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1. Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

   https://doi.org/10.1007/s12220-021-00808-9  

   Datar, Ved ; Jacob, Adam ( January 2022 , The Journal of Geometric Analysis)

   Let$$X\rightarrow {{\mathbb {P}}}^1$$$X\to P^1$be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$${\omega }_i$of Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$${\omega }_i$. Given the corresponding sequence$$\Xi _i$$${\Xi }_i$of Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$${\Xi }_i{|}_E$converges to a flat connection$$A_0$$$A_0$. Furthermore, if the restriction$$V|_E$$${V|}_E$is of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$${\oplus }_{j=1}^n{O}_E(q_j-0)$forndistinct points$$q_j\in E$$$q_j\in E$, then these points uniquely determine$$A_0$$$A_0$.

    

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2. Study of baryon number transport dynamics and strangeness conservation effects using Ω-hadron correlations

   https://doi.org/10.1007/s41365-024-01464-8  

   Dong, Wei-Jie ; Yu, Xiao-Zhou ; Ping, Si-Yuan ; Wu, Xia-Tong ; Wang, Gang ; Huang, Huan-Zhong ; Lin, Zi-Wei ( July 2024 , Nuclear Science and Techniques)

   In nuclear collisions at RHIC energies, an excess of$$\Omega$$$\Omega$hyperons over$$\bar{\Omega }$$$\overline{\Omega }$is observed, indicating that$$\Omega$$$\Omega$has a net baryon number despitesand$$\bar{s}$$$\overline{s}$quarks being produced in pairs. The baryon number in$$\Omega$$$\Omega$may have been transported from the incident nuclei and/or produced in the baryon-pair production of$$\Omega$$$\Omega$with other types of anti-hyperons such as$$\bar{\Xi }$$$\overline{\Xi }$. To investigate these two scenarios, we propose to measure the correlations between$$\Omega$$$\Omega$andKand between$$\Omega$$$\Omega$and anti-hyperons. We use two versions, the default and string-melting, of a multiphase transport (AMPT) model to illustrate the method for measuring the correlation and to demonstrate the general shape of the correlation. We present the$$\Omega$$$\Omega$-hadron correlations from simulated Au+Au collisions at$$\sqrt{s_\text{NN}} = 7.7$$$\sqrt{s_{\text{NN}}}=7.7$and$$14.6 \ \textrm{GeV}$$$14.6\text{GeV}$and discuss the dependence on the collision energy and on the hadronization scheme in these two AMPT versions. These correlations can be used to explore the mechanism of baryon number transport and the effects of baryon number and strangeness conservation on nuclear collisions.

    

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3. Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

   https://doi.org/10.1007/s00493-023-00015-w  

   Scott, Alex ; Seymour, Paul ; Spirkl, Sophie ( October 2023 , Combinatorica)

   A graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$$P_5$-free graph with clique number$$\omega \ge 3$$$\omega \ge 3$has chromatic number at most$$\omega ^{\log _2(\omega )}$$${\omega }^{{log}_2\left(\omega \right)}$. The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$$(5/27)3^{\omega }$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$$P_5$, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$$P_5$-free graphs, and our result is an attempt to approach that.

    

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4. Construction of a Dirichlet form on Metric Measure Spaces of Controlled Geometry

   https://doi.org/10.1007/s11118-024-10144-6  

   Butaev, Almaz ; Luo, Liangbing ; Shanmugalingam, Nageswari ( June 2024 , Potential Analysis)

   Given a compact doubling metric measure spaceXthat supports a 2-Poincaré inequality, we construct a Dirichlet form on$$N^{1,2}(X)$$$N^{1,2}\left(X\right)$that is comparable to the upper gradient energy form on$$N^{1,2}(X)$$$N^{1,2}\left(X\right)$. Our approach is based on the approximation ofXby a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on$$N^{1,2}(X)$$$N^{1,2}\left(X\right)$using the Dirichlet form on the graph. We show that the$$\Gamma $$$\Gamma$-limit$$\mathcal {E}$$$E$of this family of bilinear forms (by taking a subsequence) exists and that$$\mathcal {E}$$$E$is a Dirichlet form onX. Properties of$$\mathcal {E}$$$E$are established. Moreover, we prove that$$\mathcal {E}$$$E$has the property of matching boundary values on a domain$$\Omega \subseteq X$$$\Omega \subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form$$\mathcal {E}$$$E$) on a domain inXwith a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.

    

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5. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana ( March 2022 , Mathematische Annalen)

   It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$$L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$$\Gamma$of dimension$$d < n-1$$$d<n-1$, the now usual distance to the boundary$$D = D_\beta $$$D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$$D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$$L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$$D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    

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