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1. Global inviscid limit of 2D, stationary Navier-Stokes and stability of Prandtl expansions

   https://doi.org/10.1017/fmp.2026.10027  

   Iyer, Sameer; Masmoudi, Nader (April 2026, Forum of Mathematics, Pi)

   Abstract In this work, we establish the convergence of 2D, stationary Navier-Stokes flows with viscosity$$\varepsilon> 0$$,$$(u^\varepsilon , v^\varepsilon )$$to the classical Prandtl boundary layer,$$(\bar {u}_p, \bar {v}_p)$$, posed on the domain$$(0, \infty ) \times (0, \infty )$$:$$ \begin{align*} \| u^\varepsilon - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 4 + \delta}, \qquad \| v^\varepsilon - \sqrt{\varepsilon} \bar{v}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 2}. \end{align*} $$ This validates Prandtl’s boundary layer theorygloballyin thex-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as$$\varepsilon \rightarrow 0$$and (2) asymptotic as$$x \rightarrow \infty $$. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot “separate” in these stable regimes, which is very important for physical and engineering applications. 

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2. An explicit economical additive basis

   https://doi.org/10.1017/S096354832510014X  

   Jain, Vishesh; Pham, Huy Tuan; Sawhney, Mehtaab; Zakharov, Dmitrii (November 2025, Combinatorics, Probability and Computing)

   Abstract We present an explicit subset$$A\subseteq \mathbb{N} = \{0,1,\ldots \}$$such that$$A + A = \mathbb{N}$$and for all$$\varepsilon \gt 0$$,\begin{equation*}\lim _{N\to \infty }\frac {\big |\big \{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big \}\big |}{N^{\varepsilon }} = 0.\end{equation*} This answers a question of Erdős. 

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3. DERIVATIVES OF THETA FUNCTIONS AS TRACES OF PARTITION EISENSTEIN SERIES

   https://doi.org/10.1017/nmj.2024.30  

   AMDEBERHAN, TEWODROS; ONO, KEN; SINGH, AJIT (June 2025, Nagoya Mathematical Journal)

   Abstract In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences ofq-series$$\{U_{2t}(q)\}$$and$$\{V_{2t}(q)\}$$that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series$$E_{2k}(q),$$defined by$$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$ For functions$$\phi : \mathcal {P}\mapsto {\mathbb C}$$on partitions, theweight$$2n$$partition Eisenstein traceis$$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$ For allt, we prove that$$U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$$and$$V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$$where$$\phi _U$$and$$\phi _V$$are natural partition weights, giving the first explicit quasimodular formulas for these series. 

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4. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (January 2023, Forum of Mathematics, Pi)

   Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 

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5. Interaction energies in nematic liquid crystal suspensions

   https://doi.org/10.1017/fms.2025.10147  

   Bronsard, Lia; Lamy, Xavier; Stantejsky, Dominik; Venkatraman, Raghavendra (January 2026, Forum of Mathematics, Sigma)

   Abstract We establish, as$$\rho \to 0$$, an asymptotic expansion for the minimal Dirichlet energy of$$\mathbb S^2$$-valued maps outside a finite number of particles of size$$\rho $$with fixed centers$$x_j\in {\mathbb R}^3$$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form$$ \begin{align*} E_\rho = \sum_j \mu_j -4\pi\rho \sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(\rho)\,, \end{align*} $$ where$$\mu _j$$is the minimal energy after zooming in at scale$$\rho $$around each particle, and$$v_j\in {\mathbb R}^3$$is determined by the far-field behavior of the corresponding single-particle minimizer. The Coulomb-like interaction in this expansion agrees with theelectrostatic analogy: a linearized approximation commonly used in the physics literature for colloid interactions in nematic liquid crystal. We obtain here for the first time a precise estimate of the energy error introduced by that linearization, by developing new tools that address the lack of convergence rate when zooming in at scale$$\rho $$. 

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