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University of Oregon

1. Kutler, Max. Faithful tropicalization of hypertoric varieties.

Degree: PhD, Department of Mathematics, 2017, University of Oregon

URL: http://hdl.handle.net/1794/22756 

The hypertoric variety M_A defined by an arrangement A of affine hyperplanes admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. In this thesis, we explicitly describe the polyhedral structure of this tropicalization and calculate the fibers of the tropicalization map. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map. Advisors/Committee Members: Proudfoot, Nicholas (advisor).

Subjects/Keywords: Hypertoric varieties; Matroids; Non-Archimedean Geometry; Tropical geometry

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

APA (6th Edition):

Kutler, M. (2017). Faithful tropicalization of hypertoric varieties. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/22756

Chicago Manual of Style (16th Edition):

Kutler, Max. “Faithful tropicalization of hypertoric varieties.” 2017. Doctoral Dissertation, University of Oregon. Accessed January 22, 2021. http://hdl.handle.net/1794/22756.

MLA Handbook (7th Edition):

Kutler, Max. “Faithful tropicalization of hypertoric varieties.” 2017. Web. 22 Jan 2021.

Vancouver:

Kutler M. Faithful tropicalization of hypertoric varieties. [Internet] [Doctoral dissertation]. University of Oregon; 2017. [cited 2021 Jan 22]. Available from: http://hdl.handle.net/1794/22756.

Council of Science Editors:

Kutler M. Faithful tropicalization of hypertoric varieties. [Doctoral Dissertation]. University of Oregon; 2017. Available from: http://hdl.handle.net/1794/22756

2. Shenfeld, Daniel. Abelianization of Stable Envelopes in Symplectic Resolutions .

Degree: PhD, 2013, Princeton University

URL: http://arks.princeton.edu/ark:/88435/dsp01ws859f77c 

Stable envelopes, introduced by Maulik and Okounkov, form a basis for the equivariant cohomology of symplectic resolutions. We study the case of Nakajima quiver varieties, where the resolution is a hyperk"ahler quotient. We relate the stable basis to that of the associated quotient by a maximal torus, and obtain a formula for the transition between the stable basis and the fixed point basis, using the root system and combinatorial data from the torus quotient. We compute the transition matrix explicitly in the case of cotangent bundles to partial flag varieties, and show that for Grassmannians, it is given by rational Schur polynomials. This recovers in a geometric way the diagonalization of the Hamiltonian in the XXX spin chain. As a second application, we study the case of the Hilbert scheme of points on the plane, and obtain a novel formula expressing Schur polynomials in terms of Jack polynomials. Advisors/Committee Members: Okounkov, Andrei (advisor).

Subjects/Keywords: hypertoric varieties; symmetric polynomials; symplectic resolutions

…called hypertoric varieties; much like toric varieties, their geometry can be described by… …combinatorial methods, in this case by using hyperplane arrangements. Smooth hypertoric varieties are… …cohomology ring of hypertoric varieties, appears in [23]. From the point of view of the… …in [22]. In chapter 3 we introduce hypertoric varieties and give some background… …scheme of n points on C2 . 6 Hypertoric varieties and abelianization Another family of… 

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

APA (6th Edition):

Shenfeld, D. (2013). Abelianization of Stable Envelopes in Symplectic Resolutions . (Doctoral Dissertation). Princeton University. Retrieved from http://arks.princeton.edu/ark:/88435/dsp01ws859f77c

Chicago Manual of Style (16th Edition):

Shenfeld, Daniel. “Abelianization of Stable Envelopes in Symplectic Resolutions .” 2013. Doctoral Dissertation, Princeton University. Accessed January 22, 2021. http://arks.princeton.edu/ark:/88435/dsp01ws859f77c.

MLA Handbook (7th Edition):

Shenfeld, Daniel. “Abelianization of Stable Envelopes in Symplectic Resolutions .” 2013. Web. 22 Jan 2021.

Vancouver:

Shenfeld D. Abelianization of Stable Envelopes in Symplectic Resolutions . [Internet] [Doctoral dissertation]. Princeton University; 2013. [cited 2021 Jan 22]. Available from: http://arks.princeton.edu/ark:/88435/dsp01ws859f77c.

Council of Science Editors:

Shenfeld D. Abelianization of Stable Envelopes in Symplectic Resolutions . [Doctoral Dissertation]. Princeton University; 2013. Available from: http://arks.princeton.edu/ark:/88435/dsp01ws859f77c

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