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California State University – Northridge

1. Chauhan, Sonia. Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties.

Degree: MS, Department of Mathematics, 2013, California State University – Northridge

URL: http://hdl.handle.net/10211.2/2509

For m C 2; consider the m??m determinantal variety of m??(m+1) matrices mod t2:
this is the variety Zm;m+1
m;2 obtained by considering generic m?? (m+ 1) matrices over
the ring F[t]~(t2), and setting the coe cients of powers of t of all m??m minors to
zero. The corresponding object mod t is of course the classical determinantal variety
Zm;m+1
m of m?? (m+ 1) matrices of rank less than m. We refer to the mod t2 variety
Zm;m+1
m;2 as the tangent bundle of the classical variety Zm;m+1
m.
In this thesis, we begin by providing a conjectured Groebner basis for the ideal
Im;m+1
m;2 which de nes Zm;m+1
m;2 , and as well, conjectured lead terms of the Groebner
basis. These conjectures were made based on explicit computations for the cases
2 B m B 6 that were done using the computer algebra system Singular ([8]). Since our
conjectured lead terms are squarefree for all m, we can construct Im, the Stanley-
Reisner simplicial complex attached to the ideal generated by our conjectured lead
terms. This complex has an existence of its own, independent of the conjectures, and
is an interesting object in its own right. The bulk of our thesis consists of the analysis
of this simplicial complex. For all values of m, we describe the facets of Im in terms
of their intersections with certain antidiagonals of a matrix of variables. Also for all
values of m, we derive a formula that counts the number of facets of this complex. This
number turns out to be the square of the degree of the classical variety Zm;m+1
m , which
yields evidence that our conjectures about the Groebner basis and their lead terms
are correct. In addition, for 2 B m B 6, we show that Im is shellable. Using standard
results, we conclude from the shellability of Im that for 2 B m B 6, the coordinate
ring of Zm;m+1
m;2 is Cohen-Macaulay, a property of great interest in algebraic geometry.
Further, for 2 B m B 6, we compute the Hilbert function of the Stanley-Reisner ring
of Im. The corresponding Hilbert series of Zm;m+1
m;2 pleasingly turns out to be the
square of the Hilbert series of the classical variety Zm;m+1
m , giving further credibility
to our conjectures.
*Advisors/Committee Members: Sethuraman, Bharath A. (advisor), Dye, John M. (committee member).*

Subjects/Keywords: Stanley Reisner complex; Dissertations, Academic – CSUN – Mathematics.

Record Details Similar Records

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

APA (6^{th} Edition):

Chauhan, S. (2013). Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties. (Masters Thesis). California State University – Northridge. Retrieved from http://hdl.handle.net/10211.2/2509

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

Chauhan, Sonia. “Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties.” 2013. Masters Thesis, California State University – Northridge. Accessed June 25, 2019. http://hdl.handle.net/10211.2/2509.

MLA Handbook (7^{th} Edition):

Chauhan, Sonia. “Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties.” 2013. Web. 25 Jun 2019.

Vancouver:

Chauhan S. Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties. [Internet] [Masters thesis]. California State University – Northridge; 2013. [cited 2019 Jun 25]. Available from: http://hdl.handle.net/10211.2/2509.

Council of Science Editors:

Chauhan S. Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties. [Masters Thesis]. California State University – Northridge; 2013. Available from: http://hdl.handle.net/10211.2/2509

2. Balas, Kevin. Rectangle packing problems with distinct representatives.

Degree: MS, Department of Mathematics, 2015, California State University – Northridge

URL: http://hdl.handle.net/10211.3/135144

In this thesis, we deal with problems involving finding the maximum area covered when packing rectangles into a bounding box, each containing a specified representative point. Given a set of n points in the unit square, U = [0, 1]^{2}, we choose n interior-disjoint axis parallel rectangles each containing one point and seek the packing with maximal area. There are several variants of the problem, depending on the constraints put upon the rectangles. For example, arbitrary rectangles, anchored rectangles (where the representative point is a vertex) or squares. We look at how to generate all maximal anchored rectangle packings for a given point set and show that we need only compare those packings in which all vertices, from all rectangles, lie on grid points induced by the given point set. Our main result is an exponential upper bound for the number of lower left anchored maximal rectangle packings for a given point set in the unit square. We revisit some previous results using squares instead of rectangles and introduce upper and lower bounds for the area of anchored and unanchored square packings.
*Advisors/Committee Members: Toth, Csaba D. (advisor), Dye, John M. (committee member).*

Subjects/Keywords: Rectangle packing; Dissertations, Academic – CSUN – Mathematics.

Record Details Similar Records

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

APA (6^{th} Edition):

Balas, K. (2015). Rectangle packing problems with distinct representatives. (Masters Thesis). California State University – Northridge. Retrieved from http://hdl.handle.net/10211.3/135144

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

Balas, Kevin. “Rectangle packing problems with distinct representatives.” 2015. Masters Thesis, California State University – Northridge. Accessed June 25, 2019. http://hdl.handle.net/10211.3/135144.

MLA Handbook (7^{th} Edition):

Balas, Kevin. “Rectangle packing problems with distinct representatives.” 2015. Web. 25 Jun 2019.

Vancouver:

Balas K. Rectangle packing problems with distinct representatives. [Internet] [Masters thesis]. California State University – Northridge; 2015. [cited 2019 Jun 25]. Available from: http://hdl.handle.net/10211.3/135144.

Council of Science Editors:

Balas K. Rectangle packing problems with distinct representatives. [Masters Thesis]. California State University – Northridge; 2015. Available from: http://hdl.handle.net/10211.3/135144