Mirror Symmetry and Related Topics

这项研究项目涉及代数几何，这是一个研究多项式方程组解的结构的数学分支。在过去的二十年里，代数几何的发展速度加快了，因为它与数学物理，更具体地说，是弦理论的互惠关系。这个项目的前两部分旨在通过验证一些受弦理论启发的数学预测，为这一不断增长的研究领域做出贡献。该项目的第三部分预计将涉及本科生的研究。除了它在代数几何中的内在科学价值，这部分项目的目的是扩大本科生参与尖端数学研究的机会，使他们认识到不同数学领域之间的深刻联系。本课题主要研究代数几何中的双镜现象以及Eisenbud-Goto猜想的一些组合方面。本研究涉及三个主要研究方向。该项目的第一部分将进一步研究Pfaffian-Grassmanian双镜实例，目的是将在Calabi-Yau环形品种的完全交叉口背景下开发的想法和技术转移到这个新的、更复杂的环境中。该项目的第二部分继续沿着川田猜想的方向进行研究，该猜想指出，两个民族的Calabi-Yau品种是导出等价的。这是一个已有十五年历史的猜想，它对同系镜像对称性的领域具有重要意义。该项目的第三部分解决了在环状簇的特殊情况下长期存在的Eisenbud-Goto猜想，该猜想可以被重新表述为关于某些整数多面体中的整数点集的结构的问题。

This research project concerns algebraic geometry, a branch of mathematics that studies the structure of the solutions of systems of polynomial equations. In the last twenty years, the pace of development of algebraic geometry has accelerated due to its mutually beneficial connections with mathematical physics, more specifically to string theory. The first two parts of this project aim to contribute to this growing area of research by verifying some of the mathematical predictions that are inspired by string theory. A third part of the project is expected to involve research by undergraduate students. In addition to its intrinsic scientific value in algebraic geometry, this part of the project is aimed at expanding the opportunities for undergraduate students to participate in cutting-edge mathematical research in a way that makes them cognizant of deep connections between different areas of mathematics. This project focuses on the double mirror phenomenon in algebraic geometry as well as some combinatorial aspects of the Eisenbud-Goto conjecture. The research involves three main directions of study. The first part of the project will further investigate the Pfaffian-Grassmannian double mirror example, with the goal of transferring ideas and techniques developed in the context of Calabi-Yau complete intersections in toric varieties to this new, more sophisticated setting. The second part of the project continues research in the direction of the Kawamata's conjecture, which states that birational Calabi-Yau varieties are derived equivalent. This is a fifteen-year-old conjecture of key importance to the area of homological mirror symmetry. The third part of the project tackles the longstanding Eisenbud-Goto conjecture in the particular case of toric varieties, which can be reformulated as a question about the structure of the sets of integer points in certain integer polytopes.