Based on the pluripotential methods developed in Darvas and Zhang (Commun Pure Appl Math 77(12):4289–4327, 2024), we give a simplified prove for a result of Chi Li, which states that a log Fano vatiety admits a Kähler–Einstein metric if it has vanishing Futaki invariant and its reduced delta invariant is bigger than one.

We prove that the Gromov–Hausdorff limit of Kähler–Ricci flow on a ${\textbf{G}}$-spherical Fano manifold X is a ${\textbf{G}}$-spherical ${\mathbb {Q}}$-Fano variety $X_{\infty }$, which admits a (singular) Kähler–Ricci soliton. Moreover, the ${\textbf{G}}$-spherical variety structure of $X_{\infty }$ can be constructed as a center of torus ${\mathbb {C}}^*$-degeneration of X induced by an element in the Lie algebra of Cartan torus of ${\textbf{G}}$.

We consider $G=Q_8,\textrm{SD}_{16},G_{24},$ and $G_{48}$ as finite subgroups of the Morava stabilizer group which acts on the height 2 Morava E-theory ${\textbf{E}}_2$ at the prime 2. We completely compute the G-homotopy fixed point spectral sequences of ${\textbf{E}}_2$. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the $(*-\sigma _i)$-graded $Q_8$- and $\textrm{SD}_{16}$-homotopy fixed point spectral sequences, where $\sigma _i$ is a non-trivial one-dimensional representation of $Q_8$.

This paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne–Lusztig varieties (ADLV). The primary objective is to investigate the non-emptiness pattern, dimension, and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.

In the present paper, we study the properties of singular Nakano positivity of singular Hermitian metrics on holomorphic vector bundles, and establish an optimal $L^2$ extension theorem for holomorphic vector bundles with singular Hermitian metrics on weakly pseudoconvex Kähler manifolds, which is a unified version of the optimal $L^2$ extension theorems for holomorphic line bundles with singular Hermitian metrics of Guan–Zhou and Zhou–Zhu. As applications, we give a necessary condition for the holding of the equality in optimal $L^2$ extension theorem, and present singular Hermitian holomorphic vector bundle versions of some $L^2$ extension theorems with optimal estimate.

We study abelian subcategories and torsion pairs in Abramovich–Polishchuk’s heart. And we apply the construction from Liu (J Reine Angew Math 770:135–157, 2021) on a full triangulated subcategory ${\mathcal {D}}_S^{\le 1}$ in $D(X\times S)$, for an arbitrary smooth projective variety S. We also define a notion of l-th level stability, which is a generalization of the slope stability and the Gieseker stability. We show that for any object E in Abramovich–Polishchuk’s heart, there is a unique filtration whose factors are l-th level semistable, and the phase vectors are decreasing in a lexicographic order.

We classify to some extent the pairs of group morphisms $\Gamma \rightarrow \textrm{Spin}(7)$ which are element-conjugate but not globally conjugate. As an application, we study the case where $\Gamma $ is the Weil group of a p-adic local field, which is relevant to the recent approach to the local Langlands correspondence for $\textrm{G}_2$ and $\textrm{PGSp}_6$ in Gan and Savin (Forum Math Pi 11:e28, 2023). As a second application, we improve some result in Kret and Shin (J Eur Math Soc 25(1):75–152, 2023) about $\textrm{GSpin}_7$-valued Galois representations.

We show that the recent techniques developed to study the Fourier restriction problem apply equally well to the Bochner–Riesz problem. This is achieved via applying a pseudo-conformal transformation and a two-parameter induction-on-scales argument. As a consequence, we improve the Bochner–Riesz problem to the best known range of the Fourier restriction problem in all high dimensions.

Sarnak’s Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics. We prove that Möbius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost reducible, which extends (Liu and Sarnak in Duke Math J 164(7):1353–1399, 2015; Wang in Invent Math 209:175–196, 2017) to the noncommutative case. The proof relies on quantitative version of almost reducibility.

We consider a theory of noncommutative Gröbner bases on decreasingly filtered algebras whose associated graded algebras are commutative. We transfer many algorithms that use commutative Gröbner bases to this context. As a result, we have a very efficient way to compute Ext groups for a large class of graded algebras. This has many applications especially in algebraic topology.