space Outside the Cave

One of humanity's greatest intellectual tropes is Plato's ``Allegory of the Cave,'' which likens what we apprehend through our senses to shadows of forms projected on a wall from a dimly-lit fire. The forms hail from an outside world whose light is blinding to one prisoner unchained and led out of the cave. Although Plato speaking through Socrates addressed all of reality, let us just imagine this outside world as Euclidean space, in which the Platonic solids shimmer in their ideal forms. Then what Grothendieck perceived when he was led into the light is the following:

The outside of the cave is another cave.

In this outer cave the focal point is $latex {0}&fg=000000$, zero. This zero is the only solution to the equation $latex {x = 0}&fg=000000$. It is also the only solution in that cave to the equation $latex {x^2 = 0}&fg=000000$. Likewise $latex {x^3 = 0}&fg=000000$, $latex {x^4 = 0}&fg=000000$, and so on. These are different equations, but each has only the same single root in the Euclidean space of the outer cave. We can add the words ``with different multiplicities,'' but what difference do they make to the objects from which we draw our solution?

Perhaps that $latex {0}&fg=000000$ is but a projection along a beam of elements in a higher space that can furnish different solution structures to these different equations. What Grothendieck regarded as needed for ``truly natural methods in geometry,'' as related by Jacob Murre quoting a lecture by Grothendieck in 1959, is the employment of nilpotents, that is, elements $latex {e}&fg=000000$ such that the sequence $latex {e,e^2,e^3,e^4,\dots}&fg=000000$ eventually gives zero. Such elements can be as simple as $latex {2}&fg=000000$ modulo $latex {4}&fg=000000$ or the matrix $latex {[^0_0{\,}^1_0]}&fg=000000$, but organizing them is what unchains us from the single zero.

Space and Syntax

In moving essays written by Grothendieck's friend and colleague Pierre Cartier for the 40th and 50th anniversaries of the IHES, coinciding with Grothendieck's 70th and 80th birthdays, Cartier did not shrink from invoking Albert Einstein for intellectual comparison. Nor did Grothendieck, as the latter essay relates regarding the approach to space.

Einstein famously derived the core of his physical theories by working out all the logical consequences of the visions in his thought experiments. One of them is that there is no focal point of space. Nor is space a pre-existing entity, as Isaac Newton had posited, but rather space emerges from relational properties of its contents. A manifold is not made by its points, and it need not be determined by the locally Euclidean structure near any one point, but rather by how open sets around points mesh together. But in math, when we have no matter, what can we take as the content that drives the structure?

In executing mathematics we can approach contents only via definitions and formulas and proofs, which are pieces of syntax. Plato was aware of this. In his Allegory of the Divided Line, which comes right before the ``Cave'' passage in his Republic, Plato divided the mathematical world internally in the same ratio by which he divided it from the world of sense experience. Mathematical Platonists distinguish themselves from formalists by affirming reality beyond formulas and proofs, which can seem like chains on the intellect. However, all schools can alike acclaim the way major advances in the 20th Century came from treating syntax as objects. One example comes clearest in Leon Henkin's proof of Kurt Gödel's Completeness Theorem by employing logical statements as elements of the constructed model. Cartier's 1998 essay picks right up from this. Consider a model $latex {M}&fg=000000$ assigning a truth value $latex {v_M(P) \in \{0,1\}}&fg=000000$ to every proposition $latex {P}&fg=000000$ in some Boolean algebra $latex {{\cal A}}&fg=000000$. Associate to $latex {P}&fg=000000$ the set $latex {[P]}&fg=000000$ of all models $latex {M}&fg=000000$ that make $latex {v_M(P) = 1}&fg=000000$. These sets obey the rules:

$latex \displaystyle [P \wedge Q] = [P] \cap [Q],\qquad[P \vee Q] = [P] \cup [Q]. &fg=000000$

$latex \displaystyle v(P) + v(Q) = v(P \wedge Q) + v(P \vee Q), &fg=000000$

What Are the Points?

