Moduli spaces of compact RCD structures
<p>This thesis investigates RCD spaces, which are metric measure spaces with Ricci curvature bounded below and dimension bounded above in a synthetic sense. We introduce moduli spaces of compact RCD structures and study their topology. In particular, we discuss the results obtained in [MN22] (...
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| Format: | Thesis |
| Language: | English |
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2023
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| _version_ | 1826313469582901248 |
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| author | Navarro, D |
| author2 | Mondino, A |
| author_facet | Mondino, A Navarro, D |
| author_sort | Navarro, D |
| collection | OXFORD |
| description | <p>This thesis investigates RCD spaces, which are metric measure spaces with Ricci curvature bounded below and dimension bounded above in a synthetic sense. We introduce moduli spaces of compact RCD structures and study their topology. In particular, we discuss the results obtained in [MN22] (written in collaboration with Andrea Mondino) and [Nav22].</p>
<p>In Chapter 2, we present the primary tools we use in the thesis. We recall Gromov–Hausdorff type topologies and RCD spaces with their covering and moduli spaces. The main contributions of this chapter are the equivariant measured Gromov–Hausdorff topology and the equivariant theorem (both obtained in [MN22]).</p>
<p>In Chapter 3, we focus on the case of nonnegative curvature. In particular, we obtain topological invariants of RCD(0,N) spaces using the splitting theorem. In addition, we introduce the Albanese and soul maps and prove their continuity. This last result is the most technical part of the chapter and was obtained in [MN22]. Finally, we use the Albanese map to construct examples of moduli spaces with non-trivial higher homotopy groups in every dimension N ≥ 3.</p>
<p>Chapter 4 is devoted to nonnegative curvature in dimension 2 and discusses the results of [Nav22]. We obtain a classification (up to homeomorphism) of the topological spaces that admit an RCD(0, 2) structure. For every space appearing in the classification, we compute the homeomorphism type of the moduli space of RCD(0, 2) structures and show that it is contractible.</p>
<p>Finally, in Chapter 5, we apply Ricci flow techniques to study moduli spaces of RCD(−1, 2) structures. In particular, we show that if a space has a negative Euler characteristic, then its moduli space of RCD(−1, 2) structures is homotopy equivalent to its moduli space of hyperbolic metrics.</p> |
| first_indexed | 2024-03-07T07:51:47Z |
| format | Thesis |
| id | oxford-uuid:6eb13fe0-37d1-4478-afa9-f6265e5550d5 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:13:54Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | oxford-uuid:6eb13fe0-37d1-4478-afa9-f6265e5550d52024-07-15T08:46:18ZModuli spaces of compact RCD structuresThesishttp://purl.org/coar/resource_type/c_db06uuid:6eb13fe0-37d1-4478-afa9-f6265e5550d5MathematicsEnglishHyrax Deposit2023Navarro, DMondino, ABesson, GLotay, JTuschmann, W<p>This thesis investigates RCD spaces, which are metric measure spaces with Ricci curvature bounded below and dimension bounded above in a synthetic sense. We introduce moduli spaces of compact RCD structures and study their topology. In particular, we discuss the results obtained in [MN22] (written in collaboration with Andrea Mondino) and [Nav22].</p> <p>In Chapter 2, we present the primary tools we use in the thesis. We recall Gromov–Hausdorff type topologies and RCD spaces with their covering and moduli spaces. The main contributions of this chapter are the equivariant measured Gromov–Hausdorff topology and the equivariant theorem (both obtained in [MN22]).</p> <p>In Chapter 3, we focus on the case of nonnegative curvature. In particular, we obtain topological invariants of RCD(0,N) spaces using the splitting theorem. In addition, we introduce the Albanese and soul maps and prove their continuity. This last result is the most technical part of the chapter and was obtained in [MN22]. Finally, we use the Albanese map to construct examples of moduli spaces with non-trivial higher homotopy groups in every dimension N ≥ 3.</p> <p>Chapter 4 is devoted to nonnegative curvature in dimension 2 and discusses the results of [Nav22]. We obtain a classification (up to homeomorphism) of the topological spaces that admit an RCD(0, 2) structure. For every space appearing in the classification, we compute the homeomorphism type of the moduli space of RCD(0, 2) structures and show that it is contractible.</p> <p>Finally, in Chapter 5, we apply Ricci flow techniques to study moduli spaces of RCD(−1, 2) structures. In particular, we show that if a space has a negative Euler characteristic, then its moduli space of RCD(−1, 2) structures is homotopy equivalent to its moduli space of hyperbolic metrics.</p> |
| spellingShingle | Mathematics Navarro, D Moduli spaces of compact RCD structures |
| title | Moduli spaces of compact RCD structures |
| title_full | Moduli spaces of compact RCD structures |
| title_fullStr | Moduli spaces of compact RCD structures |
| title_full_unstemmed | Moduli spaces of compact RCD structures |
| title_short | Moduli spaces of compact RCD structures |
| title_sort | moduli spaces of compact rcd structures |
| topic | Mathematics |
| work_keys_str_mv | AT navarrod modulispacesofcompactrcdstructures |