相关论文: Computational Geometry Column 37

Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the…

This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum…

This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.

In this survey paper, we present open problems and conjectures on visibility graphs of points, segments and polygons along with necessary backgrounds for understanding them.

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…

The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from…

Motivated by the fact that not all nonconvex optimization problems are difficult to solve, we survey in this paper three widely-used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems.…

We examine a few problems of enumerative geometry and present their solutions in the framework of deformed (quantum) cohomology rings.

This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic…

We give a quick survey of the various fixed point theorems in computability theory, partial combinatory algebra, and the theory of numberings, as well as generalizations based on those. We also point out several open problems connected to…

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by…

We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our…

In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the…

This is a collection of open problems in geometry that I think of as puzzles: they stick to my brain -- I see many grips, but no spare hands. Puzzle-charm is the only criterion for including a problem here; importance is ignored.

We survey some ideas from the subject of Random Algebraic Geometry, a field that introduces a probabilistic perspective on classical topics in real algebraic geometry. This offers a modern approach to classical problems, such as Hilbert's…

Modelling, simulation and optimization form an integrated part of modern design practice in engineering and industry. Tremendous progress has been observed for all three components over the last few decades. However, many challenging issues…

This is an extended abstract for a presentation at The 17th International Conference on CUPUM - Computational Urban Planning and Urban Management in June 2021. This study presents an interdisciplinary synthesis of the state-of-the-art…

These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in G\"ottingen, June 2007. The first part gives an overview of the methods. The main new result is the…