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University of Minnesota

1. Mak, Cheuk Yu. Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences.

Degree: PhD, Mathematics, 2016, University of Minnesota

URL: http://hdl.handle.net/11299/182326

We present three different aspects of symplectic geometry in connection to complex geometry. Convex symplectic manifolds, symplectic divisors and Lagrangians are central objects to study on the symplectic side. The focus of the thesis is to establish relations of these symplectic objects to the corresponding complex analytic objects, namely Stein fillings, divisors and coherent sheaves, respectively. Using pseudoholomorphic curve techniques and Gauge theoretic results, we system- atically study obstructions to symplectic/Stein fillings of contact 3-manifolds arising from the rigidity of closed symplectic four-manifolds with non-positive Kodaira dimen- sion. This perspective provides surprising consequences which, in particular, captures a new rigidity phenomenon for exact fillings of unit cotangent bundle of orientable surfaces and recovers many known results in a uniform way. The most important source of Stein fillings comes from smoothing of a complex isolated singularities. This motivates us to study when a symplectic divisor admits a convex/concave neighborhood and we obtain a complete and very computable answer to this local behaviour of symplectic divisors. Globally speaking, symplectic divisors in a closed symplectic manifold that represent its first Chern class are of particular importance in mirror symmetry. Such a symplectic divisor, together with the closed symplectic manifold together is called a symplectic log Calabi-Yau surface. We obtain a complete classification of symplectic log Calabi-Yau surface up to isotopy of symplectic divisors. Finally, we study algebraic properties of Fukaya category on the functor level and uti- lize Biran-Cornea’s Lagrangian cobordism theory and Mau-Wehrheim-Woodward func- tor to provide a partial proof of Huybrechts-Thomas’s conjecture.

Subjects/Keywords: Dehn twists; Log Calabi-Yau surfaces; Symplectic fillings; Symplectic geometry

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Mak, C. Y. (2016). Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences. (Doctoral Dissertation). University of Minnesota. Retrieved from http://hdl.handle.net/11299/182326

Chicago Manual of Style (16^{th} Edition):

Mak, Cheuk Yu. “Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences.” 2016. Doctoral Dissertation, University of Minnesota. Accessed August 11, 2020. http://hdl.handle.net/11299/182326.

MLA Handbook (7^{th} Edition):

Mak, Cheuk Yu. “Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences.” 2016. Web. 11 Aug 2020.

Vancouver:

Mak CY. Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences. [Internet] [Doctoral dissertation]. University of Minnesota; 2016. [cited 2020 Aug 11]. Available from: http://hdl.handle.net/11299/182326.

Council of Science Editors:

Mak CY. Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences. [Doctoral Dissertation]. University of Minnesota; 2016. Available from: http://hdl.handle.net/11299/182326

2.
Mandel, Travis Glenn.
Tropical theta functions and *log* *Calabi*-*Yau* surfaces.

Degree: PhD, Mathematics, 2014, University of Texas – Austin

URL: http://hdl.handle.net/2152/24935

We describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher rank cluster varieties.
*Advisors/Committee Members: Keel, Seán (advisor).*

Subjects/Keywords: Theta function; Log Calabi-Yau; Surfaces; Tropical; Toric; Cluster; Mirror symmetry; Polytope

…thesis is that *log* *Calabi*-*Yau* surfaces (or in another
language, fibers of rank 2 cluster X… …of a *log* *Calabi*-*Yau* surface U .
They then use toric degenerations, modified by scattering… …diagrams, to construct a
mirror family V of *log* *Calabi*-*Yau* surfaces, with the integer points of U… …order to
better understand the *log* *Calabi*-*Yau* surface.
1.0.1
Some Main Results
As mentioned… …conjectures that tropicalizations of regular functions are
tropical for any *log* *Calabi*-*Yau* variety…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Mandel, T. G. (2014). Tropical theta functions and log Calabi-Yau surfaces. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/24935

Chicago Manual of Style (16^{th} Edition):

Mandel, Travis Glenn. “Tropical theta functions and log Calabi-Yau surfaces.” 2014. Doctoral Dissertation, University of Texas – Austin. Accessed August 11, 2020. http://hdl.handle.net/2152/24935.

MLA Handbook (7^{th} Edition):

Mandel, Travis Glenn. “Tropical theta functions and log Calabi-Yau surfaces.” 2014. Web. 11 Aug 2020.

Vancouver:

Mandel TG. Tropical theta functions and log Calabi-Yau surfaces. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2014. [cited 2020 Aug 11]. Available from: http://hdl.handle.net/2152/24935.

Council of Science Editors:

Mandel TG. Tropical theta functions and log Calabi-Yau surfaces. [Doctoral Dissertation]. University of Texas – Austin; 2014. Available from: http://hdl.handle.net/2152/24935

3. -4617-5386. GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients.

Degree: PhD, Mathematics, 2017, University of Texas – Austin

URL: http://hdl.handle.net/2152/63035

I prove that the full Fock-Goncharov conjecture holds for Conf₃[superscript x] ([mathcal] A) – the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau-Ginzburg potential W on Conf₃[superscript x] ([mathcal] A)[superscript vee] associated to the partial minimal model Conf₃[superscript x] ([mathcal] A) [subset] Conf₃ ([mathcal] A). The integral points of the associated "cone" [Xi] [does not equal] {W[superscript T] [less than or equal to] 0] [subset] Conf₃[superscript x] ([mathcal] A)[superscript vee] ([mathbb R][superscript T]) parametrize a basis for [mathcal O] (Conf₃[superscript x] ([mathcal] A) )= [big o plus] (V[subscript alpha] [o times] V[supscript beta] [o times] V[subscript gamma])[subscript G] and encode the Littlewood-Richardson coefficients c[superscript gamma][subscript alpha beta]. I exhibit a unimodular p[superscript *] map that identifies W with the potential of Goncharov-Shen on Conf₃[superscript x] ([mathcal] A) and Xi with the Knutson-Tao hive cone.
*Advisors/Committee Members: Keel, Seán (advisor), Neitzke, Andrew (committee member), Allcock, Daniel (committee member), Speyer, David (committee member).*

Subjects/Keywords: Log Calabi-Yau; Cluster variety; Knutson-Tao hive cone; Littlewood-Richardson coefficients

…*log* *Calabi*-*Yau* with maximal boundary,
we can’t expect to get a canonical basis for O(Y… …made to pick out the *log* *Calabi*-*Yau* open subset U = Conf ×
3 (A). It is simply the… …Looijenga pair5 with D ample, and let U = Y \ D. U is an affine *log* *Calabi*
*Yau* with maximal… …flags. It is *log* *Calabi*-*Yau*– its complement is an anticanonical divisor D
in Conf 3 (B… …of D. We could
in principle use the *log* *Calabi*-*Yau* mirror symmetry machinery to study the…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

-4617-5386. (2017). GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/63035

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

Chicago Manual of Style (16^{th} Edition):

-4617-5386. “GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients.” 2017. Doctoral Dissertation, University of Texas – Austin. Accessed August 11, 2020. http://hdl.handle.net/2152/63035.

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

MLA Handbook (7^{th} Edition):

-4617-5386. “GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients.” 2017. Web. 11 Aug 2020.

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

Vancouver:

-4617-5386. GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2017. [cited 2020 Aug 11]. Available from: http://hdl.handle.net/2152/63035.

Author name may be incomplete

Council of Science Editors:

-4617-5386. GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients. [Doctoral Dissertation]. University of Texas – Austin; 2017. Available from: http://hdl.handle.net/2152/63035

Author name may be incomplete