arXiv research

The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.

Extends results on smoothability of singular Fano and Calabi-Yau varieties.

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singu…

Easy-to-assemble 3D model of Boy's surface.

This is an introductory article about the Boy surface. Boy found 1901 that $\mathbb{R}P^2$ can be immersed into $\mathbb{R}^3$, and published it. (The image of) the immersion is called the Boy surface after Boy's discovery. We have created a way to construct the Boy surface by using a pair of scissors, a piece of paper…

Study symplectic cohomology of certain singularities using homological mirror symmetry.

In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} Δu$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the induced metric to establish two-sided bounds on the singular values of Du, which shows tha…

Smooth resolutions found for quotient of R^2 by infinite discrete groups.

BOIS optimizes complex systems by leveraging structural knowledge.

BOIS optimizes complex systems by combining known and unknown functions, improving efficiency.

Paper explores weak solutions' regularity in critical dimensions without conservation law.

We give a mathematical computation of the number of solutions of Apollonius problem, by use of Lie Sphere Geometry. Unlike in higher dimensions, the number of solutions depends only on the topology of the configuration of the 3 objects. It appears that our classification is non redundant, and far simpler than those obt…

Motivated by the importance and universal character of phase singularities which are clarified recently, we study the local structure of equi-phase loci near the dislocation locus of complex valued planar and spatial waves, from the viewpoint of singularity theory of differentiable mappings, initiated by H. Whitney and…

Study on homology groups of cDV singularity links, identifying their topology.

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a fam…

Du, Kakade, Wang, and Yang recently established intriguing lower bounds on sample complexity, which suggest that reinforcement learning with a misspecified representation is intractable. Another line of work, which centers around a statistic called the eluder dimension, establishes tractability of problems similar to t…

We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $\int_M|du_p|^p=O(\frac{1}{2-p}).$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the sup…

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and unif…

The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.

We determine explicitly the foliated cohomology $H_{\cal F}^\ast (M)$ of the affine Reeb flow ${\cal F}$ on the Hopf manifold ${\Bbb S}^n\times {\Bbb S}^1$. The vector space $H_{\cal F}^1(M)$ contains exactly the obstructions to solve the cohomological equation $X\cdot f=g$ where $f$ and $g$ are $C^\infty$-functions a…

Paper analyzes GP-EI for Bayesian optimization with no regret and provides guidance on choosing incumbents.

Gradient flow method solves isoperimetric inequality for maps.

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

We give criteria for Morin singularities into higher dimensions. As an application, we study the number of A-isotopy classes of Morin singularities.

Higher-dimensional Ricci flows are shown to have unique and stable solutions.

Removes singularities for Yang-Mills-Higgs fields in higher dimensions.

The paper classifies ancient ovals in higher dimensions and proves their symmetry and uniqueness.

New maxfaces with Enneper ends found.

Classifies surfaces translating under specific curvature flows.

We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function $u:M\to\mathbb{R}$ evolve in such a way whenever u solves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfying a geom…

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

For any compact manifold of dimension n>=5, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian acting on diffential forms of degree 1<p<n-1 (exept for p=n/2 if n is even), within a given conformal class. When n<5 and when p=0,1,n-1,n, and p=n/2 if n is even, this simultaneous prescriptio…

Let $M^n$ be a n-dimensional compact manifold, with $n\geq3$. For any conformal class C of riemannian metrics on M, we set $μ_k^c(M,C)=\inf_{g\in C}μ_{[\frac n2],k}(M,g)\Vol(M,g)^{\frac2n}$, where $μ_{p,k}(M,g)$ is the k-th eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that $0<μ_k^c(M,C)\leqμ_k^…

Existence of singular gradient Ricci solitons proved in higher dimensions.

Classifies ancient noncollapsed flows in 4D space.

Study on mean curvature flow through singularities in 3D and 4D.

The Gannon-Lee singularity theorems give well-known restrictions on the spatial topology of singularity-free (i.e., nonspacelike geodesically complete), globally hyperbolic spacetimes. In this paper, we revisit these classic results in the light of recent developments, especially the failure in higher dimensions of a c…

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…

This is a short survey of Cheeger and Kleiner's nonembeddability theorem for Heisenberg group into $L^1$.

Deterministic bounds for tensor singular values and vectors, differing from matrix cases.

We improve the over-parametrization size over two beautiful results [Li and Liang' 2018] and [Du, Zhai, Poczos and Singh' 2019] in deep learning theory.

New framework for higher-order singular-value derivatives of rectangular matrices.

Minimal graphs grow slowly on curved spaces, proving constant solutions.

New distance comparison principle for curve shortening flow in higher dimensions.

This paper constructs metrics with constant fractional higher order curvature on punctured spheres.

Study derived Lie ∞-groupoids and algebroids in higher differential geometry.

We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero

In a previous article, we generalised the classical four-dimensional Chern-Gauss-Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conica…