Weekly Progress: Tame Geometry, Pregeometries, and Dynamical Complexity (10–16 July 2026)

This week in one sentence

The strongest new signals are not a single o-minimality theorem, but a useful cluster around tame geometry: Zariski topology recovering algebraic structure, dense-codense expansions of geometric classes, finite pregeometric tests for stable forking, and two adjacent applications where resolution or bifurcation forces quantitative control.

This first issue covers arXiv submissions dated 10–16 July 2026 in the requested math.LO and math.AG orbit. The papers are preprints; the summaries below report the authors’ stated results, not peer-review status.

1. Topological reconstruction reaches quasi-projective varieties

Benjamin Castle and Ronan O’Gorman, “Topological reconstruction theorems over uncountable algebraically closed fields” — submitted 16 July.

The paper extends the Kollár–Lieblich–Olsson–Sawin reconstruction picture from normal projective varieties to arbitrary quasi-projective varieties over uncountable algebraically closed fields. In the irreducible case, for varieties of dimension at least two, a homeomorphism of the underlying Zariski spaces factors through a field isomorphism and a universal homeomorphism between the normalizations. 1

That formulation is close to optimal. Normalization is unavoidable because bijective normalization maps can be homeomorphisms without being isomorphisms. In positive characteristic, Frobenius produces further purely inseparable homeomorphisms, so the conclusion is a universal homeomorphism rather than an isomorphism. In characteristic zero, the normal case recovers the cleaner isomorphism statement. 1

The model-theoretic point is the method: the authors place the reconstruction problem in a framework they call the “Zilber trichotomy for ACF-relics.” For this channel, the important takeaway is that model theory is not only classifying definable sets here; it is supplying the mechanism that turns a topological equivalence into algebraic structure.

2. Dense-codense expansions make pregeometry measurable

Alexander Berenstein and Evgueni Vassiliev, “Dense-codense expansions of quasiminimal pregeometry structures” — submitted 13 July.

The authors study two unary-predicate expansions of quasiminimal pregeometry structures: beautiful pairs and HH-structures. They show that each can be axiomatized by a single sentence of

L_{ω_{1} ω} (Q)

, and that both expansions are

ω

-stable. They also construct natural independence relations and relate ranks in the expansions to the complexity of the underlying pregeometry. 2

The bridge to tame geometry is explicit rather than incidental. The introduction places dense o-minimal structures among the geometric theories whose algebraic closure behaves like a pregeometry, and treats dense-codense expansions as a common generalization of dense pairs of o-minimal structures and related pair constructions. 2

Two structural results are especially worth tracking. For HH-structures, the paper distinguishes trivial from non-trivial pregeometry by whether the associated Lascar-splitting rank is 1 or at least

ω

. For beautiful pairs, in the trivial and locally modular non-trivial cases the corresponding ranks are 1 and 2, respectively. The paper also characterizes local modularity through weak one-basedness and several equivalent properties of beautiful pairs. 2

This is not a new theorem about a particular o-minimal expansion of the real field. Its value for the channel is broader: it gives a clean framework for asking how much of a tame pregeometry survives after adding a dense predicate.

3. Stable forking becomes a finite pregeometry problem

Scott Mutchnik, “Reducing stable forking dependence to finitely many pregeometries” — submitted 10 July.

In finite-rank supersimple theories, the paper proves that the relevant case of the stable forking conjecture is determined by finitely many pregeometries in each rank. Equivalently, for rank

n

, the instability of forking can be detected by embedding one of a finite collection of matroids into the pregeometry of a rank-one partial type over a finite set. 3

The result sits on the stable/simple side of the model-theoretic map rather than directly in NIP or dp-minimality. That makes it useful context for this channel: it turns a conjectural dependence phenomenon into a finite combinatorial obstruction problem, while keeping the geometry of types in view. The proof combines a rank-three result of Peretz, a higher-rank existence substitute, and an argument adapted from multi-experiment parameter identifiability in applied model theory. 3

The practical research signal is the shift in scale. Instead of searching through an uncontrolled family of possible matroid obstructions, one can ask whether a finite obstruction basis exists rank by rank.

4. Real log-canonical thresholds control a PDE boundary

Nivaldo Grulha and Andréa Prokopczyk, “Geometric Criteria for Morrey Admissibility via the Real Log-Canonical Threshold” — submitted 16 July; cross-listed in math.AG.

For singular interaction kernels whose gradient is comparable near an isolated analytic zero to

∣ \nabla f ∣ / ∣ f ∣^{2}

, the authors give an exact local integrability criterion. If

(N_{i}, M_{i}, a_{i})

are the vanishing-order and discrepancy data produced by a log resolution, then

\nabla K \in L_{loc}^{p} ⟺ p < i min \frac{a _{i} + 1}{2 N _{i} - M _{i}} .

The real log-canonical threshold supplies a computable lower bound for this exact Morrey-admissibility threshold, and the two coincide for Newton non-degenerate singularities. 4

The paper is an adjacent real-algebraic-geometry item rather than a model-theory paper. Its conceptual relevance is the same one that repeatedly appears in tame settings: complicated local behavior is reduced to finite divisorial data after resolution. Here that data controls analytic admissibility for aggregation equations, including anisotropic singular kernels, rather than definability or cell decomposition. 4

5. Bifurcation forces gonality growth in dynamical families

Zhuchao Ji and Junyi Xie, “Genus and Gonality of Small Curves, Dynamical Uniform Boundedness, and Bifurcation” — submitted 14 July; cross-listed in math.AG.

For a non-isotrivial one-parameter family of rational maps on

P^{1}

, the authors prove the Gonality Conjecture: distinct dynatomic curves have gonality tending to infinity. Outside the flexible Lattès family, they prove more generally that every small sequence of horizontal curves has gonality tending to infinity and genus growing superlinearly relative to its degree over the parameter curve. 5

The paper also derives uniform boundedness statements for iterated preimages over number fields and geometric uniform boundedness for preperiodic points over function fields. In higher dimension, the analogous theorem requires a non-empty bifurcation set and, from dimension three onward, a periodic-multiplier genericity condition. 5

For tame-topology readers, the connection is indirect but instructive: arithmetic smallness is converted into geometric complexity. The obstruction is not definability, but bifurcation and ramification; the output is still a quantitative statement about how structured families can be.

What to watch next

Direct papers on o-minimality, NIP, and dp-minimality remain the highest-priority scan for the next issue.
The most promising adjacent threads this week are dense expansions of geometric structures, model-theoretic reconstruction over algebraically closed fields, and resolution-theoretic invariants that turn local singular behavior into computable thresholds.
As always with arXiv, a new version or a paper’s eventual journal publication may change the status or formulation of a result.