Mathematical Sciences: Research Experience in Computational Group Theory and Hyperbolic Geometry

为期七周的六名学生的本科生研究体验。用计算群论方法研究了由三角形拼接的双曲Riemann曲面的几何和双曲平面的诱导拼接。所考虑的所有平铺都会生成曲面的大对称群，并且平铺的几何和组合性质强烈地反映在群的结构和群对几何的作用上。几何和组合问题需要通过在对称群中进行大量计算来解决，因为很难直观地表示这些平铺的复杂性。在这些问题的激励下，学生们将通过使用MAGMA软件包进行计算机计算来进行群论实验、发现和提出猜想。最终目标将是发现和证明关于曲面的几何和组合学的定理，当然，还有任何关于在此过程中发现的群的定理。除了这个技术项目，学生还将参加一个伙伴项目，以发展他们的口头和书面数学沟通技能，协作和其他专业技能。这将通过要求的口头陈述和关于其工作的技术报告以及学生和私人侦查员之间的密切、积极的工作环境来实现，剧院是专门用于REU的计算机实验室/工作场所。

An Undergraduate research experience for six students over seven weeks. Computational group theory methods are employed to explore the geometry of hyperbolic Riemann surfaces tiled by triangles and the induced tilings of the hyperbolic plane. All tilings considered generate large symmetry groups of the surface, and the geometrical and combinatorial properties of the tiling are strongly reflected in the structure of the group and the groupOs action on the geometry. The geometric and combinatorial problems need to be solved by massive computations in the symmetry groups, as it is very difficult to visually represent the complexities of these tilings. Motivated by these problems, students will perform group theoretic experiments, make discoveries and formulate conjectures by carrying out computer calculations using the software package MAGMA. The end goal will be to discover and prove theorems about the geometry and combinatorics of the surfaces and, of course, anything about groups discovered along the way. In addition to this technical program, students will be engaged in a companion program to develop their oral and written mathematical communication skills, collaborative, and other professional skills. This will be accomplished through required oral presentations and technical reports on their work and a close, positive working environment among the students and the P.I. in the Theorodrome, the computer laboratory/workplace devoted to the REU.