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1. Affine isoperimetric inequalities for higher-order projection and centroid bodies

   https://doi.org/10.1007/s00208-025-03271-x  

   Haddad, Julián; Langharst, Dylan; Putterman, Eli; Roysdon, Michael; Ye, Deping (August 2025, Mathematische Annalen)

   Abstract In 1970, Schneider introduced the$$m$$ $m$th order difference body of a convex body, and also established the$$m$$ $m$th-order Rogers–Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang’s projection inequality, Petty’s projection inequality, the Busemann–Petty centroid inequality and Busemann’s random simplex inequality). We also establish a new proof of Schneider’s$$m$$ $m$th-order Rogers–Shephard inequality. As an application, a$$m$$ $m$th-order affine Sobolev inequality for functions of bounded variation is provided. 

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2. Small-data global existence of solutions for the Pitaevskii model of superfluidity

   https://doi.org/10.1088/1361-6544/ad3cae  

   Jang, Juhi; Chaitanya Jayanti, Pranava; Kukavica, Igor (April 2024, Nonlinearity)

   Abstract We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959Sov. Phys. JETP8282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in ${\mathbb{T}}^2$—strong in wavefunction and velocity, and weak in density. 

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3. On the Maximum F‐Free Induced Subgraphs in Kt‐Free Graphs

   https://doi.org/10.1002/rsa.21273  

   Balogh, József; Chen, Ce; Luo, Haoran (January 2025, Random Structures & Algorithms)

   ABSTRACT For graphs and , let be the minimum possible size of a maximum ‐free induced subgraph in an ‐vertex ‐free graph. This notion generalizes the Ramsey function and the Erdős–Rogers function. Establishing a container lemma for the ‐free subgraphs, we give a general upper bound on , assuming the existence of certain locally dense ‐free graphs. In particular, we prove that for every graph with , where , we have For the cases where is a complete multipartite graph, letting , we prove that We also make an observation which improves the bounds of by a polylogarithmic factor. 

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4. On the order of the classical Erdős–Rogers functions

   https://doi.org/10.1112/blms.13214  

   Mubayi, Dhruv; Verstraete, Jacques (December 2024, Bulletin of the London Mathematical Society)

   Abstract For an integer , the Erdős–Rogers function is the maximum integer such that every ‐vertex ‐free graph has a ‐free induced subgraph with vertices. It is known that for all , as . In this paper, we show that for all , there exists a constant such thatThis improves previous bounds of order by Dudek, Retter and Rödl and answers a question of Warnke. 

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5. Inhomogeneous Diophantine approximation for generic homogeneous functions

   https://doi.org/10.1142/S1793042123500628  

   Kleinbock, Dmitry; Skenderi, Mishel (July 2023, International Journal of Number Theory)

   This paper is a sequel to [Monatsh. Math. 194 (2021) 523–554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers [Formula: see text] and [Formula: see text], any [Formula: see text], and any homogeneous function [Formula: see text] that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function [Formula: see text] for a generic element [Formula: see text] in the [Formula: see text]-orbit of [Formula: see text] to be (respectively, not to be) [Formula: see text]-approximable at [Formula: see text]: that is, for there to exist infinitely many (respectively, only finitely many) [Formula: see text] such that [Formula: see text] for each [Formula: see text]. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of [Formula: see text] that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace [Formula: see text] above by any closed subgroup of [Formula: see text] that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper. 

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