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1. Koonz, Jennifer. Properties of Singular Schubert Varieties.

Degree: PhD, Mathematics, 2013, U of Massachusetts : PhD

URL: https://scholarworks.umass.edu/open_access_dissertations/839

This thesis deals with the study of Schubert varieties, which are subsets of flag varieties indexed by elements of Weyl groups. We start by defining Lascoux elements in the Hecke algebra, and showing that they coincide with the Kazhdan-Lusztig basis elements in certain cases. We then construct a resolution (<em>Z_{w}, π</em>) of the Schubert variety* X*_{w} for which <em>Rπ_{*}(C[l(w)])</em> is a sheaf on *X*_{w} whose expression in the Hecke algebra is closely related to the Lascoux element. We also define two new polynomials which coincide with the intersection cohomology Poincar\'e polynomial in certain cases. In the final chapter, we discuss some interesting combinatorial results concerning Bell and Catalan numbers which arose throughout the course of this work.
*Advisors/Committee Members: Eric Sommers, Tom Braden, Julianna Tymoczko.*

Subjects/Keywords: Combinatorics; Hecke Algebra; Intersection Cohomology; Kazhdan-Lusztig Polynomials; Schubert Varieties; Mathematics

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APA (6^{th} Edition):

Koonz, J. (2013). Properties of Singular Schubert Varieties. (Doctoral Dissertation). U of Massachusetts : PhD. Retrieved from https://scholarworks.umass.edu/open_access_dissertations/839

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

Koonz, Jennifer. “Properties of Singular Schubert Varieties.” 2013. Doctoral Dissertation, U of Massachusetts : PhD. Accessed December 05, 2020. https://scholarworks.umass.edu/open_access_dissertations/839.

MLA Handbook (7^{th} Edition):

Koonz, Jennifer. “Properties of Singular Schubert Varieties.” 2013. Web. 05 Dec 2020.

Vancouver:

Koonz J. Properties of Singular Schubert Varieties. [Internet] [Doctoral dissertation]. U of Massachusetts : PhD; 2013. [cited 2020 Dec 05]. Available from: https://scholarworks.umass.edu/open_access_dissertations/839.

Council of Science Editors:

Koonz J. Properties of Singular Schubert Varieties. [Doctoral Dissertation]. U of Massachusetts : PhD; 2013. Available from: https://scholarworks.umass.edu/open_access_dissertations/839

2. Boland, Patrick Michael. Geometry of Satake and Toroidal Compactifications.

Degree: PhD, Mathematics, 2010, U of Massachusetts : PhD

URL: https://scholarworks.umass.edu/open_access_dissertations/270

In [JM02, section 14], Ji and MacPherson give new constructions of the Borel – Serre and reductive Borel – Serre compactifications [BS73, Zuc82] of a locally symmetric space. They use equivalence classes of eventually distance minimizing (EDM) rays to describe the boundaries of these compactications. The primary goal of this thesis is to construct the Satake compactifications of a locally symmetric space [Sat60a] using finer equivalence relations on EDM rays. To do this, we first construct the Satake compactifications of the global symmetric space [Sat60b] with equivalence classes of geodesics in the symmetric space. We then define equivalence relations on EDM rays using geometric properties of their lifts in the symmetric space. We show these equivalence classes are in one-to-one correspondence with the points of the Satake boundary. As a secondary goal, we outline the construction of the toroidal compactifications of Hilbert modular varieties [Hir71, Ehl75] using a larger class of "toric curves" and equivalence relations that depend on the compactications' defining combinatorial data.
*Advisors/Committee Members: Paul E. Gunnells, Tom Braden, David Kastor.*

Subjects/Keywords: Mathematics; Statistics and Probability

Record Details Similar Records

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

APA (6^{th} Edition):

Boland, P. M. (2010). Geometry of Satake and Toroidal Compactifications. (Doctoral Dissertation). U of Massachusetts : PhD. Retrieved from https://scholarworks.umass.edu/open_access_dissertations/270

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

Boland, Patrick Michael. “Geometry of Satake and Toroidal Compactifications.” 2010. Doctoral Dissertation, U of Massachusetts : PhD. Accessed December 05, 2020. https://scholarworks.umass.edu/open_access_dissertations/270.

MLA Handbook (7^{th} Edition):

Boland, Patrick Michael. “Geometry of Satake and Toroidal Compactifications.” 2010. Web. 05 Dec 2020.

Vancouver:

Boland PM. Geometry of Satake and Toroidal Compactifications. [Internet] [Doctoral dissertation]. U of Massachusetts : PhD; 2010. [cited 2020 Dec 05]. Available from: https://scholarworks.umass.edu/open_access_dissertations/270.

Council of Science Editors:

Boland PM. Geometry of Satake and Toroidal Compactifications. [Doctoral Dissertation]. U of Massachusetts : PhD; 2010. Available from: https://scholarworks.umass.edu/open_access_dissertations/270

3. McDaniel, Chris Ray. Geometric and Combinatorial Aspects of 1-Skeleta.

Degree: PhD, Mathematics, 2010, U of Massachusetts : PhD

URL: https://scholarworks.umass.edu/open_access_dissertations/250

In this thesis we investigate 1-skeleta and their associated cohomology rings. 1-skeleta
arise from the 0- and 1-dimensional orbits of a certain class of manifold admitting a
compact torus action and many questions that arise in the theory of 1-skeleta are rooted
in the geometry and topology of these manifolds. The three main results of this work
are: a lifting result for 1-skeleta (related to extending torus actions on manifolds), a
classification result for certain 1-skeleta which have the Morse package (a property of
1-skeleta motivated by Morse theory for manifolds) and two constructions on 1-skeleta
which we show preserve the Lefschetz package (a property of 1-skeleta motivated by
the hard Lefschetz theorem in algebraic geometry). A corollary of this last result is a
conceptual proof (applicable in certain cases) of the fact that the coinvariant ring of a
finite reflection group has the strong Lefschetz property.
*Advisors/Committee Members: Tom Braden, Eduardo Cattani, Paul Gunnells.*

Subjects/Keywords: 1-Skeleta; Equivariant cohomology; GKM manifolds; Mathematics; Statistics and Probability

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

APA (6^{th} Edition):

McDaniel, C. R. (2010). Geometric and Combinatorial Aspects of 1-Skeleta. (Doctoral Dissertation). U of Massachusetts : PhD. Retrieved from https://scholarworks.umass.edu/open_access_dissertations/250

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

McDaniel, Chris Ray. “Geometric and Combinatorial Aspects of 1-Skeleta.” 2010. Doctoral Dissertation, U of Massachusetts : PhD. Accessed December 05, 2020. https://scholarworks.umass.edu/open_access_dissertations/250.

MLA Handbook (7^{th} Edition):

McDaniel, Chris Ray. “Geometric and Combinatorial Aspects of 1-Skeleta.” 2010. Web. 05 Dec 2020.

Vancouver:

McDaniel CR. Geometric and Combinatorial Aspects of 1-Skeleta. [Internet] [Doctoral dissertation]. U of Massachusetts : PhD; 2010. [cited 2020 Dec 05]. Available from: https://scholarworks.umass.edu/open_access_dissertations/250.

Council of Science Editors:

McDaniel CR. Geometric and Combinatorial Aspects of 1-Skeleta. [Doctoral Dissertation]. U of Massachusetts : PhD; 2010. Available from: https://scholarworks.umass.edu/open_access_dissertations/250