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1. Counterexamples to maximal regularity for operators in divergence form

   https://doi.org/10.1007/s00013-024-02014-9  

   Bechtel, Sebastian; Mooney, Connor; Veraar, Mark (August 2024, Archiv der Mathematik)

   Abstract In this paper, we present counterexamples to maximal$$L^p$$ $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal$$L^2$$ $L^2$-regularity on$$H^{-1}$$ $H^{-1}$under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal$$L^p$$ $L^p$-regularity on$$H^{-1}(\mathbb {R}^d)$$ $H^{-1}\left(R^d\right)$or$$L^2$$ $L^2$-regularity on$$L^2(\mathbb {R}^d)$$ $L^2\left(R^d\right)$. 

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2. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ $ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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3. Curvature and sharp growth rates of log-quasimodes on compact manifolds

   https://doi.org/10.1007/s00222-025-01315-2  

   Huang, Xiaoqi; Sogge, Christopher D (March 2025, Inventiones mathematicae)

   Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

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4. 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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5. Spin density matrix elements in exclusive $$\rho ^0$$ meson muoproduction

   https://doi.org/10.1140/epjc/s10052-023-11359-4  

   Alexeev, G D; Alexeev, M G; Alice, C; Amoroso, A; Andrieux, V; Anosov, V; Augsten, K; Augustyniak, W; Azevedo, C_D R; Badelek, B; et al (October 2023, The European Physical Journal C)

   Abstract We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$ ${\rho }^0$meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$ ${\mu }^{+}$and$$ \mu ^{-}$$ ${\mu }^{-}$beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$ $c^2$$$< W<$$ $<W<$17.0 GeV/$$c^2$$ $c^2$, 1.0 (GeV/c)$$^2$$ ${}^2$$$< Q^2<$$ $<Q^2<$10.0 (GeV/c)$$^2$$ ${}^2$and 0.01 (GeV/c)$$^2$$ ${}^2$$$< p_{\textrm{T}}^2<$$ $<p_{\text{T}}^2<$0.5 (GeV/c)$$^2$$ ${}^2$. Here,Wdenotes the mass of the final hadronic system,$$Q^2$$ $Q^2$the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$ $p_{\text{T}}$the transverse momentum of the$$\rho ^0$$ ${\rho }^0$meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$ ${\gamma }_T^{\ast }\to V_L^{}$) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$ ${\rho }^0$production. 

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