Finitely presented groups at The City College of New York, Spring 2009 conference

纽约城市学院 2009 年春季会议上的有限展示小组

• 批准号: 0854902
• 负责人: Gilbert Baumslag
• 金额: $ 1.5万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-03-01 至 2010-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854902&HistoricalAwards=false
• 关键词: Finitely; presented; groups; City; College

How can you tell whether a knot is really knotted or whether it is possible to find a formula for the solutions of certain equations or how to make senseof the nature of the elementary particles of physics or why there are only 17 really different wall paper patterns? These and many other such questions depend on an algebraic structure called a group. Groups arise in many parts of mathematics and in science. They provide, in particular, the means to understand and exploitsymmetry. This makes it possible for them to helpin our understanding of the nature of the space we live in, the structure of crystals and various molecules and in the design of new cryptographic protocols. Indeed the security of information transmitted electronically over the internet depends crucially on a particular kind of group called a finite group. This group provides a wayto encode messages using the product of two very large primenumbers. The encoding is the easy part of the process.The hard part is the decoding and this is where group theoryis used. The present proposal deals with the current state ofknowledge of certain groups, called finitely presented groups.These finitely presented groups hold in many instances the keysto solving important problems in mathematics and scienceand the primary objective of this conference is for a numberof very prominent mathematicians to map out future directions of study.

你如何判断一个结是否真的结在一起，或者是否有可能找到某个方程的解的公式，或者如何理解物理基本粒子的性质，或者为什么只有17种真正不同的墙纸图案？这些问题和许多其他类似的问题都依赖于一种称为群的代数结构。群体出现在数学和科学的许多领域。它们尤其提供了理解和利用对称性的方法。这使得它们有可能帮助我们理解我们生活的空间的本质，晶体和各种分子的结构，以及新的加密协议的设计。事实上，通过互联网传输的电子信息的安全性在很大程度上取决于一种被称为有限群的特定群体。该组提供了一种使用两个非常大的素数的乘积对消息进行编码的方法。编码是这个过程中比较容易的部分。困难的部分是解码，这就是使用群体理论的地方。目前的建议处理的是某些群体的当前知识状态，这些群体被称为有限呈现群体。在许多情况下，这些有限的群体掌握着解决数学和科学中重要问题的关键，这次会议的主要目的是让一些非常杰出的数学家规划出未来的研究方向。