Characters in Low-Dimensional Topology

该奖项支持美国的参与-基于初级研究人员在会议“低维拓扑中的字符”2018年6月2日至6日在蒙特利尔举行，加拿大蒙特利尔的魁北克大学，由中心德Recherches数学主办。会议提供了一个国际论坛，传播和讨论低维拓扑和几何的最新重要进展。这是纯数学中一个庞大而活跃的研究领域，其基本目标是理解被称为流形的数学对象的结构，流形在建模许多科学现象时自然出现。流形是“空间”，其小尺度结构仅取决于自由度的数量，称为维数：例如，在小尺度上，一维流形看起来就像一条线，二维流形看起来就像一个平面，三维流形看起来就像我们生活的空间的一个区域，虽然所有给定维数的流形在小尺度上都是一样的，但它们在大尺度上的结构可能非常不同，而且相当复杂;许多数学技术已经被开发出来来研究这一点。“低维”指的是3维和4维，这两个维度特别有趣：如果我们只考虑空间宇宙，我们生活在一个有3维的流形中，但如果把时间也包括在内，我们生活在一个有4维的流形中。会议汇集了专家和新兴研究人员，报告最近的成果，并探讨未来的发展方向。这个奖项使来自美国的年轻数学家，研究生和博士后研究人员，参加。一个特别的努力将作出支持多样性，鼓励妇女和其他群体的出席，传统上在数学代表性不足。会议的重点是最近开发的许多不同的几何和拓扑技术，研究三维流形。现在我们知道3-流形具有刚性几何结构，最近的相关发展不仅给低维拓扑学带来了新的见解，而且也给邻近领域带来了新的见解。与此同时，起源于量子物理学和规范理论的新的同调理论，在纽结理论和三维、四维拓扑学的老问题上取得了巨大的进展。尽管在几个方面取得了这些进展，但人们对三维空间中分析、几何和代数不变量之间的相互作用知之甚少。会议强调了正在进行的研究，重点是寻找三维拓扑结构的这几个方面之间的直接联系。会议汇集了专家和早期职业数学家工作的新的不变量的3流形，几何群论应用于3流形，并在更经典的几何和拓扑技术。会议的网站是cirget.uqam.ca/boyerfest//enThis奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

This award supports participation by U.S.-based junior researchers in the conference "Characters in Low-Dimensional Topology" held June 2-6, 2018 in Montreal, Canada at the University of Quebec at Montreal, hosted by the Centre de Recherches Mathematiques. The conference provides an international forum for the dissemination and discussion of recent important advances in low-dimensional topology and geometry. This is a large and active area of research in pure mathematics, whose basic goal is to understand the structure of mathematical objects known as manifolds, which arise naturally in modeling many scientific phenomena. Manifolds are "spaces" whose small-scale structure depends only on the number of degrees of freedom, called the dimension: for example, on a small scale a 1-dimensional manifold looks just like a line, a 2-dimensional manifold looks just like a flat plane, a 3-dimensional manifold looks like a region of the space in which we live, and so on. Although all manifolds of a given dimension are the same on a small scale, their large-scale structures can be very different and quite complicated; many mathematical techniques have been developed to investigate this. "Low-dimensional" means dimensions 3 and 4, which are of special interest: we live in a manifold that has three dimensions if we consider only the spatial universe, but four dimensions if time is included as well. The conference brings together experts and emerging researchers to report on recent results and explore future directions. This award enables young mathematicians from the U.S., graduate students and postdoctoral researchers, to participate. A particular effort will be made to support diversity by encouraging the attendance of women and other groups that are traditionally underrepresented in mathematics.The conference focuses on the many different geometric and topological techniques that have recently been developed to study 3-dimensional manifolds. It is now known that 3-manifolds possess rigid geometric structures, and recent related developments have brought new insights not only to low-dimensional topology but also to neighboring fields. Meanwhile, new homology theories, having their origins both in quantum physics and in gauge theory, have led to dramatic progress on old questions in knot theory and 3-and 4-dimensional topology. Despite these advances on several fronts, relatively little is known about the interplay between analysis, geometry, and algebraic invariants in dimension 3. The conference highlights ongoing research focused on finding direct connections between these several aspects of 3-dimensional topology. The meeting brings together experts and early-career mathematicians working on the new invariants of 3-manifolds, on geometric group theory applied to 3-manifolds, and on more classical geometric and topological techniques. The website for the conference is cirget.uqam.ca/boyerfest//enThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.