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1. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza; Galkowski, Jeffrey (June 2023, Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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2. Sectorial descent for wrapped Fukaya categories

   https://doi.org/10.1090/jams/1035  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (October 2023, Journal of the American Mathematical Society)

   We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful. 

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3. On Newton polytopes of Lagrangian augmentations

   https://doi.org/10.1112/blms.12992  

   Capovilla‐Searle, Orsola; Casals, Roger (February 2024, Bulletin of the London Mathematical Society)

   Abstract This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many nonorientable exact Lagrangian fillings. 

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4. Existentially closed W*-probability spaces

   https://doi.org/10.1007/s00209-022-03038-z  

   Goldbring, Isaac; Houdayer, Cyril (August 2022, Mathematische Zeitschrift)

   Abstract We study several model-theoretic aspects of W $$^*$$ ∗ -probability spaces, that is, $$\sigma $$ σ -finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W $$^*$$ ∗ -spaces and prove several structural results about such spaces, including that they are type III $$_1$$ 1 factors that tensorially absorb the Araki–Woods factor $$R_\infty $$ R ∞ . We also study the existentially closed objects in the restricted class of W $$^*$$ ∗ -probability spaces with Kirchberg’s QWEP property, proving that $$R_\infty $$ R ∞ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III $$_1$$ 1 factors forms a $$\forall _2$$ ∀ 2 -axiomatizable class. We show that for $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , the class of III $$_\lambda $$ λ factors is not $$\forall _2$$ ∀ 2 -axiomatizable but is $$\forall _3$$ ∀ 3 -axiomatizable; this latter result uses a version of Keisler’s Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III $$_\lambda $$ λ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , there is a family of pairwise non-elementarily equivalent III $$_\lambda $$ λ factors of size continuum. While we cannot prove the same result for III $$_1$$ 1 factors, we show that there are at least three pairwise non-elementarily equivalent III $$_1$$ 1 factors by showing that the class of full factors is preserved under elementary equivalence. 

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5. $$\mathrm {K_S}^{0}$$- and (anti-)$$\Lambda $$-hadron correlations in pp collisions at $${\sqrt{s}} = 13$$ TeV

   https://doi.org/10.1140/epjc/s10052-021-09678-5  

   Acharya, S.; Adamová, D.; Adler, A.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S. U.; Ahuja, I.; et al (October 2021, The European Physical Journal C)

   Abstract Two-particle Azimuthal correlations are measured with the ALICE apparatus in pp collisions at $$\sqrt{s} = 13$$ s = 13 TeV to explore strangeness- and multiplicity-related effects in the fragmentation of jets and the transition regime between bulk and hard production, probed with the condition that a strange meson ( $$\mathrm {K_S}^{0}$$ K S 0 ) or baryon ( $$\Lambda $$ Λ ) with transverse momentum $$p_{\mathrm T} >3$$ p T > 3 GeV/ $$c$$ c is produced. Azimuthal correlations between kaons or $$\Lambda $$ Λ hyperons with other hadrons are presented at midrapidity for a broad range of the trigger ( $$3< p_\mathrm {T}^\mathrm {trigg} < 20$$ 3 < p T trigg < 20 GeV/ $$c$$ c ) and associated particle $$p_{\mathrm T}$$ p T (1 GeV/ $$c$$ c $$< p_\mathrm {T}^\mathrm {assoc} < p_\mathrm {T}^\mathrm {trigg} $$ < p T assoc < p T trigg ), for minimum-bias events and as a function of the event multiplicity. The near- and away-side peak yields are compared for the case of either $$\mathrm {K_S}^{0}$$ K S 0 or $$\Lambda $$ Λ ( $${\overline{\Lambda }}$$ Λ ¯ ) being the trigger particle with that of inclusive hadrons (a sample dominated by pions). In addition, the measurements are compared with predictions from PYTHIA 8 and EPOS LHC event generators. 

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