Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or posit...

My research uses algebraic and geometric methods to prove theorems in extremal combinatorics. Going the other way, I also use combinatorial methods to prove algebraic results. Algebraic methods are deeply embedded in my work and nearly all of my success in graph theoretic research has come from attacking purely combinatorial problems through the lens of algebra, combinatorial number theory, or ...

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 The modern face of enumerative combinatorics. . . . . . . . . . . . . . . . . . . . . . . . 3 2 Algebraic invariants and combinatorial structures . . . . . . . . . . . . . . . . . . . . . 4 3 Combinatorics and geometry. . . . . . . . . . . . . . . . . . . . . . ...

Digital Geometry Processing (DGP) is concerned with the construction of signal processing style algorithms that operate on surface geometry, typically specified by an unstructured triangle mesh. An active subfield of study involves the utilization of discrete mesh Laplacian operators for eigenvalue decomposition, mimicking the effect of discrete Fourier analysis on mesh geometry. In this paper,...

Any geometric algorithm can be dissected into combinatorial parts and geometric parts. The two parts intertwine in both the description and the implementation of the algorithm. We consider the benefits of relaxing this tight coupling. Coordinate freedom is widely considered the most important principle in designing and implementing geometric systems. By ensuring that client code not manipulate ...

These Smarandache spaces are right theories for objectives by logic. However, the mathematical combinatorics is a combinatorial theory for branches in classical mathematics motivated by a combinatorial speculation. Both of them are unifying theories for sciences and contribute more and more to mathematics in the 21st century. In this paper, I introduce these two subjects and mainly concentrate ...

The c-Balanced Separator problem is a graph-partitioning problem in which given a graph G, one aims to find a cut of minimum size such that both the sides of the cut have at least cn vertices. In this paper, we present new directions of progress in the c-Balanced Separator problem. More specifically, we propose a family of mathematical programs, that depend upon a parameter p > 0, and is an ext...