相关论文: Computational Geometry Column 42

We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.

Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…

We consider $\G$-graded commutative algebras, where $\G$ is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on…

$ $[This paper is a (self contained) chapter in a new book, Mathematics and Computation, whose draft is available on my homepage at https://www.math.ias.edu/avi/book ]. We survey some concrete interaction areas between computational…

We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.

By systematically applying ten well-known and inequivalent two-part relations between hypergeometric sums 3F2(...|1) to the published database of all such sums, 62 new sums are obtained. The existing literature is summarized, and many…

This article is a survey of recent developments in, and a tutorial on, the approach to P v. NP and related questions called Geometric Complexity Theory (GCT). It is written to be accessible to graduate students. Numerous open questions in…

Distance geometry problem belongs to a class of hard problems in classical computation that can be understood in terms of a set of inputs processed according to a given transformation, and for which the number of possible outcomes grows…

Below are the problems that I formulated at Open Problems Session of {\it Workshop on Group Actions on Rational Varieties}, McGill University and University of Montreal, Canada, March 2002. To appear in: "Affine Algebraic Geometry"…

We consider several basic questions pertaining to the geometry of image of a general quadratic map. In general the image of a quadratic map is non-convex, although there are several known classes of quadratic maps when the image is convex.…

An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it…

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…

This PhD dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such…

This is the list of open problems in topological algebra posed on the conference dedicated to the 20th anniversary of the Chair of Algebra and Topology of Lviv National University, that was held on 28 September 2001.

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…

Algebraic computing in relativity and gravitation dates back more than thirty years, but only relatively recently has hardware of sufficient power to tackle large scale calculations become commonplace. Whereas it is generally understood…

One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena like: high temperature superconductivity, solid-state…

We discuss several open problems on spectrally bounded operators, some new, some old, adding in a few new insights.

This is a study of terminating and ill-defined Gauss hypergeometric functions. Corresponding hypergeometric equations have a degenerate set of of 24 Kummer's solutions. We describe those solutions and relations between them.

Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.