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1. Measurement of the mass dependence of the transverse momentum of lepton pairs in Drell–Yan production in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te\hspace{-.08em}V} $$

   https://doi.org/10.1140/epjc/s10052-023-11631-7  

   Tumasyan, A.; Adam, W.; Andrejkovic, J. W.; Bergauer, T.; Chatterjee, S.; Dragicevic, M.; Valle, A. Escalante; Frühwirth, R.; Jeitler, M.; Krammer, N.; et al (July 2023, The European Physical Journal C)

   Abstract The double differential cross sections of the Drell–Yan lepton pair ($$\ell ^+\ell ^-$$ ${\ell }^{+}{\ell }^{-}$ , dielectron or dimuon) production are measured as functions of the invariant mass$$m_{\ell \ell }$$ $m_{\ell \ell }$ , transverse momentum$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$ , and$$\varphi ^{*}_{\eta }$$ ${\phi }_{\eta }^{\ast }$ . The$$\varphi ^{*}_{\eta }$$ ${\phi }_{\eta }^{\ast }$ observable, derived from angular measurements of the leptons and highly correlated with$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$ , is used to probe the low-$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$ region in a complementary way. Dilepton masses up to 1$$\,\text {Te\hspace{-.08em}V}$$ $\text{Te}\text{V}$ are investigated. Additionally, a measurement is performed requiring at least one jet in the final state. To benefit from partial cancellation of the systematic uncertainty, the ratios of the differential cross sections for various$$m_{\ell \ell }$$ $m_{\ell \ell }$ ranges to those in the Z mass peak interval are presented. The collected data correspond to an integrated luminosity of 36.3$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$ of proton–proton collisions recorded with the CMS detector at the LHC at a centre-of-mass energy of 13$$\,\text {Te\hspace{-.08em}V}$$ $\text{Te}\text{V}$ . Measurements are compared with predictions based on perturbative quantum chromodynamics, including soft-gluon resummation. 

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2. Transversality of holomorphic maps into hyperquadrics

   https://doi.org/10.1007/s00208-025-03134-5  

   Huang, Xiaojun; Zhu, Weixia (June 2025, Mathematische Annalen)

   Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ $M_{\ell }\subset C^n$ into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ $H_{{\ell }^{\prime }}^N$ with signatures$$ \ell \le (n-1)/2 $$ $\ell \le (n-1)/2$ and$$ \ell '\le (N-1)/2,$$ ${\ell }^{\prime }\le (N-1)/2,$ respectively. Assuming that$$ N - n < n - 1,$$ $N-n<n-1,$ we prove that if$$ \ell = \ell ',$$ $\ell ={\ell }^{\prime },$ thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ $H_{\ell }^N$ at every point of$$ M_{\ell },$$ $M_{\ell },$ or it maps a neighborhood of$$ M_{\ell } $$ $M_{\ell }$ in$$ {\mathbb {C}}^n $$ $C^n$ into$$ {\mathbb {H}}_{\ell }^N.$$ $H_{\ell }^N.$ Furthermore, in the case where$$ \ell ' > \ell ,$$ ${\ell }^{\prime }>\ell ,$ we show that ifFis not CR transversal at$$0\in M_\ell ,$$ $0\in M_{\ell },$ then it must be transversally flat. The latter is best possible. 

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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. Scarcity of partition congruences on semiprime progressions

   https://doi.org/10.1007/s11139-025-01245-w  

   Ahlgren, Scott; Beckwith, Olivia (November 2025, The Ramanujan Journal)

   Abstract In recent work with Raum, the authors considered congruences for the ordinary partition functionp(n) of the form$$p(\ell Q^r n+\beta )\equiv 0\pmod \ell $$ $p(\ell Q^{r}n+\beta )\equiv 0(mod\ell )$ where$$\ell , Q\ge 5$$ $\ell ,Q\ge 5$ are prime and$$r\in \{1,2\}$$ $r\in \{1,2\}$ , and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when$$r=1$$ $r=1$ ; in particular, we prove (outside of trivial cases) that the set of primesQsuch that there exists$$\beta \in \mathbb {Z}$$ $\beta \in Z$ with$$p(\ell Q n+\beta )\equiv 0\pmod \ell $$ $p(\ell Qn+\beta )\equiv 0(mod\ell )$ for allnhas density zero. The proof involves a modification of part of our previous argument and an application of a recent theorem of Dicks regarding modular forms of half-integral weight and level one modulo$$\ell $$ $\ell$ . 

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5. The shifted convolution problem in function fields

   https://doi.org/10.1007/s00208-026-03340-9  

   Florea, Alexandra; Lalín, Matilde; Malik, Amita; Sahay, Anurag (May 2026, Mathematische Annalen)

   Abstract We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of$$d(f) d(f+h)$$ $d\left(f\right)d(f+h)$ wherefruns over monic polynomials in$$\mathbb {F}_q[T]$$ $F_q\left[T\right]$ of a given degree, andhis a given monic polynomial. We prove an asymptotic formula in the range$$\deg (h) < (2-\epsilon )\deg (f)$$ $deg\left(h\right)<(2-ϵ)deg\left(f\right)$ . We also consider mixed correlations and self-correlations of$$r_\chi = 1 \star \chi $$ $r_{\chi }=1\star \chi$ , the convolution of 1 with a Dirichlet character mod$$\ell $$ $\ell$ , where$$\ell $$ $\ell$ is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of$$\mathbb {F}_q[T]$$ $F_q\left[T\right]$ . A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in$$\mathbb {F}_q[T]$$ $F_q\left[T\right]$ which was not previously available. 

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