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1. The unbounded denominators conjecture

   https://doi.org/10.1090/jams/1053  

   Calegari, Frank; Dimitrov, Vesselin; Tang, Yunqing (February 2025, Journal of the American Mathematical Society)

   We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right)$. Our result includes also Mason’s generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna second main theorem, the congruence subgroup property of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right[1/p\left]\right)$, and a close description of the Fuchsian uniformization $D(0,1)/{\mathrm{\Gamma <\#comment/>}}_N$of the Riemann surface $\mathbf{C}\setminus <\#comment/>{\mathrm{\mu <\#comment/>}}_N$. 

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2. Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

   https://doi.org/10.1090/memo/1377  

   Bedrossian, Jacob; Germain, Pierre; Masmoudi, Nader (September 2022, Memoirs of the American Mathematical Society)

   This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re . In this work, we show that there is constant 0 > c 0 ≪ 1 0 > c_0 \ll 1 , independent of R e \mathbf {Re} , such that sufficiently regular disturbances of size ϵ ≲ R e − 2 / 3 − δ \epsilon \lesssim \mathbf {Re}^{-2/3-\delta } for any δ > 0 \delta > 0 exist at least until t = c 0 ϵ − 1 t = c_0\epsilon ^{-1} and in general evolve to be O ( c 0 ) O(c_0) due to the lift-up effect. Further, after times t ≳ R e 1 / 3 t \gtrsim \mathbf {Re}^{1/3} , the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ϵ − 1 t \approx \epsilon ^{-1} . Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ \Rightarrow streak growth ⇒ \Rightarrow streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature. 

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3. COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION

   https://doi.org/10.1017/jsl.2022.78  

   CHAN, WILLIAM; JACKSON, STEPHEN; TRANG, NAM (November 2022, The Journal of Symbolic Logic)

   Abstract Assume $$\mathsf {ZF} + \mathsf {AD}$$ and all sets of reals are Suslin. Let $$\Gamma $$ be a pointclass closed under $$\wedge $$ , $$\vee $$ , $$\forall ^{\mathbb {R}}$$ , continuous substitution, and has the scale property. Let $$\kappa = \delta (\Gamma )$$ be the supremum of the length of prewellorderings on $$\mathbb {R}$$ which belong to $$\Delta = \Gamma \cap \check \Gamma $$ . Let $$\mathsf {club}$$ denote the collection of club subsets of $$\kappa $$ . Then the countable length everywhere club uniformization holds for $$\kappa $$ : For every relation $$R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$$ with the property that for all $$\ell \in {}^{<{\omega _1}}\kappa $$ and clubs $$C \subseteq D \subseteq \kappa $$ , $$R(\ell ,D)$$ implies $$R(\ell ,C)$$ , there is a uniformization function $$\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$$ with the property that for all $$\ell \in \mathrm {dom}(R)$$ , $$R(\ell ,\Lambda (\ell ))$$ . In particular, under these assumptions, for all $$n \in \omega $$ , $$\boldsymbol {\delta }^1_{2n + 1}$$ satisfies the countable length everywhere club uniformization. 

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4. Measurement of the CP-violating phase $$\phi _s$$ in $$ B_{s}^{0} \rightarrow J/\psi \phi $$ decays in ATLAS at 13 TeV

   https://doi.org/10.1140/epjc/s10052-021-09011-0  

   Aad, G.; Abbott, B.; Abbott, D. C.; Abdinov, O.; Abud, A. Abed; Abeling, K.; Abhayasinghe, D. K.; Abidi, S. H.; AbouZeid, O. S.; Abraham, N. L.; et al (April 2021, The European Physical Journal C)

   null (Ed.)

   Abstract A measurement of the $$ B_{s}^{0} \rightarrow J/\psi \phi $$ B s 0 → J / ψ ϕ decay parameters using $$ 80.5\, \mathrm {fb^{-1}} $$ 80.5 fb - 1 of integrated luminosity collected with the ATLAS detector from 13  $$\text {Te}\text {V}$$ Te proton–proton collisions at the LHC is presented. The measured parameters include the CP -violating phase $$\phi _{s} $$ ϕ s , the width difference $$ \Delta \Gamma _{s}$$ Δ Γ s between the $$B_{s}^{0}$$ B s 0 meson mass eigenstates and the average decay width $$ \Gamma _{s}$$ Γ s . The values measured for the physical parameters are combined with those from $$ 19.2\, \mathrm {fb^{-1}} $$ 19.2 fb - 1 of 7 and 8  $$\text {Te}\text {V}$$ Te data, leading to the following: $$\begin{aligned} \phi _{s}= & {} -0.087 \pm 0.036 ~\mathrm {(stat.)} \pm 0.021 ~\mathrm {(syst.)~rad} \\ \Delta \Gamma _{s}= & {} 0.0657 \pm 0.0043 ~\mathrm {(stat.)}\pm 0.0037 ~\mathrm {(syst.)~ps}^{-1} \\ \Gamma _{s}= & {} 0.6703 \pm 0.0014 ~\mathrm {(stat.)}\pm 0.0018 ~\mathrm {(syst.)~ps}^{-1} \end{aligned}$$ ϕ s = - 0.087 ± 0.036 ( stat . ) ± 0.021 ( syst . ) rad Δ Γ s = 0.0657 ± 0.0043 ( stat . ) ± 0.0037 ( syst . ) ps - 1 Γ s = 0.6703 ± 0.0014 ( stat . ) ± 0.0018 ( syst . ) ps - 1 Results for $$\phi _{s} $$ ϕ s and $$ \Delta \Gamma _{s}$$ Δ Γ s are also presented as 68% confidence level contours in the $$\phi _{s} $$ ϕ s – $$ \Delta \Gamma _{s}$$ Δ Γ s plane. Furthermore the transversity amplitudes and corresponding strong phases are measured. $$\phi _{s} $$ ϕ s and $$ \Delta \Gamma _{s}$$ Δ Γ s measurements are in agreement with the Standard Model predictions. 

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5. MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

   https://doi.org/10.1017/S1474748020000225  

   Le, Daniel; Morra, Stefano; Schraen, Benjamin (May 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$F$$ be a totally real field in which $$p$$ is unramified. Let $$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $$v$$ above $$p$$ . Let $$\mathfrak{m}$$ be the corresponding Hecke eigensystem. We describe the $$\mathfrak{m}$$ -torsion in the $$\text{mod}\,p$$ cohomology of Shimura curves with full congruence level at $$v$$ as a $$\text{GL}_{2}(k_{v})$$ -representation. In particular, it only depends on $$\overline{r}|_{I_{F_{v}}}$$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $$\text{GL}_{2}(\mathbf{F}_{q})$$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.   200 (1) (2015), 1–96]. 

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