L3

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.
 

The dynamical Phi^4_3 model is a stochastic partial differential equation that arises in quantum field theory and statistical physics. Owing to the singular nature of the driving noise and the presence of a nonlinear term, the equation is inherently ill-posed. Nevertheless, it can be given a rigorous meaning, for example, through the framework of regularity structures. On compact domains, standard arguments show that any solution converges to the equilibrium state described by the unique invariant measure. Extending this result to infinite volume is highly nontrivial: even for the lattice version of the model, uniqueness holds only in the high-temperature regime, whereas at low temperatures multiple phases coexist.

We prove that, when the mass is sufficiently large or the coupling constant sufficiently small (that is, in the high-temperature regime), all solutions of the dynamical Phi^4_3 model in infinite volume converge exponentially fast to the unique stationary solution, uniformly over all initial conditions. In particular, this result implies that the invariant measure of the dynamics is unique, exhibits exponential decay of correlations, and is invariant under translations, rotations, and reflections.

Joint work with Martin Hairer, Jaeyun Yi, and Wenhao Zhao.

We study a prototypical geometric wave equation, given by wave maps from the Minkowski space R 1+d into the sphere S d , under the influence of additive stochastic forcing, in all energy-supercritical dimensions d ≥ 3. In the deterministic setting, self-similar finite-time blowup is expected for large data, but remains open beyond perturbative regimes. We show that adding a non-degenerate Gaussian noise provokes finite-time blowup with positive probability for arbitrary initial data. Moreover, the blowup is governed by the explicit self-similar profile originally identified in the deterministic theory. Our approach combines local well-posedness for stochastic wave equations, a Da Prato-Debussche decomposition, and a stability analysis in self-similar variables. The result corroborates the conjecture that the self-similar blowup mechanism is robust and represents the generic large-data behavior in the deterministic problem.

This is joint work with M. Hofmanova and E. Luongo (Bielefeld)

Fracture is usually treated as an outcome to be avoided; here we see it as something we may write into a lattice's microstructure. Maxwell lattices sit at the edge of mechanical stability, where robust topological properties provide a way on how stress localises and delocalises across the structure with directional preference. Building on this, we propose a direct relationship between lattice topology and damage propagation. We identify a set of topology- and geometry-dependent parameters that gives a simple, predictive framework for nonideal Maxwell lattices and their damage processes. We will discuss how topological polarisation and domain walls steer and arrest damage in a repeatable way. Experiments confirm the theoretical predicted localisation and the resulting tuneable progression of damage and show how this control mechanism can be used to enhance dissipation and raise the apparent fracture energy.

The mitigation of bacterial adhesion to surfaces and subsequent biofilm formation is a key challenge in healthcare and manufacturing processes. To accurately predict biofilm formation you must determine how changes to bacteria behaviours and dynamics alter their ability to adhere to surfaces. In this talk, I will present a framework for incorporating microscale behaviour into continuum models using techniques from statistical mechanics at the microscale combined with boundary-layer theory at the macroscale.

We will examine the flow of a dilute suspension of motile bacteria over a flat absorbing surface, developing an effective model for the bacteria density near the boundary inspired by the classical Lévêque boundary layer problem. We use our effective model to derive analytical solutions for the bacterial adhesion rate as a function of fluid shear rate and individual motility parameters of the bacteria, validating against stochastic numerical simulations of individual bacteria. We find that bacterial adhesion is greatest at intermediate flow rates, since at higher flow rates shear-induced upstream swimming limits adhesion.

Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. Notably, these estimates require the computation of determinants of differential operators, which in relevant cases are not traceclass and require appropriate renormalization. We demonstrate that the Carleman--Fredholm operator determinant is the correct choice. Throughout, I will demonstrate the applicability of these methods to high-dimensional real-world systems, for example coming from atmosphere and ocean dynamics.