arXiv research

Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.

Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.

Study convexity of Mabuchi functional in big cohomology classes.

Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.

Compact Kähler manifold minus a divisor is projective space.

Global solutions found for certain reaction-diffusion equations on specific manifolds.

Introduces valuative stability for polarised varieties, equivalent to K-stability.

This is an expository article on the equivariant local index developed by Fujita, Furuta, and the author in arXiv:1008.5007.

The paper studies foliations on smooth projective varieties and their properties.

New approach finds analytic interpretation of algebraic invariants for balanced metrics.

Solves Cheltsov-Rubinstein problem for complex surfaces with two boundary components.

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fa…

New invariants prove existence of Kahler-Einstein metrics on big classes.

New stability criterion for Fano manifolds using anticanonically balanced metrics.

We propose a statistical approach to tornadoes modeling for predicting and simulating occurrences of tornadoes and accumulated cost distributions over a time interval. This is achieved by modeling the tornadoes intensity, measured with the Fujita scale, as a stochastic process. Since the Fujita scale divides tornadoes …

Study Miyaoka-Yau inequality for certain projective manifolds.

New stability criteria for Fano varieties using generalized b-divisors.

We present a model for random simple graphs with a degree distribution that obeys a power law (i.e., is heavy-tailed). To attain this behavior, the edge probabilities in the graph are constructed from Bertoin-Fujita-Roynette-Yor (BFRY) random variables, which have been recently utilized in Bayesian statistics for the c…

Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corr…

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kahler-Einstein edge metrics on all but finitely…

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

We show that the anti-canonical volume of an $n$-dimensional Kähler-Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\mathrm{lct}^n\cdot\mathrm{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fuji…

Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.

New insights into Kähler Ricci solitons and Calabi-Yau cones.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

Sharp bounds on Fano varieties' heights proven for specific cases.

We first notice in this article that if a compact Kähler manifold has the same integral cohomology ring and Pontrjagin classes as the complex projective space $\mathbb{C}P^n$, then it is biholomorphic to $\mathbb{C}P^n$ provided $n$ is odd. The same holds for even $n$ if we further assume that $M$ is simply-connected. …

Study shows symplectic hypersurfaces transform complex projective spaces.

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our for…

We are concerned in this article with a classical question in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of holomorphic sectional curvature of a complex $n$-dimensional compact Kähler manifold can be completely determined by the eigenvalues of its $p$-Laplacian for a …

For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\rm vol}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\rm vol}$. We prove that the normalized volume has a uniform positive lower bound by pro…

The paper proves K-stability of special Gushel-Mukai manifolds.

Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that t…

Sharp bounds on K-semistable Fano varieties for low dimensions.

Global stability proved for Navier-Stokes equations on hyperbolic space.

The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.

The paper extends stability theorem for Navier-Stokes equations to negatively curved manifolds.

The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.

Geometric Gaussian approximations capture any distribution.

Method approximates Riemannian barycenter on manifolds.

We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of approximating linear pieces? We establish the necessary and sufficient conditions and uniqu…

Efficiently reduces tensor ranks using mean-field approximation.

Study approximates unknown function levels with queries.

We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such …

Softmax attention approximates complex functions and subsumes many known universal approximators.

Improved matrix approximation using randomized algorithms.

Deviation inequalities for stochastic approximation methods.

Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…