The Valley

1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

“I see a massive valley before me. The only question is… Should I cross it?”

This is how, about three years ago, and living in Moscow, I posed to a friend the dilemma of whether I should attend graduate school for math.

The analogy held up well. I often envisioned my progress through the program in terms of the same imposing valley. “The brambles are getting so thick, at this depth, that the light which once flowed liberally from the surface is becoming damp and attenuated,” I once thought, as I entered the later stages of the notorious first-year program in algebra and analysis. “I’m so far from charted land that I’m encountering species wholly unknown to the outside world,” I thought later, during my second year – as I first began to encounter Hodge Theory – envisioning passing into a dark forest where marvelous, strange creatures chirped and glowed.

Once in while I’d perceive myself on an unexpected trail towards a promising discovery. “It’s as if I’ve glimpsed light on the other side much earlier than I had anticipated,” I’d think.

The analogy seems to have lost some of its power now. I’ve made my way to the other side, sure. But much of the valley remains unexplored – by me or by anyone else. Indeed, I was struck by just how much math we simply don’t understand – how little we know about higher-dimensional algebraic varieties, for example.

It stands to reason then that I could discover just one little intriguing, fascinating, and yet surprisingly deep pattern. In this post, I’ll try to describe what I found. (The full text of my paper, which is currently under review at the Journal of Pure and Applied Algebra, is available here.)

Surfaces in a fourfold in an ambient space

Our starting point will be a smooth, four-dimensional algebraic variety, also called a smooth fourfold.

We observe first that when a smooth fourfold is embedded inside a larger ambient space – say, n-dimensional projective space – surfaces inside this fourfold can be produced in a convenient way by intersecting the fourfold with subvarieties of the appropriate dimension in the ambient space.

An illustration of this process is given below. For practical reasons, I’ve divided the relevant dimensions by two, so that instead of drawing surfaces in a fourfold, I’ve drawn curves in a surface.

Each ambient subvariety, when intersected with this surface, yields a curve inscribed in it.

In reality, the dimensions involved make this process all but impossible to visualize. We can study it nonetheless, though. Surfaces produced in this special way will be our primary object of study.

Families of surfaces

We want to start “counting” things. But the smooth objects we’re dealing with can be shifted and deformed in a continuous way, covering infinitely many intermediate states, so that it doesn’t make sense to count them per se. We must group things together in a sensible way, so that we can count groups instead of individual objects.

We will put subvarieties in the ambient space into “bins” by associating together into the same bin any two subvarieties that can be smoothly deformed into each other. If two subvarieties are so different that they cannot be smoothly shifted into each other, then we will keep them separate.

The same grouping procedure can be performed within the fourfold itself, so that any two surfaces that can be morphed into each other will be likewise counted as one family. This way, we can count families of surfaces in the fourfold instead of individual surfaces.

The above-described intersection procedure plays nicely with the grouping procedure. Indeed, if two subvarieties can be morphed into each other in the ambient space, then their corresponding intersections can also be morphed into each other within the fourfold. The mathematical way of saying this is that the deformation relation is preserved across intersection.

The lowermost two subvarieties can be smoothly deformed into each other, as can the curves they yield.

The associated function

After we package together subvarieties in the ambient space into equivalence classes as described above, the equivalence classes themselves assemble into the structure of a lattice of dots (that is, an infinite row, an infinite grid, or, generally, a grid of some arbitrary fixed dimension). It might help to imagine the separate families of subvarieties arranged discretely in an infinitely tall stack (indeed, when the ambient variety is a projective space, the lattice is just one-dimensional, though for other ambient varieties this dimension can vary). (In reality, the deformation relation has nothing to do with the positions of the subvarieties but rather of their shapes, but the visual aid is helpful nonetheless.)

The families of ambient subvarieties assemble into a lattice.

