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This paper begins by reviewing numerous theoretical advancements in the field of multivariate splines, primarily contributed by Professor Larry L. Schumaker. These foundational results have paved the way for a wide range of applications and…
The paper reviews recent developments in the study of Alexander invariants of quasi-projective manifolds using methods of singularity theory. Several results in topology of the complements to singular plane curves and hypersurfaces in…
The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.
The main question we target is the following: If one fixes a topological type of a complex normal surface singularity then what are the possible analytic types supported by it, and/or, what are the possible values of the geometric genus? We…
We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces Jan Steven's list [Manuscripta Math. 1993] of the numbers…
Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use…
We give an explicit description of adjoint quotient maps for Jacobson-Witt algebra Wn and special algebra Sn. An analogue of Kostant's differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of adjoint…
K\"ahler's paper "\"Uber die Verzweigung einer algebraischen Funktion zweier Ver\"anderlichen in der Umgebung einer singul\"aren Stelle" offered a more perceptual view of the link of a complex plane curve singularity than that provided…
In this paper, we give two direct applications of the theory of singular connections developped by Harvey-Lawson [10]. The first one is a version of Lelong-Poincar\'e formula for vector bundle over an almost complex manifold. The second is…
A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact…
This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit…
One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank…
A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods…
The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our…
We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume. Known exact results for fluctuations of…
Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning…
A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional…
This is the abstract prepared for Workshop on Topology and Geometry (Zhang jiang, China, October 1994), and is a review of my recent works. What kinds of combinations of singularities can appear in small deformation fibers of a fixed…