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Abstract Let$$(R,\mathfrak {m})$$ $(R,m)$ be a complete local ring, and$$G=\textrm{gr}_{\mathfrak {m}}(R)$$ $G={\text{gr}}_m\left(R\right)$ be its associated graded ring. We introduce a homogenization technique which allows to relateGto the special fiber andRto the generic fiber of a “Gröbner-like” deformation. Using this technique we prove sharp results concerning the connectedness ofRandG. We also construct a family of local domains which fail to satisfy Abhyankar’s inequality for the Hilbert–Samuel multiplicity. However, we prove a version of the inequality which holds whenRis connected in codimension one. 

Abstract This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with construction of cohomogeneity one models with singular orbits. There is a partial result for the shrinking case. We also study a particular ansatz of popular interest and obtain asymptotic behaviors. 

Abstract We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3‐manifold . In particular, we describe invariant measures on the Grassmann bundle which arise as limits of random minimal surfaces. We show that if contains at least one totally geodesic subsurface then every topological limiting measure is totally scarring (i.e., supported on the totally geodesic locus), while we prove that geometrical limiting measures are never totally scarring. 

Abstract We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen–Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo–Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus. 

Abstract Apportionment refers to the well-studied problem of allocating legislative seats among parties or groups with different entitlements. We present a multi-level generalization of apportionment where the groups form a hierarchical structure, which gives rise to stronger versions of the upper and lower quota notions. We show that running Adams’ method level-by-level satisfies upper quota, while running Jefferson’s method or the quota method level-by-level guarantees lower quota. Moreover, we prove that both quota notions can always be fulfilled simultaneously. 

Abstract The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori–Vauquelin–Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous’ continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere. 

Abstract We study unknottedness for free boundary minimal surfaces in a three-dimensional Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary, and for self-shrinkers in the three-dimensional Euclidean space. For doing so, we introduce the concepts of boundary graph for free boundary minimal surfaces and of graph at infinity for self-shrinkers. We prove that these surfaces are unknotted in the sense that any two such surfaces with isomorphic boundary graph or graph at infinity are smoothly isotopic. 

Abstract This work investigates the ambient potential identification problem in inverse mean-field games (MFGs), where the goal is to recover the unknown potential from the value function at equilibrium. We propose a simple yet effective iterative strategy, equilibrium correction iteration (ECI), that leverages the structure of MFGs rather than relying on generic optimization formulations. ECI uncovers hidden information from equilibrium measurements, offering a new perspective on inverse MFGs. To improve computational efficiency, two acceleration variants are introduced: best response iteration (BRI), which uses inexact forward solvers, and hierarchical ECI (HECI), which incorporates multilevel grids. While BRI performs efficiently in general settings, HECI proves particularly effective in recovering low-frequency potentials. We also highlight a connection between the potential identification problem in inverse MFGs and inverse linear parabolic equations, suggesting promising directions for future theoretical analysis. Finally, comprehensive numerical experiments demonstrate how viscosity, terminal time, and interaction costs can influence the well-posedness of the inverse problem. 

Abstract We study the random connection model on hyperbolic space in dimension . Vertices of the spatial random graph are given as a Poisson point process with intensity . Upon variation of , there is a percolation phase transition: there exists a critical value such that for , all clusters are finite, but infinite clusters exist for . We identify certain critical exponents that characterise the clusters at (and near) , and show that they agree with the mean‐field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram. 

Abstract Let be a graph on vertices, independence number , Lovász theta function and Shannon capacity . We define to be the minimum number of non‐negative eigenvalues taken over all Hermitian weighted adjacency matrices of . It is well known that and . Continuing a long line of work, we investigate the relationships between , , and . We prove a conjecture of Kwan and Wigderson, showing that for every integer , there exists a graph with and . In addition, we prove that for every integer , there exists a graph with and . Both results rely on a new observation: if the complement of contains a good spectral expander, then must be large. Moreover, we show that the multiplicative gap between and can be . We also show that can be exponentially larger than , improving a recent result of Ihringer.