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1. Theta functions, fourth moments of eigenforms and the sup-norm problem II

   https://doi.org/10.1017/fmp.2024.9  

   Khayutin, Ilya; Nelson, Paul D; Steiner, Raphael S (January 2024, Forum of Mathematics, Pi)

   Abstract Letfbe an$$L^2$$-normalized holomorphic newform of weightkon$$\Gamma _0(N) \backslash \mathbb {H}$$withNsquarefree or, more generally, on any hyperbolic surface$$\Gamma \backslash \mathbb {H}$$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$$\mathbb {Q}$$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform$$\varphi $$of eigenvalue$$\lambda $$on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras. 

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2. 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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3. 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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4. Lower bounds on Bourgain’s constant for harmonic measure

   https://doi.org/10.4153/S0008414X2300069X  

   Badger, Matthew; Genschaw, Alyssa (December 2024, Canadian Journal of Mathematics)

   Abstract For every$$n\geq 2$$, Bourgain’s constant$$b_n$$is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most$$n-b_n$$for every domain in$$\mathbb {R}^n$$on which harmonic measure is defined. Jones and Wolff (1988,Acta Mathematica161, 131–144) proved that$$b_2=1$$. When$$n\geq 3$$, Bourgain (1987,Inventiones Mathematicae87, 477–483) proved that$$b_n>0$$and Wolff (1995,Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing$$b_n<1$$. Refining Bourgain’s original outline, we prove that$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$for all$$n\geq 3$$, where$$c>0$$is a constant that is independent ofn. We further estimate$$b_3\geq 1\times 10^{-15}$$and$$b_4\geq 2\times 10^{-26}$$. 

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5. 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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