[ { "id": "thesis:10170", "collection": "thesis", "collection_id": "10170", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05122017-151540545", "primary_object_url": { "basename": "seunghee_ye_2017_thesis.pdf", "content": "final", "filesize": 501743, "license": "other", "mime_type": "application/pdf", "url": "/10170/1/seunghee_ye_2017_thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "The Geometry of Moduli Spaces of Maps from Curves", "author": [ { "family_name": "Ye", "given_name": "Seunghee", "orcid": "0000-0002-1250-2935", "clpid": "Ye-Seunghee" } ], "thesis_advisor": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Solis", "given_name": "Pablo R.", "clpid": "Solis-P-R" }, { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A class of moduli spaces that has long been the interest of many algebraic geometers is the class of moduli spaces parametrizing maps from curves to target spaces. Different such moduli spaces have distinct geometry and also invariants associated to them. In this thesis, we will study the geometry of three such moduli spaces. By understanding the global geometry of each moduli space, we will produce a stratification, which plays a central role in proving a result about invariants associated to the space.", "doi": "10.7907/Z9MP5191", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:7807", "collection": "thesis", "collection_id": "7807", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312013-164406051", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 416503, "license": "other", "mime_type": "application/pdf", "url": "/7807/1/thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Relative Mirror Symmetry and Ramifications of a Formula for Gromov-Witten Invariants", "author": [ { "family_name": "van Garrel", "given_name": "Michel", "clpid": "van-Garrel-Michel" } ], "thesis_advisor": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Tian", "given_name": "Zhiyu", "clpid": "Tian-Z" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "For a toric Del Pezzo surface S, a new instance of mirror symmetry, said relative, is introduced and developed. On the A-model, this relative mirror symmetry conjecture concerns genus 0 relative Gromov-Witten of maximal tangency of S. These correspond, on the B-model, to relative periods of the mirror to S. Furthermore, for S not necessarily toric, two conjectures for BPS state counts are related. It is proven that the integrality of BPS state counts of the total space of the canonical bundle on S implies the integrality for the relative BPS state counts of S. Finally, a prediction of homological mirror symmetry for the open complement is explored. The B-model prediction is calculated in all cases and matches the known A-model computation for the projective plane.", "doi": "10.7907/9EQP-PD83", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:5820", "collection": "thesis", "collection_id": "5820", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212010-060144793", "primary_object_url": { "basename": "DissCrepFlop.pdf", "content": "final", "filesize": 873224, "license": "other", "mime_type": "application/pdf", "url": "/5820/1/DissCrepFlop.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Gromov-Witten Invariants: Crepant Resolutions and Simple Flops", "author": [ { "family_name": "Cheong", "given_name": "Wan Keng", "clpid": "Cheong-Wan-Keng" } ], "thesis_advisor": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Aschbacher", "given_name": "Michael", "clpid": "Aschbacher-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "

Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Symn(S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilbn(S) of n points in S. This proves a generalization of Ruan's Cohomological Crepant Resolution Conjecture for the case of Symn(S).

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Moreover, we determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Symn(Ar)], where Ar is the minimal resolution of the cyclic quotient singularity C2/Zr+1. Under the assumption of the nonderogatory conjecture, these operators completely determine the quantum ring structure, which gives an affirmative answer to Bryan-Graber's Crepant Resolution Conjecture on [Symn(Ar)] and Hilbn(Ar). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Symn(Ar)]/Hilbn(Ar) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar x P1.

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Finally, we prove a closed formula for an excess integral over the moduli space of degree d stable maps from unmarked curves of genus one to the projective space Pr for positive integers r and d. The result generalizes the multiple cover formula for Pr and reveals that any simple Pr flop of smooth projective varieties preserves the theory of extremal Gromov-Witten invariants of arbitrary genus. It also provides examples for which Ruan's Minimal Model Conjecture holds.

\r\n", "doi": "10.7907/6KZZ-MT72", "publication_date": "2010", "thesis_type": "phd", "thesis_year": "2010" } ]