Let’s assume now that the smooth fourfold satisfies a technical condition whereby its first Chern class vanishes (the Calabi-Yau manifolds used in string theory satisfy this condition). My work defines, for any such fourfold (embedded in an ambient variety), a function associated to the embedding — defined on the set of equivalence classes of ambient subvarieties (so that it associated a number to each such class) — with many interesting properties. (Defining the function and showing that that it actually has these properties takes some difficult math.)

For one, the value of the function on any particular class (meaning the number it associates to this class) confers topological information about the surface produced by intersecting any member of this class with the fourfold (provided that the surface is smooth, a technicality I’ll ignore here). This topological information is this surface’s so-called holomorphic Euler characteristic, which measures in some sense how “twisted” the surface is. More precisely, the function bounds this quantity from below, so that if we determine its value on any particular equivalence class of subvarieties, we can establish that the Euler characteristic of any surface corresponding to a member of this class is at least that high.

This function’s second key property is that it is integer-valued and quadratic, when viewed as a function on the underlying lattice. This means that it’s a polynomial, with at the highest second powers. The following functions are examples of quadratic functions:

1. $f(x) = 6x^2 - 90x$ (so that, e.g., $1 \mapsto -54$)
2. $f(x, y) = 45x^2 + 36xy + 27y^2 - 171x - 9y$ (so that, e.g., $(1,1) \mapsto -72$)
3. $f(x,y,z) = 16xy + 48xz + 24y^2 + 48yz + 8z^3 - 72x - 128y - 112z$

In each case, the dimension of the underlying lattice is indicated by function’s bullet point. (All of these are actual associated functions attached to specific examples I’ve studied.)

Asymptotic behavior of the associated function

These two properties serve to establish rich information about the surfaces which arise in the fourfold through intersection with an ambient subvariety.

One sort of case is particularly illustrative: That in which the fourfold’s associated function is positive definite, which means that as one travels away from the lattice’s origin the value of the function gets progressively higher and higher. (Functions 1 and 2, but not 3, above are positive definite.)

What these properties show is that in such a case, for any integer r the surfaces whose twistedness is not greater than r belong to at most finitely many families. Indeed, if the associated function tends to infinity away from the origin, it can attain a value less than or equal to r on only finitely many lattice points. But because the value of this function on a lattice point bounds from below the Euler characteristic of any smooth surface attached to the family of ambient subvarieties this lattice point represents, the smooth surfaces with Euler characteristic also lower than or equal to r must belong to even fewer families. (Think about this: While it’s tricky, it doesn’t depend on any advanced ideas.)

What’s even more interesting is trying to determine more exactly just how many such families there can be as r grows asymptotically. Indeed, as the value of the function grows progressively away from the origin, the “fronts” at which it attains any particular value r grow, as r grows, as an ever-expanding family of nested ellipsoids (that is, intervals, ellipses, or ellipsoids of arbitrary dimension). The problem of estimating how many families of surfaces have twistedness less than or equal to r reduces to estimating how many lattice points live within an ellipsoid whose size is determined by r.

This question is none other than a generalization of the celebrated, deep, Gauss circle problem, in which the dimension of the lattice is 2 and the ellipse is actually a circle. Indeed, though the number of lattice points within a circle is clearly related to the circle’s area, estimating the number exactly is a truly deep problem, whose solution involves infinite series and which is related to the Riemann hypothesis.

When the associated function is not positive definite, it tends towards either positive infinity or negative infinity (or somewhere in between) as one moves away from the origin, and things become even harder to estimate. Nonetheless, the “fronts” in this case are shaped like hyperbolic sheets, and in these cases too many interesting estimates can be achieved.

Thus, my work produces a “tunnel” or “wormhole” between the smooth world of complex algebraic geometry and the discrete, number-theoretic world of lattice point counting. This journey in fact occupies a sequence of steps (many of which are glossed over here), in each of which information is packaged and translated from more smooth to more discrete forms. That this is possible to begin with attests to the intrigue of algebraic geometry, in which, though smooth objects abound, discrete structure lurks within. To extract this hidden organization is the magic and the promise of the field.

I found something deep within the valley and brought it back.