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1. Spectral Stability of the $${\overline{\partial }}-$$Neumann Laplacian: Domain Perturbations

   https://doi.org/10.1007/s12220-021-00769-z  

   Fu, Siqi; Zhu, Weixia (January 2022, The Journal of Geometric Analysis)

   Abstract We study spectral stability of the$${\bar{\partial }}$$ $\overline{\partial }$-Neumann Laplacian on a bounded domain in$${\mathbb {C}}^n$$ $C^n$when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the$${\bar{\partial }}$$ $\overline{\partial }$-Neumann Laplacian on bounded pseudoconvex domains in$${\mathbb {C}}^n$$ $C^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in$${\mathbb {C}}^n$$ $C^n$. 

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2. Transversality of holomorphic maps into hyperquadrics

   https://doi.org/10.1007/s00208-025-03134-5  

   Huang, Xiaojun; Zhu, Weixia (March 2025, Mathematische Annalen)

   Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ $M_{\ell }\subset C^n$into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ $H_{{\ell }^{\prime }}^N$with signatures$$ \ell \le (n-1)/2 $$ $\ell \le (n-1)/2$and$$ \ell '\le (N-1)/2,$$ ${\ell }^{\prime }\le (N-1)/2,$respectively. Assuming that$$ N - n < n - 1,$$ $N-n<n-1,$we prove that if$$ \ell = \ell ',$$ $\ell ={\ell }^{\prime },$thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ $H_{\ell }^N$at every point of$$ M_{\ell },$$ $M_{\ell },$or it maps a neighborhood of$$ M_{\ell } $$ $M_{\ell }$in$$ {\mathbb {C}}^n $$ $C^n$into$$ {\mathbb {H}}_{\ell }^N.$$ $H_{\ell }^N.$Furthermore, in the case where$$ \ell ' > \ell ,$$ ${\ell }^{\prime }>\ell ,$we show that ifFis not CR transversal at$$0\in M_\ell ,$$ $0\in M_{\ell },$then it must be transversally flat. The latter is best possible. 

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3. An algorithm and computation to verify Legendre’s conjecture up to $$7\cdot 10^{13}$$

   https://doi.org/10.1007/s40993-024-00589-4  

   Sorenson, Jonathan; Webster, Jonathan (March 2025, Research in Number Theory)

   Abstract We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integern, there exists a prime between$$n^2$$ $n^2$and$$(n+1)^2$$ ${(n+1)}^2$. Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between$$n^2$$ $n^2$and$$n(n+1)$$ $n(n+1)$and also between$$n(n+1)$$ $n(n+1)$and$$(n+1)^2$$ ${(n+1)}^2$. Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all$$n\le N$$ $n\le N$in time$$O( N \log N \log \log N)$$ $O(NlogNloglogN)$and space$$N^{O(1/\log \log N)}$$ $N^{O(1/loglogN)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann’s conjecture from the previous$$N = 2\cdot 10^{9}$$ $N=2·{10}^9$up to$$N = 7.05\cdot 10^{13} > 2^{46}$$ $N=7.05·{10}^{13}>2^{46}$, so we were finding 27 digit primes. The computation ran for about half a year on each of two platforms: four Intel Xeon Phi 7210 processors using a total of 256 cores, and a 192-core cluster of Intel Xeon E5-2630 2.3GHz processors. 

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4. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric; Müller-Hermes, Alexander (February 2023, Letters in Mathematical Physics)

   Abstract Let$$\phi $$ $\varphi$be a positive map from the$$n\times n$$ $n\times n$matrices$$\mathcal {M}_n$$ $M_n$to the$$m\times m$$ $m\times m$matrices$$\mathcal {M}_m$$ $M_m$. It is known that$$\phi $$ $\varphi$is 2-positive if and only if for all$$K\in \mathcal {M}_n$$ $K\in M_n$and all strictly positive$$X\in \mathcal {M}_n$$ $X\in M_n$,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ $\varphi \left(K^{\ast }{X}^{-1}K\right)⩾\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)$. This inequality is not generally true if$$\phi $$ $\varphi$is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ $\text{Tr}\left[\varphi \left(K^{\ast }{X}^{-1}K\right)\right]⩾\text{Tr}\left[\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)\right]$holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 

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5. Carleson’s $$\varepsilon ^{2}$$ conjecture in higher dimensions

   https://doi.org/10.1007/s00222-025-01337-w  

   Fleschler, Ian; Tolsa, Xavier; Villa, Michele (July 2025, Inventiones mathematicae)

   Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

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