Related papers: Comparing Teichm\"uller and curve graph translatio…
Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves'…
This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory" (from 4…
Consider a family $\mathcal F$ of $C_{2r+1}$-free graphs, where $r\geq 2$. Suppose that each graph in $\mathcal F$ has minimum degree linear in its number of vertices. Thomassen showed that such a family has bounded chromatic number, or,…
Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has…
The Erd\H{o}s-Gallai Theorem states that every graph of average degree more than $l-2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd\H{o}s-Gallai Theorem in terms of minimum degree. Let…
We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the…
Given a unit vector $\textbf{v}\in\mathbb{R}^3$ and $\lambda\in\mathbb{R}$, a translating $\lambda$-soliton is a surface in $\mathbb{R}^3$ whose mean curvature $H$ satisfies $H=\langle N,\textbf{v}\rangle+\lambda,\ |\textbf{v}|=1$, where…
In this work, we study long-time wave transport in correlated and uncorrelated disordered 2D arrays. When a separation of dimensions is applied to the model, we find that the predicted 1D random dimer phenomenology also appears in so-called…
We consider classes of pseudo-random graphs on $n$ vertices for which the degree of every vertex and the co-degree between every pair of vertices are in the intervals $(np - Cn^\delta,np+Cn^\delta)$ and $(np^2- C n^\delta, np^2 +C…
We give nearly optimal bounds on the sample complexity of $(\widetilde{\Omega}(\epsilon),\epsilon)$-tolerant testing the $\rho$-independent set property in the dense graph setting. In particular, we give an algorithm that inspects a random…
This is the second of a three part study of relative free splitting complexes $\mathcal{FS}(\Gamma;\mathscr A)$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths $\tau_\phi \ge 0$ of the…
A filling curve $\gamma$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(\gamma)$ of $S$ via the process of "point-pushing along $\gamma$." We consider the relationship between the self-intersection number $i(\gamma)$ of…
We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of…
Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary…
Recently, the author and Melentijevi\'c resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{\pi^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the…
An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations…
In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the…
Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including which spaces allow such a diffeomorphism. It is…
We study the stability of spectrally stable, strictly monotone, smooth shear flows in the 2D Navier-Stokes equations on $\mathbb{T} \times \mathbb{R}$ with small viscosity $\nu$. We establish nonlinear stability in $H^s$ for $s \geq 2$ with…
The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…