Geometric methods in group theory and group-based cryptography

群论和群密码学中的几何方法

• 批准号: 298965-2011
• 负责人: Bumagin, Inna
• 金额: $ 0.8万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571524
• 关键词: Geometric; methods; group; theory; based

My research is in Pure Mathematics, more precisely, in Geometric Group Theory. This flourishing and rapidly developing field of study attracts many brilliant researchers. It is interdisciplinary in nature. It is rooted in the work of many mathematicians. It started getting its shape in the 1980s when Gromov clearly articulated the idea that infinite groups, usually regarded as algebraic objects, can be characterized by their geometric properties, and that tools from topology can be used in group theory. Since then, research in this area made its way also to harmonic analysis, logic, probability and computer science. A wealth of available techniques and potential applications ensures that the field will remain a mainstream in mathematics for a long while. There are two strands in geometric group theory: groups can be studied via their intrinsic geometry, or via their actions on appropriate spaces. In my research I am interested in both. The intrinsic geometry of a group reveals algebraic, algorithmic and asymptotic properties of the group. Group actions on spaces, such as real trees or tree-graded spaces, allow one to gain insight into the structure of the group. It is fascinating to see how quickly the development of new methods leads to striking applications, both in pure mathematics and in the real life. An example of the former is recent solution to the following two Tarski's problems from the 1940s: Are the 2-generated free group and the 3-generated free group elementarily equivalent? Is the elementary theory of a free group decidable? On the practical side, the interest in these methods is motivated by the needs of the group-based cryptography. These seemingly very different areas of research can be approached using the versatile theory of solving systems of equations and inequations over groups. A canonical description of the solution set is of importance to a logician, while a cryptographer would focus on decidability and complexity. My interest and passion lie in the study of the systems of equations over relatively hyperbolic groups, using a variety of available techniques from different areas, borrowing existing methods and developing new ones.

我的研究是在纯数学，更准确地说，在几何群论。这一蓬勃发展的研究领域吸引了许多杰出的研究人员。它是跨学科的性质。它植根于许多数学家的工作。它开始得到它的形状在20世纪80年代时格罗莫夫清楚地阐述了这样的想法，即无限群，通常被视为代数对象，可以由其几何性质，并从拓扑工具可以用于群论。从那时起，这一领域的研究也进入了调和分析、逻辑、概率和计算机科学。丰富的可用技术和潜在的应用确保了该领域将在很长一段时间内保持数学的主流。 几何群论中有两条链：群可以通过它们的内在几何来研究，或者通过它们在适当空间上的作用来研究。在我的研究中，我对两者都感兴趣。群的内在几何揭示了群的代数、算法和渐进性质。空间上的群作用，如真实的树或树分级空间，允许人们深入了解群的结构。无论是在纯数学还是在真实的生活中，新方法的发展都带来了惊人的应用，这是令人着迷的。前一个例子是最近的解决方案以下两个塔斯基的问题从1940年代：是2生成的自由群和3生成的自由群elementarily等价？自由群的基本理论是可判定的吗？在实践方面，对这些方法的兴趣是由基于组的密码学的需求所激发的。这些看起来非常不同的研究领域可以使用多功能的理论来解决方程组和不等式组。解集的规范描述对逻辑学家来说很重要，而密码学家则专注于可判定性和复杂性。我的兴趣和激情在于研究相对双曲群的方程组，使用各种不同领域的可用技术，借用现有方法并开发新方法。