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1. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan; Smirnov, Ilya (February 2023, Proceedings of the American Mathematical Society)

   Let $(R,\mathfrak{m})$be a Noetherian local ring of dimension $d\ge <\#comment/>2$. We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$, then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$for all $\mathfrak{m}$-primary ideals $I\subseteq <\#comment/>R$. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of $I$. 

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2. Hereditary uniform property Γ

   https://doi.org/10.1007/s11425-022-2005-x  

   Lin, Huaxin (February 2023, Science China Mathematics)
3. Counting non-uniform lattices

   https://doi.org/10.1007/s11856-019-1868-4  

   Belolipetsky, Mikhail; Lubotzky, Alexander (August 2019, Israel Journal of Mathematics)
4. Small-Constant Uniform Rectifiability

   https://doi.org/10.1007/s12220-024-01567-z  

   Jeznach, Cole (May 2024, The Journal of Geometric Analysis)
5. Quotients of Uniform Positroids

   https://doi.org/10.37236/10056  

   Benedetti, Carolina; Chavez, Anastasia; Tamayo Jiménez, Daniel (January 2022, The Electronic Journal of Combinatorics)

   Two matroids $$M$$ and $$N$$ are said to be concordant if there is a strong map from $$N$$ to $$M$$. This also can be stated by saying that each circuit of $$N$$ is a union of circuits of $$M$$. In this paper, we consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorics to determine concordance among some of them. More precisely, given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that is concordant with it. We do this by means of their associated decorated permutations. As a byproduct of our work, we describe completely the collection of circuits of this particular subset of positroids. 

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