arXiv research

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

Uniform volume estimate for Kähler metrics in big cohomology classes.

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…

Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.

Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.

Study convexity of Mabuchi functional in big cohomology classes.

This thesis introduces big mapping class groups and their structure.

Paper defines new stability and metrics for complex spaces.

Extends K-stability theory to projective klt pairs with a big anticanonical class.

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

Estimates Kaehler metrics' diameter in big cohomology classes.

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

Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.

We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass a…

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

Classifies big mapping classes on infinite type surfaces.

The paper tackles machine unlearning by designing efficient algorithms for adaptive query classes.

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-…

We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomolog…

The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

We show that if $M$ is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of …

Unified view on big bang singularities from initial data.

The study examines when mapping class groups are quasi-isometric to graphs of curves.

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

The paper studies deformations of Kähler manifolds to normal bundles and restricted volumes of big classes.

Two Calabi-Yau theorems for Kähler manifold degenerations.

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite ou…

Explosive growth in data and availability of cheap computing resources have sparked increasing interest in Big learning, an emerging subfield that studies scalable machine learning algorithms, systems, and applications with Big Data. Bayesian methods represent one important class of statistic methods for machine learni…

Proves left-orderability of mapping class groups of infinite-type surfaces.

Paper shows mapping class groups are not extremely amenable except for specific cases.

In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum $\mathcal{H} (α)$ of Abelian differentials. The first is the saddle connection Siegel-Veech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (α) \big)$ counting saddle conne…

This note displays an interesting phenomenon for percentiles of independent but non-identical random variables. Let $X_1,\cdots,X_n$ be independent random variables obeying non-identical continuous distributions and $X^{(1)}\geq \cdots\geq X^{(n)}$ be the corresponding order statistics. For any $p\in(0,1)$, we investig…

Study on self-similar surfaces and their mapping class groups generated by involutions.

We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing…

Suppose $(X,ω)$ is a compact Kähler manifold of dimension $n$, and $θ$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,θ)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog f…

Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.

Differences in data size per class, also known as imbalanced data distribution, have become a common problem affecting data quality. Big Data scenarios pose a new challenge to traditional imbalanced classification algorithms, since they are not prepared to work with such amount of data. Split data strategies and lack o…

Overview of infinite surface mapping class groups.

Given $(X,ω)$ compact Kähler manifold and $ψ\in\mathcal{M}^{+}\subset PSH(X,ω)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(ω+dd^{c}ψ)^{n}>0$, we prove that the $ψ-$relative finite energy class $\mathcal{E}^{1}(X,ω,ψ)$ becomes a complete metric space…

We consider spacetimes $N$ satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime $\hat N$ by switching the light cone and using reflection to define a new time function, such that the two…

Big Data is one of the major challenges of statistical science and has numerous consequences from algorithmic and theoretical viewpoints. Big Data always involve massive data but they also often include online data and data heterogeneity. Recently some statistical methods have been adapted to process Big Data, like lin…

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…

Study shows volumes of complex classes can be represented by convex bodies.