Suppose we have a system $latex {B}&fg=000000$ of equations $latex {f_1(X) = 0,\dots,f_s(X) = 0}&fg=000000$, where $latex {X}&fg=000000$ ranges over some space $latex {S}&fg=000000$. Consider all objects of the form

$latex \displaystyle f = g_1 \cdot f_1 + g_2 \cdot f_2 + \cdots + g_s \cdot f_s, &fg=000000$

Now consider any point $latex {a = (a_1,\dots,a_n)}&fg=000000$ in an $latex {n}&fg=000000$-dimensional Euclidean space $latex {R^n}&fg=000000$. It is the unique solution to the simple system $latex {B = B_a}&fg=000000$ of equations

$latex \displaystyle f_1 = x_1 - a_1,\; f_2 = x_2 - a_2,\;\dots\; f_n = x_n - a_n, &fg=000000$

The concepts of ideal and prime and maximal can be applied even in simpler spaces $latex {S}&fg=000000$ such as the set of integers. Every integer $latex {a}&fg=000000$ generates the ideal $latex {I_a}&fg=000000$ of multiples of $latex {a}&fg=000000$. If $latex {a}&fg=000000$ factors properly as $latex {bc}&fg=000000$, then $latex {b \notin I_a}&fg=000000$ and $latex {c \notin I_a}&fg=000000$, so $latex {I_a}&fg=000000$ is not prime, while $latex {I_a \subset I_b}&fg=000000$ and $latex {I_a \subset I_c}&fg=000000$, so $latex {I_a}&fg=000000$ is not maximal. But when $latex {a}&fg=000000$ is prime, $latex {I_a}&fg=000000$ is both prime and maximal, and these are the only prime or maximal ideals of $latex {\mathbb{Z}}&fg=000000$. For other spaces such as our polynomials over $latex {R}&fg=000000$, however, the concepts of prime and maximal do not coincide. Grothendieck culminated a long list of people who realized that while maximality is the ``pointy'' property, primality is the ``special'' one.

With respect to our original points $latex {a \in R^n}&fg=000000$, for any ideal $latex {I}&fg=000000$ we can identify the set

$latex \displaystyle V(I) = \{a: f(a) = 0 \text{ for each } f \in I\}, &fg=000000$

$latex \displaystyle V(I \cap J) = V(I) \cup V(J),\qquad V(I \oplus J) = V(I_B) \cap V(I_{B'}). &fg=000000$

We must skip over some wonderful finiteness theorems by David Hilbert and his students---and over distinctions such as ``base ring'' $latex {R}&fg=000000$ being a field that is/is-not algebraically closed and projective versus affine space---to say only that algebraic sets are primitively defined in first-order arithmetic and hence are ``neat'' in many senses. Specially neat are those $latex {V}&fg=000000$ that cannot be written as $latex {V = V(I) \cup V(J)}&fg=000000$ in a nontrivial way, that is without one of $latex {V(I)}&fg=000000$ or $latex {V(J)}&fg=000000$ being all of $latex {V}&fg=000000$. Then we can't have $latex {V = V(IJ)}&fg=000000$ or $latex {V = V(I \cap J)}&fg=000000$ in a nontrivial manner, and exactly what this means is that the ideal $latex {I(V)}&fg=000000$ of all functions that vanish on $latex {V}&fg=000000$ is prime. Such a $latex {V}&fg=000000$ is called irreducible, and sometimes the term ``variety'' is still reserved for this case. In an abstract but natural way, irreducible varieties of all dimensions up to $latex {n-1}&fg=000000$ can be made to behave like points. To quote Cartier on ``the meaning of the word scheme'' (his emphasis):

``One must, of course, understand that the space Grothendieck associated with an algebraic variety is not the set of its own points, but the set of its irreducible subvarieties.''

Syntax Goes it Alone

To see where the nilpotents come in, and how Grothendieck unchained us not only from zero but from Euclidean points overall, we can begin by perceiving how evaluating a function is like doing long division with remainder. With respect to any ideal $latex {I}&fg=000000$ of $latex {S}&fg=000000$, the relation $latex {f - g \in I}&fg=000000$ is an equivalence relation, and allows us to write the quotient $latex {S/I}&fg=000000$. For our Euclidean point $latex {a}&fg=000000$, evaluating $latex {f(a)}&fg=000000$ can be achieved syntactically by reducing $latex {f}&fg=000000$ modulo (the ideal $latex {I_a}&fg=000000$ generated by) $latex {B_a}&fg=000000$. Taking $latex {f}&fg=000000$ modulo $latex {x_1 - a_1}&fg=000000$ by long division works out the same as substituting $latex {x_1 := a_1}&fg=000000$, and the same goes iteratively with the other elements of $latex {B_a}&fg=000000$.

