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In this article, we introduce and study a natural notion of coarse separation for metric spaces, with an emphasis on coarse separation by subspaces of polynomial or subexponential growth. For instance, we show that symmetric spaces of…
We describe a versatile mechanism that provides tight-binding models with an enriched, topologically nontrivial bandstructure. The mechanism is algebraic in nature, and leads to tight-binding models that can be interpreted as a non-trivial…
The bidimensionality of a set of vertices $X$ in a graph $G$ is the maximum $k$ for which $G$ contains as a $X$-rooted minor the $(k \times k)$-grid. This notion allows for the following version of the Graph Minors Structure Theorem (GMST)…
We use the Riemann-Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers…
We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised…
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary…
We explicitly construct a symplectomorphism that relates magnetic twists to the invariant hyperk\"ahler structure of the tangent bundle of a Hermitian symmetric space. This symplectomorphism reveals foliations by (pseudo-) holomorphic…
For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$ such that images of every pairwise distinct $k$ points are affinely (linearly) independent. Such maps are called affinely (linearly) $k$-regular embeddings. We…
We strengthen the connection between the Ahlfors-regular (AR) conformal dimension Confdim$(Z)$ of a compact AR metric space $Z$ and a certain critical exponent of the Poincar\'e profiles $p_{\Lambda}$ of its hyperbolic cone $X$ in the sense…
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…
We discuss various aspects of "braid spaces'' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of…
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links…
We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that…
We develop a new framework to study minimum $d$-degree conditions in $k$-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path cover and connecting…
We study difference equations which are obtained from the asymptotic expansion of topological string theory on the deformed and the resolved conifold geometries as well as for topological string theory on arbitrary families of Calabi-Yau…
A Riemannian manifold is a called a good rational expander in dimension $i$ if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume. We construct 3-manifolds which are good expanders in all dimensions. On the other…
Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially…
Exploiting the special features of four-dimensional Riemannian geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5. First, we provide a complete description of the Weyl tensor for…
A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi shows that every large $n$-vertex graph with minimum degree at least $(1/2+\gamma)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning…
We employ partitioning methods, in the spirit of Montiel--Ros but here recast for general actions of compact Lie groups, to prove effective lower bounds on the Morse index of certain families of closed minimal hypersurfaces in the round…