Computational tropical geometry and its applications

计算热带几何及其应用

• 批准号: MR/S034463/1
• 负责人: Yue Ren
• 金额: $ 88.41万
• 依托单位: Swansea University
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=MR%2FS034463%2F1
• 关键词: Computational; tropical; geometry; its; applications

Tropical geometry is a young area of mathematics which studies combinatorial objects arising from polynomial equations. These so-called tropical varieties arise naturally in many areas of mathematics and beyond, such as phylogenetics in biology, celestial mechanics in physics, and auction theory in economics. Wherever they arise, tropical varieties often allow new computational approaches to existing problems. In the UK, the Bank of England has been using tropical geometry since the financial crises to allocate money to the UK financial system. In France, tropical geometry is used for optimisation of load balancing of mobile networks, and performance analysis of emergency call centres.This research projects aims at establishing tropical geometry as a powerful and versatile tool for computational questions in applied sciences and industry beyond optimisation. To this end, we pursue concrete applications as well as improvements of computational methods. The final deliverable is a comprehensive open source software system for tropical algebraic geometry with strong emphasis on its wide spectrum of applications. We focus on three main problems, which were chosen to maximise the impact and the range of techniques that they encompass.The first problem revolves around systems of polynomial equations, which are ubiquitous in applied science. They describe the steady states of chemical reaction networks, the range of movement of a robot arm, or the binding behaviour of ligands in a biochemical system. For over two decades, the state of the art for solving such systems has been homotopy continuation, which works by carefully deforming an easy start system to the target system while tracing all solutions along the way.We seek to improve the existing capabilities, in particular for the type of polynomial systems which arise in the aforementioned applications. While ideas to apply tropical geometry to homotopy continuation have already been studied, all past approaches have failed due to questions of efficiency. However, the last couple of years have seen significant algorithmic breakthroughs in tropical geometry, which we will exploit and build upon.The second problem involves p-adic numbers, which are an indispensable class of fields for number theory. This not only makes them important for the applications of tropical geometry in number theory, but also entails a vast array of number theoretic tools available exclusively over them. Hence a good grasp on tropical geometry over p-adics numbers is an imperative for both theory and practice.That being said, computationally, tropical geometry over p-adic numbers has been neglected due to the unique algorithmic challenges they pose. We seek to remedy this situation and explore computational aspects of tropical geometry specifically over p-adic numbers, facilitated by recent trends in computer algebra.The third problem involves Gröbner bases, which have long history in computational algebraic geometry and adjacent fields such as cryptography. Furthermore, the past decade featured an explosion of algebro-geometric techniques in areas outside of mathematics. As such, Gröbner bases have gained traction both as tool for studying polynomial systems and as object of interest themselves, e.g., as Markov bases in algebraic statistics. However, Gröbner bases are notoriously hard to compute, which severely inhibits their use in practical applications.We will investigate so-called saturating Gröbner bases. In general, polynomial unknowns represent arbitrary elements of the coefficient field, and all operations within a Gröbner basis computation respect this ambiguity. In practice, one is often only interested in specific solutions, e.g. strictly positive real solutions. Saturating Gröbner basis algorithm are symbolic algorithms which are capable of exploiting this numerical information that is abundant in many applications and use it to speed up its performance.

热带几何是数学的一个年轻领域，它研究由多项式方程产生的组合对象，这些所谓的热带变体自然出现在数学及其他领域的许多领域，例如生物学中的系统发育学、物理学中的天体力学和物理学中的拍卖理论。无论热带品种出现在哪里，它们通常都会为现有问题提供新的计算方法。自金融危机以来，英国央行一直在使用热带几何来向英国金融体系分配资金。用于优化移动网络的负载平衡以及紧急呼叫中心的性能分析。该研究项目旨在将热带几何学建立为一种强大且多功能的工具，用于解决应用科学和工业中超越优化的计算问题。以及计算方法的改进，最终的成果是一个全面的热带代数几何开源软件系统，重点关注其广泛的应用，我们选择这些问题是为了最大限度地提高其影响和范围。它们包含的技术。第一个问题围绕着应用科学中普遍存在的多项式方程组，它们描述了化学反应网络的稳态、机器人手臂的运动范围或生化系统中配体的结合行为。解决此类系统的最先进技术是同伦延拓，它的工作原理是仔细地将一个简单的启动系统变形为目标系统，同时跟踪沿途的所有解决方案。我们寻求提高现有的能力，特别是多项式类型虽然已经研究了将热带几何应用于同伦延拓的想法，但由于效率问题，过去的所有方法都失败了。然而，在过去的几年里，热带几何在算法上取得了重大突破。第二个问题涉及 p 进数，它是数论中不可或缺的一类域，这不仅使它们对于热带几何在数论中的应用很重要，而且还涉及大量的问题。可用的数论工具因此，对 p 进数数的热带几何的良好掌握对于理论和实践都是必要的。话虽如此，在计算上，由于 p 进数数带来的独特算法挑战，热带几何已被忽视。纠正这种情况，并在计算机代数的最新趋势的推动下，探索热带几何的计算方面，特别是在 p-adic 数上。第三个问题涉及 Gröbner 基，它在计算代数几何和密码学等邻近领域有着悠久的历史。此外，过去十年，代数几何技术在数学以外的领域出现了爆炸式增长，因此，格罗布纳基作为研究多项式系统的工具和本身的兴趣对象（例如代数统计中的马尔可夫基）而受到关注。然而，Gröbner 基是出了名的难以计算，这严重限制了它们在实际应用中的使用。我们将研究所谓的饱和 Gröbner 基。未知数表示系数域的任意元素，并且 Gröbner 基计算中的所有运算都考虑到这种模糊性。在实践中，人们通常只对特定的解感兴趣，例如，饱和严格正实数解是能够实现的符号算法。利用许多应用程序中丰富的数字信息并使用它来加速其性能。

项目成果

Cooperativity, absolute interaction, and algebraic optimization.

合作性、绝对相互作用和代数优化。

• DOI: http://dx.10.1007/s00285-020-01540-8
• 发表时间: 2020
• 期刊: Journal of mathematical biology
• 影响因子: 1.9
• 作者: Kaihnsa N

Sharp Bounds for the Number of Regions of Maxout Networks and Vertices of Minkowski Sums

Maxout 网络区域数和 Minkowski 和顶点数的锐界

• DOI: http://dx.10.1137/21m1413699
• 发表时间: 2022
• 期刊: SIAM Journal on Applied Algebra and Geometry
• 影响因子: 1.2
• 作者: Montúfar G

On intersections and stable intersections of tropical hypersurfaces

热带超曲面的交集和稳定交集

• DOI: http://dx.10.5802/alco.327
• 发表时间: 2024
• 期刊: Algebraic Combinatorics
• 作者: Ren Y

Computing zero-dimensional tropical varieties via projections

通过投影计算零维热带品种

• DOI: http://dx.10.1007/s00037-022-00222-9
• 发表时间: 2022
• 期刊: computational complexity
• 影响因子: 1.4
• 作者: Görlach P

Computing GIT-fans with symmetry and the Mori chamber decomposition of \overline{}_{0,6}

计算具有对称性的 GIT-fans 和 overline{}_{0,6} 的 Mori 室分解

• DOI: http://dx.10.1090/mcom/3546
• 发表时间: 2020
• 期刊: Mathematics of Computation
• 影响因子: 2
• 作者: Böhm J