The reduction process works for any ideal $latex {I}&fg=000000$ and gives a unique result $latex {f'}&fg=000000$, provided a special kind of basis $latex {B}&fg=000000$ giving $latex {I = I_B}&fg=000000$ named after Wolfgang Gröbner is used for the iterated long division. Thus the ``evaluation'' $latex {f' = f \bmod{I}}&fg=000000$ is well-defined for any ideal $latex {I}&fg=000000$ and can be carried out by an algorithm that first expands any initial set of generators for $latex {I}&fg=000000$ into a Gröbner basis. Alas all known algorithms have doubly exponential worst-case time complexity, perhaps unavoidably since deciding whether $latex {f \bmod{I} = 0}&fg=000000$ is complete for exponential space even when $latex {f}&fg=000000$ is linear and the initial generators for $latex {I}&fg=000000$ have constant degree. Nevertheless, these algorithms are run all the time for important equation-solving applications, and impress on us this philosophical fact:

We can do richer kinds of evaluation in the space delineated by our syntax than in the external Euclidean space.

This holds even when we return to our simple equations $latex {f_1 = x}&fg=000000$, $latex {f_2 = x^2}&fg=000000$ with $latex {n = 1}&fg=000000$, only one variable. The ideal $latex {I_1 = I_{\{x\}}}&fg=000000$ is prime---indeed maximal---but $latex {I_2 = I_{\{x^2\}}}&fg=000000$ is not prime. Also $latex {I_2 = I_1\cdot I_1}&fg=000000$, and the ideal $latex {I_3}&fg=000000$ generated by $latex {x^3}&fg=000000$ equals $latex {I_1\cdot I_2}&fg=000000$. Nevertheless, when we reduce a polynomial like $latex {f = x^3 + x^2 + x + 1}&fg=000000$ modulo these respective ideals, we get different results.

Thus we can dispense with the original points, even the origin in $latex {R^n}&fg=000000$. However, we would still like to preserve our primitive idea of ``evaluation'' in some kind of external space. How can we do this, and in what kind of space?

Laying Out Spaces

We cannot squeeze these answers out of our Euclidean space. We can interpret a quotient $latex {S/I}&fg=000000$ as endowing $latex {V(I)}&fg=000000$ with coordinates as a space in its own right, but that only works up to the prime ideals. Once we connected irreducible varieties to prime ideals, that much was one-and-done. We can't get multiple function values $latex {f \bmod{I_1}}&fg=000000$, $latex {f \bmod{I_2}}&fg=000000$, $latex {f \bmod{I_3}}&fg=000000$ out of our single, irreducible zero. There is no ``square root of zero'' different from zero. To go further, Grothendieck drew inspiration from how multiple-valued complex functions such as square-root and $latex {\log}&fg=000000$ can still be treated as holomorphic, by ``snipping'' and then ``layering'' $latex {\mathbb{C}}&fg=000000$ to riffle out their branches.

For square-root, let us snip the non-negative real axis out of the complex plane. This leaves an open subset $latex {O}&fg=000000$, on which every $latex {z \in O}&fg=000000$ has a unique square root $latex {s_+(z)}&fg=000000$ with positive real part. The function $latex {s_{+}}&fg=000000$ is analytic on $latex {O}&fg=000000$, as is the other branch $latex {s_{-}}&fg=000000$. To get them to coexist as a single entity with the essence of being holomorphic, however, requires a way of building ``layers'' on $latex {O}&fg=000000$, and on other open subsets as needed to cover the part of $latex {\mathbb{C}}&fg=000000$ that was snipped out.

Here is where the edifices become tall and the abstraction too steep to cover in a single post. Considering the infinitely-branching $latex {\log}&fg=000000$ function on one hand, and infinitely many degrees of equations on the other, we can expect that infinite structures will be employed. Indeed, Grothendieck built them above every open subset $latex {O}&fg=000000$ in a ``glued-together'' manner. We cannot even easily resort to our usual sign-off to ``see the paper for details.''

Yet we can say that the structures carry the idea of ``becoming'' points in $latex {O}&fg=000000$ via the concepts of fibres and sheaves, and that nilpotent elements are employed. Einstein's relational foundation is actuated by what Grothendieck termed his ``relative view'' of defining morphisms between representations as regulated by category theory, rather than defining stand-alone objects. The category of sheaves is abstracted to topos theory, by which the Greek word for ``place'' supplants the original idea of ``point.'' His French word étale described a flat sea as ``spreading'' like these layerings. It also reflects his heritage by deriving from a German word estal meaning ``place,'' whereby it connotes spreading out goods in layers in a stall that can be one of many spread out over a marketplace.

Personal Spaces

All this became massive, so much that Grothendieck's manuscripts before and after leaving IHES spread to hundreds and thousands of pages, as well as his personal memoir in the 1980s. Indeed, as related by Winfried Scharlau in a 2008 article for the AMS Notices, some of Grothendieck's colleagues believed that he tired at the prospect of climbing his own mountains.

