Related papers: On the quantization of some polynomial minimal sur…
An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total…
Chapters : Old and new inequalities; Surfaces with $\chi=1$ and the bicanonical map; Surfaces with $p_g=4$; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF…
We survey what is known about minimal surfaces in $\bold R^3 $ that are complete, embedded, and have finite total curvature. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa, early…
We establish a generic weak uniqueness result and partial regularity of the free boundary and of minimizers for the composite membrane problem.
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations…
Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit…
Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for…
We present examples of prequantizations over integral symplectic manifolds which admit infinitely many smoothly trivial contact mapping classes. These classes are given by the connected components of the strict contactomorphism group which…
We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the…
We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit…
In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles…
In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by De Marchis, Ianni and Pacella [6, Remark 1.2].
We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions $f:\{1, ..., N\}\to\{1, ..., M\}$, its polynomial degree is the same for all…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
We define the notion of a minimal affinization of an irreducible representation of $U_q(g)$. We prove that minimal affinizations exist and establish their uniqueness in the rank 2 case.
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portoro\v{z} on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of…
We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral…
We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…
In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of…