Cluster Algebras and Mirror Symmetry for Homogeneous Spaces- [electronic resource]
Cluster Algebras and Mirror Symmetry for Homogeneous Spaces- [electronic resource]
- 자료유형
- 학위논문파일 국외
- 최종처리일시
- 20240214100427
- ISBN
- 9798379603601
- DDC
- 510
- 저자명
- Wang, Charles.
- 서명/저자
- Cluster Algebras and Mirror Symmetry for Homogeneous Spaces - [electronic resource]
- 발행사항
- [S.l.]: : Harvard University., 2023
- 발행사항
- Ann Arbor : : ProQuest Dissertations & Theses,, 2023
- 형태사항
- 1 online resource(180 p.)
- 주기사항
- Source: Dissertations Abstracts International, Volume: 84-12, Section: B.
- 주기사항
- Advisor: Williams, Lauren;Sturmfels, Bernd.
- 학위논문주기
- Thesis (Ph.D.)--Harvard University, 2023.
- 사용제한주기
- This item must not be sold to any third party vendors.
- 초록/해제
- 요약Homogeneous spaces lie at the intersection of various fields of study, and as a result possess an incredibly rich structure. This thesis focuses in particular on the interactions between cluster algebras and mirror symmetry in the context of homogeneous spaces.Rietsch's construction of Landau-Ginzburg (LG) models for homogeneous spaces motivated studies of various mirror symmetry statements for homogeneous spaces, such as Grassmannians Gr(k, n) which are homogeneous spaces for the special linear group SLn. In nearly all of these studies, the authors preferred to work with an equivalent LG model presented in terms of coordinates rather than with Rietsch's original Lie-theoretic formulation. However, coordinate formulations of Rietsch's LG models were only available for certain special cases, and were difficult to generalize. In joint work with Peter Spacek, which forms the first part of this thesis, we give a coordinate presentation of Rietsch's LG models for the Cayley plane and Freudenthal variety, which are homogeneous spaces for the exceptional Lie groups of types E6 and E7, respectively. Furthermore, we are currently working to extend our methods to the more general family of cominuscule homogeneous spaces, and present preliminary results in this direction in this thesis.In some works mentioned above, cluster algebras were used to facilitate proofs or explain certain phenomena. These interesting connections between mirror symmetry and cluster algebras for homogeneous spaces are most well-studied for the Grassmannians, for example in the works of Marsh and Rietsch and of Rietsch and Williams. We expect this connection is not specific to Grassmannians, but rather a general feature of homogeneous spaces, and in this direction we present exploratory work for the Lagrangian and orthogonal Grassmannians, which are homogeneous spaces for the symplectic group Sp2n and the special orthogonal group SO2n+1, respectively. We hope to greatly develop this connection further. There have also been remarkable connections between cluster algebras and integrability in the context of Grassmannians Gr(k, n). The recent works of Kodama and Williams as well as of Abenda and Grinevich study the relationship between soliton solutions to the KP equation and the structure of the (totally nonnegative) Grassmannian. These connections remain somewhat mysterious, and in order to gain further insight into this relationship, we study the problems of identifying commuting differential operators and reconstructing solutions to the KP equation from water waves. Commuting differential operators, particularly in the context of of pseudo-differential operators and the Sato Grassmannian, have deep connections to the KP equation and algebraic curves. Furthermore, Krichever showed how to construct solutions to the KP equation using algebraic curves, and understanding the inverse problem will be helpful in studying the combinatorial structure of solutions to the KP equation.
- 일반주제명
- Mathematics.
- 일반주제명
- Applied mathematics.
- 키워드
- Cluster algebras
- 키워드
- Integrability
- 키워드
- Mirror symmetry
- 기타저자
- Harvard University Mathematics
- 기본자료저록
- Dissertations Abstracts International. 84-12B.
- 기본자료저록
- Dissertation Abstract International
- 전자적 위치 및 접속
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