We can ``morph'' the description of Grothendieck's ``rising sea'' approach in an essay by Colin McLarty to say that Grothendieck preferred to harness surveyors, engineers, and dam-builders so he could float to the top on rising waters, rather than do the ascent by ``hammer and chisel.'' He decried Pierre Deligne's 1974 completion of their program of proving the famous conjectures by André Weil by methods he and others felt were not ``morally right'' on account of bypassing Grothendieck's still-open ``standard conjectures.''

A 2004 article by Allyn Jackson reproduces a cartooned abstract by Grothendieck for a colloquium in 1971 by which he warned that doing his lecture ``in black-and-white detail for Springer Lecture Notes would likely take 400--500 pages,'' ending by writing that from ``a life-enrapturing logical delirium'' it was ``high time to change course.'' Today's mathematical community has in two years still barely touched a similarly-motivated though procedurally different theory erected by Shinichi Mochizuki on foundations named for Oswald Teichmüller, despite its ``mere'' 512 pages in first drafts.

Where it Touches Complexity

So what can all this mean for us who work in what Grothendieck described as a ``mansion'' in which ``the windows and blinds are all closed,'' while he was one of those ``whose spontaneous and joyful vocation it has been to be ceaseless building new mansions''? At least he did not call our dwelling a cave. However, in complexity theory we have it worse than Plato's cave-prisoners in not merely missing the blinding world outside, but sensing its impact as a negative image in our present ignorance of lower bounds.

Much of complexity theory translates naturally to questions about polynomials over finite fields. This goes not only to $latex {\mathbb{F}_2}&fg=000000$ for Boolean functions but also to $latex {\mathbb{F}_p}&fg=000000$ and $latex {\mathbb{F}_{p^k}}&fg=000000$ for higher primes $latex {p}&fg=000000$ and $latex {k > 1}&fg=000000$, which in turn yield questions about Boolean solutions to equations over these fields. There are possible advances to be had by improving the partial correspondence to problems in zero characteristic. The larger program out of the Weil conjectures seeks to transfer geometry to positive characteristic. Can we see how its further development might allow us to extract combinatorial results needed to put bounds on complexity?

Polynomials modulo composite numbers give us a more immediate frontier, one represented by the complexity class $latex {\mathsf{ACC^0}}&fg=000000$, whose nonuniform version was only recently separated from nondeterministic exponential time. These polynomials behave badly in manners stemming from nilpotent elements in the rings $latex {\mathbb{Z}_m}&fg=000000$ for composite $latex {m}&fg=000000$. Can we somehow supply ``extra points'' and valuations to raise their structure toward that of polynomials over finite fields, and thus at least achieve bounds known for polynomials modulo primes?

A third example, my favorite and most immediate for the theme of this post, concerns the famous $latex {\Omega(n\log n)}&fg=000000$ lower bound of Volker Strassen and Walter Baur on the size of arithmetic circuits computing some natural families of polynomial functions $latex {f(x_1,\dots,x_n)}&fg=000000$ in zero characteristic. Its proof, as we recounted in 2010, turns on a property of geometric degree that pertains only to affine Euclidean space (or to its projective cousin). It employs the ideal $latex {J_f}&fg=000000$ generated by

$latex \displaystyle y_1 - \frac{\partial f}{\partial x_1},\;\dots,\; y_n - \frac{\partial f}{\partial x_n}, &fg=000000$

Higher algebraic geometry has yielded notions of ``algebraic degree'' that can go as high as $latex {d^{2^n}}&fg=000000$. If the Strassen-Baur technique could be transferred to their higher spaces, then we could hope for strong lower bounds. The étale idea and related facets of algebraic geometry and representation theory also animate Ketan Mulmuley's ``Geometric Complexity Theory'' programme. I once tried to find a combinatorial shortcut using a degree-like measure $latex {\mu}&fg=000000$ of counting ``minimal monomials'' in ideals, which we described here. It is striking that the determinant polynomials score zero on this measure, whereas the permanents score astronomically even for $latex {n=5}&fg=000000$, but as with the other degree measures $latex {\mu}&fg=000000$, there are counterexamples to $latex {\log\mu}&fg=000000$ being a circuit size lower bound.

Open Problems

Can the answer to $latex {\mathsf{P}}&fg=000000$ versus $latex {\mathsf{NP}}&fg=000000$ be caught up in the ``rising sea''? Or will it need something even stronger than ``hammer and chisel''? What can we learn from his work? A sign of hope is that for all their heft and abstraction, his schemes can be programmed.

Our condolences to his relations and friends.