Approximate subrings are subsets $A$ of a ring $R$ satisfying \[ A + A + AA \subset F + A \] for some finite $F \subset R$. They encode the failure of sum-product phenomena, much like approximate subgroups encode failure of growth in groups.

I will discuss how approximate subrings mirror approximate subgroups and how model-theoretic tools, such as a stabilizer lemma for approximate subrings due to Krupiński, lead to structural results implying a general, non-effective sum-product phenomenon in arbitrary rings: either sets grow rapidly under sum and product, or nilpotent ideals govern their structure. I will also outline related results for infinite approximate subrings and conjectures unifying known (effective) sum-product phenomena.

Based on joint work with Krzysztof Krupiński.

We generalize the seminal polynomial partitioning theorems of Guth and Katz [1, 2] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n - k - r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemeredi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves.

These results, together with some of my other recent work (e.g., bounding the number of distinct distances on plane Pfaffian curves), are steps in a larger program - pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk.

This talk is based on multiple joint works with Saugata Basu, Antonio Lerario, Martin Lotz, Adam Sheffer, and Nicolai Vorobjov.

[1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge
Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.

[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of
mathematics (2015), 155–190.
 

In this ``journal club''-style advanced class, I will present some material from a recent paper of Tom Scanlon https://arxiv.org/abs/2508.17485 . Motivated by the question of decidability of the field C(t) of complex rational functions in one variable, he considers the structure $(\mathbb{C};+,\cdot,CM)$ of the complex field expanded by a predicate for the set CM of j-invariants of elliptic curves with complex multiplication (the "special points"). Analogous to Zilber's result from the 90s on stability of the expansion by a predicate for the roots of unity, Scanlon shows that Pila's solution to the André-Oort conjecture implies that this structure is stable, and moreover that effectivity in this conjecture due to Binyamini implies decidability. I aim to explain Scanlon's proof of this result in some detail.