arXiv research

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

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

Compact Kähler manifold minus a divisor is projective space.

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

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

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

Study convexity of Mabuchi functional in big cohomology classes.

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.

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.

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

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. …

Paper explains why plateau phenomenon is rare in modern deep learning.

Study shows symplectic hypersurfaces transform complex projective spaces.

The two phase behavior in financial markets actually means the bifurcation phenomenon, which represents the change of the conditional probability from an unimodal to a bimodal distribution. In this paper, the bifurcation phenomenon in Hang-Seng index is carefully investigated. It is observed that the bifurcation phenom…

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…

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 paper explains how neural networks learn less salient frequency components during training.

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

Proves a formula linking LMO invariants of spliced 3-spheres.

In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fo…

Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.

Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.

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…

Study gap phenomenon in flat manifolds with Ricci curvature.

New transport method simplifies cutoff phenomenon for Markov processes.

Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principle components analysis is performed separately on the data sets to reduce their dimensionality. In some circumstances it may happen that the two lower-dimensional data sets have an inordinat…

Representation stability is a phenomenon whereby the structure of certain sequences $X_n$ of spaces can be seen to stabilize when viewed through the lens of representation theory. In this paper I describe this phenomenon and sketch a framework, the theory of FI-modules, that explains the mechanism behind it.

New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.

Cohen et al. (2021) show GD trajectories align on a bifurcation diagram.

New phenomenon found in Gothen components' boundary.

It was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces. We link this framework to Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable unpunc…

For the supervised least squares classifier, when the number of training objects is smaller than the dimensionality of the data, adding more data to the training set may first increase the error rate before decreasing it. This, possibly counterintuitive, phenomenon is known as peaking. In this work, we observe that a s…