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1. Nonnegative Ricci curvature, nilpotency, and Hausdorff dimension

   https://doi.org/10.1007/s00208-026-03378-9  

   Pan, Jiayin (January 2026, Mathematische Annalen)

   Abstract LetMbe an open (complete and non-compact) manifold with$$\textrm{Ric}\ge 0$$ $\text{Ric}\ge 0$ and escape rate not 1/2. It is known that under these conditions, the fundamental group$$\pi _1(M)$$ ${\pi }_1\left(M\right)$ has a finitely generated torsion-free nilpotent subgroup$$\mathcal {N}$$ $N$ of finite index, as long as$$\pi _1(M)$$ ${\pi }_1\left(M\right)$ is an infinite group. We show that the nilpotency step of$$\mathcal {N}$$ $N$ must be reflected in the asymptotic geometry of the universal cover$$\widetilde{M}$$ $\tilde{M}$ , in terms of the Hausdorff dimension of an isometric$$\mathbb {R}$$ $R$ -orbit: there exist an asymptotic cone (Y, y) of$$\widetilde{M}$$ $\tilde{M}$ and a closed$$\mathbb {R}$$ $R$ -subgroupLof the isometry group ofYsuch that its orbitLyhas Hausdorff dimension at least the nilpotency step of$$\mathcal {N}$$ $N$ . This resolves a question raised by Wei and the author (see Pan and Wei in Geom Funct Anal 32:676–685, 2022, Remark 1.7 and Pan in Geom Topol 28:1409–1436, 2024, Conjecture 0.2). 

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2. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven J; Romero_Acosta, Francisco; et al (April 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$ be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$ and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$ , let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$ , where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$ is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$ . Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$ that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$ andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$ is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$ isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$ with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$ . 

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3. Sunflowers in Set Systems with Small VC-Dimension

   https://doi.org/10.1007/s00493-025-00186-8  

   Balogh, József; Bernshteyn, Anton; Delcourt, Michelle; Ferber, Asaf; Pham, Huy Tuan (December 2025, Combinatorica)

   Abstract A family ofrdistinct sets$$\{A_1,\ldots , A_r\}$$ $\{A_1,\dots ,A_r\}$ is anr-sunflower if for all$$1 \leqslant i < j\leqslant r$$ $1⩽i<j⩽r$ and$$1 \leqslant i' < j'\leqslant r$$ $1⩽i^{\prime }<j^{\prime }⩽r$ , we have$$A_i\cap A_j = A_{i'}\cap A_{j'}$$ $A_i\cap A_j=A_{i^{\prime }}\cap A_{j^{\prime }}$ . Erdős and Rado conjectured in 1960 that every family$$\mathcal {H}$$ $H$ of$$\ell $$ $\ell$ -element sets of size at least$$K(r)^\ell $$ $K{\left(r\right)}^{\ell }$ contains anr-sunflower, whereK(r) is some function that depends only onr. We prove that if$$\mathcal {H}$$ $H$ is a family of$$\ell $$ $\ell$ -element sets of VC-dimension at mostdand$$|\mathcal H| > (C r(\log d+\log ^*\ell ))^\ell $$ $\left|H\right|>{\left(Cr(logd+{log}^{\ast }\ell )\right)}^{\ell }$ for some absolute constant$$C > 0$$ $C>0$ , then$$\mathcal {H}$$ $H$ contains anr-sunflower. This improves a recent result of Fox, Pach, and Suk. When$$d=1$$ $d=1$ , we obtain a sharp bound, namely that$$|\mathcal H| > (r-1)^\ell $$ ${\left|H\right|>(r-1)}^{\ell }$ is sufficient. Along the way, we establish a strengthening of the Kahn–Kalai conjecture for set families of bounded VC-dimension, which is of independent interest. 

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4. Weak Brill–Noether on abelian surfaces

   https://doi.org/10.1007/s00029-025-01044-7  

   Coskun, Izzet; Nuer, Howard; Yoshioka, Kōta (July 2025, Selecta Mathematica)

   Abstract We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill–Noether if the general sheaf has at most one non-zero cohomology group. Let (X, H) be a polarized abelian surface and let$${\textbf{v}}=(r, \xi , a)$$ $v=(r,\xi ,a)$ be a Mukai vector onXwith$${\textbf{v}}^2 \geqslant 0$$ $v^2⩾0$ ,$$r>0$$ $r>0$ and$$\xi \cdot H>0$$ $\xi ·H>0$ . We show that if$$\rho (X)=1$$ $\rho \left(X\right)=1$ or$$\rho (X)=2$$ $\rho \left(X\right)=2$ andXcontains an elliptic curve, then all the moduli spaces$$M_{X,H}({\textbf{v}})$$ $M_{X,H}\left(v\right)$ satisfy weak Brill–Noether. Conversely, if$$\rho (X)>2$$ $\rho \left(X\right)>2$ or$$\rho (X)=2$$ $\rho \left(X\right)=2$ andXdoes not contain an elliptic curve, we show that there are infinitely many moduli spaces$$M_{X,H}({\textbf{v}})$$ $M_{X,H}\left(v\right)$ that fail weak Brill–Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces. 

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5. Generalized Parking Function Polytopes

   https://doi.org/10.1007/s00026-023-00671-1  

   Hanada, Mitsuki; Lentfer, John; Vindas-Meléndez, Andrés R (June 2024, Annals of Combinatorics)

   Abstract A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$ $(a_1,a_2,\dots ,a_n)$ whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$ $b_1\le b_2\le \cdots \le b_n$ satisfies$$b_i \le i$$ $b_i\le i$ . The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$ $R^n$ , which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$ $x$ -parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$ $x=(a,b,\cdots ,b)$ , which we refer to as$${\textbf{x}}$$ $x$ -parking function polytopes. We explore connections between these$${\textbf{x}}$$ $x$ -parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$ $x$ -parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary. 

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