Conference: Motivic and non-commutative aspects of enumerative geometry, Homotopy theory, K-theory, and trace methods

该奖项资助职业生涯早期的美国数学家参加于2023年7月3日至7日在荷兰奈梅亨举行的“列举几何的动机和非对易方面”和“同伦理论、K理论和迹方法”会议的旅费和当地费用。这些会议集中在激动人心的、相互关联的数学领域，这些领域广泛地研究代数K-理论中的基本问题。这些活动旨在将美国的数学家与欧洲的研究人员联系起来，并提供建立新合作的机会。除了与该领域更知名的研究人员互动外，还设计了简短的演示课程，以允许研究生、博士后研究人员和职业生涯早期的教师分享他们自己的工作。代数K-理论记录了环或方案的重要和微妙的不变量。不幸的是，在少数例子之外计算这些K-群是出了名的困难。为了解决这个问题，已经出现了两种主要的方法：Motivic Homoty，它建立在代数几何的稳定同伦理论的许多经典结构中，以及迹方法，它通过经典和等变稳定同伦中的结构来逼近代数K-群。这些会议将汇集代表所有这些领域和数学中几个相关领域的专家，建立在令人兴奋的最新发展基础上，并提供一个论坛，让代数几何和同伦理论方面的早期数学家接触这些领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award provides support for the travel and local expenses of early-career U.S.-based mathematicians to attend the conferences "Motivic and non-commutative aspects of enumerative geometry" and "Homotopy theory, K-theory, and trace methods" in Nijmegen, the Netherlands, held from July 3-7th, 2023. The conferences focuses on exciting, interconnected areas of mathematics which broadly study foundational questions in algebraic K-theory. The events aim to connect U.S.-based mathematicians with European researchers and provide opportunities to build new collaborations. In addition to the interaction with more established researchers in the field, sessions of short presentations are designed to allow graduate students, postdoctoral researchers, and early career faculty to share their own work.Algebraic K-theory records important and subtle invariants of a ring or scheme. Unfortunately, computing these K-groups outside of a handful of examples is notoriously difficult. Two dominant approaches have arisen to tackle this problem: motivic homotopy, which builds in algebraic geometry many classical constructions from stable homotopy theory, and trace methods, which approximate the algebraic K-groups by constructions in classical and equivariant stable homotopy. These conferences will bring together experts representing all of these areas and several related fields in mathematics, building on exciting recent developments and providing a forum to expose early-career mathematicians in algebraic geometry and homotopy theory to these areas.This 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.