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1. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$is an$$\ell $$ $\ell$-tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$is an$$\ell $$ $\ell$-tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$along$$\Sigma _j$$ ${\Sigma }_j$,$$1\le j\le \ell $$ $1\le j\le \ell$. If$$Y\subset X$$ $Y\subset X$is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$. Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$, we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$. WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$as$$p\rightarrow \infty $$ $p\to \infty$. 

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2. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (June 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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3. 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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4. Carleson’s $$\varepsilon ^{2}$$ conjecture in higher dimensions

   https://doi.org/10.1007/s00222-025-01337-w  

   Fleschler, Ian; Tolsa, Xavier; Villa, Michele (July 2025, Inventiones mathematicae)

   Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

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5. Planar chemical reaction systems with algebraic and non-algebraic limit cycles

   https://doi.org/10.1007/s00285-025-02221-0  

   Craciun, Gheorghe; Erban, Radek (June 2025, Journal of Mathematical Biology)

   Abstract The Hilbert numberH(n) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most$$n \in {{\mathbb {N}}}$$ $n\in N$. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, wherenis equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number H(n) for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to then-th order; (ii) systems with up ton-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve$$h(x,y)=0$$ $h(x,y)=0$of degree$$n_h \in {{\mathbb {N}}}$$ $n_h\in N$and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most$$n=2\,n_h+1.$$ $n=2n_h+1.$Considering$$n_h \ge 4,$$ $n_h\ge 4,$the algebraic curve$$h(x,y)=0$$ $h(x,y)=0$can contain multiple closed components with the maximum number of ovals given by Harnack’s curve theorem as$$1+(n_h-1)(n_h-2)/2$$ $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for$$n_h=4.$$ $n_h=4.$Algebraic curve$$h(x,y)=0$$ $h(x,y)=0$with$$n_h=4$$ $n_h=4$and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles. 

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