Research

Existential Closedness (EC) asks for geometric conditions ensuring that systems of equations involving transcendental functions have solutions. A familiar starting point is the fundamental theorem of algebra: every nonconstant polynomial in one complex variable has a root. For exponential equations, one must also account for identities such as e^{z + w} = e^ze^w and the fact that an exponential never vanishes. The guiding idea is that systems should have solutions unless algebraic or functional relations make them overdetermined.

The exponential case

Replacing exponential expressions by new variables turns the polynomial part of a system into an algebraic variety in ℂⁿ × (ℂ^×)ⁿ. Solving the system means finding a point on this variety of the form (z₁, …, z_n, e^z₁, …, e^z_n). Zilber’s Exponential Closedness conjecture predicts such a point for every irreducible variety that is free and rotund. Freeness excludes constant nontrivial integer linear combinations of the input coordinates and constant nontrivial multiplicative combinations of the output coordinates. Rotundity requires the image under each integer matrix of rank r, acting additively on inputs and multiplicatively on outputs, to have dimension at least r. This captures constraints that a simple count of equations and variables can miss.

The strong version asks for solutions that are algebraically generic over the parameters defining the equations. It therefore concerns how independent a solution can be, as well as whether one exists. Variants of these questions apply to exponential maps of abelian and semiabelian varieties and to other transcendental functions.

The modular j-function

The modular j-function assigns an invariant to each complex elliptic curve represented by a point of the upper half-plane. Its transformation rules play the role of the exponential identities. In particular, rational linear fractional transformations of the input give rise to modular-polynomial relations between the corresponding j-values. These relations encode isogenies between elliptic curves.

The geometric formulation uses an irreducible variety V ⊆ ℂⁿ × ℂⁿ. It is broad if its projection onto any k input–output pairs has dimension at least k. It is free if it has no constant coordinates, rational Möbius relations between inputs, or modular-polynomial relations between outputs.

Modular EC asks for points of the form (z₁, …, z_n, j(z₁), …, j(z_n)) on every broad, free irreducible variety. The domain is part of the formulation: one can consider j on the upper half-plane or its extension by Schwarz reflection to both half-planes, ℂ ∖ ℝ. The same geometric hypotheses need not give equivalent existence statements for these two domains.

Derivatives and connections with Zilber–Pink

Adding j′ and j″ leads to systems involving the inputs, the j-values, and their derivatives. The problem is again to identify the algebraic and differential constraints that are forced by the functions, and then to find solutions whenever the remaining equations are compatible with those constraints. A strong version asks for algebraically generic solutions, while differential versions study the corresponding equations in differential fields.

One of my broad research directions is to study EC and Zilber–Pink (ZP) for j together with its derivatives. This reveals connections between the two problems and gives a more general picture of each: EC supplies solutions, while ZP helps control exceptional algebraic relations among them. Differential-algebraic and functional-transcendence methods provide a common framework for exploring these connections.

Selected results

• Full Modular Existential Closedness. I prove MEC for j extended by Schwarz reflection to both half-planes, obtaining Zariski-dense solutions on every broad, free irreducible variety.
• MEC with derivatives for n = 1. With Sebastian Eterović and Vincenzo Mantova, I prove Zariski-dense solvability for polynomial equations in z, j, j′, and j″, apart from explicit exceptional cases.
• Abelian EC with dominant projection. With Jonathan Kirby and Vincenzo Mantova, I prove EC for varieties projecting dominantly onto the domain of the exponential map, for abelian varieties and algebraic tori.
• Differential EC. With Sebastian Eterović and Jonathan Kirby, I prove EC for the differential equation of j and its derivatives in differentially closed fields.

Zilber–Pink (ZP) concerns intersections that are larger than a dimension count would suggest. If algebraic varieties V and T lie in a smooth ambient variety S, the expected intersection dimension is dim V + dim T − dim S. An intersection component of greater dimension is called atypical. The conjecture studies this phenomenon when T belongs to an arithmetically distinguished collection of special subvarieties.

Semiabelian varieties

A semiabelian variety is an algebraic group that is an extension of an abelian variety by an algebraic torus. These groups provide a natural setting for Diophantine questions: examples include powers of the multiplicative group and products of elliptic curves. In the torsion formulation, special subvarieties are translates of connected algebraic subgroups by torsion points.

The conjecture predicts that a fixed algebraic variety contains only finitely many maximal atypical subvarieties arising from intersections with these special subvarieties. Thus, exceptional intersections should be accounted for by finitely many larger geometric structures. This encompasses the Manin–Mumford phenomenon, while broader formulations connect with Mordell–Lang. Finiteness of maximal atypical subvarieties does not assert finiteness of all special points: infinitely many may lie in a positive-dimensional special subvariety.

The modular setting

For products of modular curves, the special subvarieties are defined by modular-polynomial equations, expressing isogenies between elliptic curves, and by coordinates fixed at j-values of elliptic curves with complex multiplication. ZP predicts the same kind of finiteness for maximal atypical intersections. The André–Oort conjecture is an important part of this picture, describing algebraic varieties containing a Zariski-dense set of special points.

Adding derivatives

I study modular and automorphic versions of these questions, especially ZP and EC for j together with j′ and j″. Derivatives retain information that is invisible if one considers only the j-values. The appropriate functional and differential versions of special subvarieties and atypical intersections bring out links between EC and ZP and provide a more general picture of both problems.

Ax–Schanuel theorems constrain the algebraic dependencies that can occur among these functions and their arguments. They are central to this approach: together with differential algebra and model theory, they help separate the relations forced by the functions from additional, exceptional relations.

Selected results

• Modular ZP for geometrically generic varieties. With Sebastian Eterović and Guy Fowler, I prove Pink’s formulation for varieties V ⊆ Y(1)ⁿ whose projections onto any dim V + 2 coordinates are not defined over the algebraic numbers.
• ZP with derivatives. I prove functional and differential analogues for j and its derivatives, including a weak form of Modular ZP with Derivatives and a functional André–Oort statement.
• Atypical intersections in semiabelian and modular settings. I prove finiteness results for certain atypical intersections by combining weak forms of ZP with Mordell–Lang and its variants.