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University of Illinois – Urbana-Champaign

1. DiPasquale, Michael Robert. Splines on polytopal complexes.

Degree: PhD, Mathematics, 2015, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/87949

This thesis concerns the algebra C^{r}(\PC) of C^{r} piecewise polynomial functions (splines) over a subdivision by convex polytopes \PC of a domain Ω\subset\R^{n}. Interest in this algebra arises in a wide variety of contexts, ranging from approximation theory and computer-aided geometric design to equivariant cohomology and GKM theory. A primary goal in approximation theory is to construct bases of the vector space C^{r}_{d}(\PC) of splines of degree at most d on \PC, although even computing the dimension of this space proves to be challenging. From the perspective of GKM theory it is more important to have a good description of the generators of C^{r}(\PC) as an algebra; one would especially like to know the multiplication table for these generators (the case r=0 is of particular interest). For certain choices of \PC and r there are beautiful answers to these questions, but in most cases the answers are still out of reach.
In the late 1980s Billera formulated an approach to spline theory using the tools of commutative algebra, homological algebra, and algebraic geometry~, but focused primarily on the simplicial case. This thesis details a number of results that can be obtained using this algebraic perspective, particularly for splines over subdivisions by convex polytopes.
The first three chapters of the thesis are devoted to introducing splines and providing some background material.
In Chapter~\ref{ch:Introduction} we give a brief history of spline theory. In Chapter~\ref{ch:CommutativeAlgebra} we record results from commutative algebra which we will use, mostly without proof. In Chapter~\ref{ch:SplinePreliminaries} we set up the algebraic approach to spline theory, along with our choice of notation which differs slightly from the literature.
In Chapter~\ref{ch:Continuous} we investigate the algebraic structure of continous splines over a central polytopal complex (equivalently a fan) in \R^{3}. We give an example of such a fan where the link of the central vertex is homeomorphic to a 2-ball, and yet the C^{0} splines on this fan are not free as an algebra over the underlying polynomial ring in three variables, providing a negative answer to a question of Schenck~\cite[Question~3.3]{Chow}. This is interesting for several reasons. First, this is very different behavior from the case of simplicial fans, where the ring of continuous splines is always free if the link of the central vertex is homeomorphic to a disk. Second, from the perspective of GKM theory and toric geometry, it means that the multiplication tables of generators will be much more complicated. In the remainder of the chapter we investigate criteria that may be used to detect freeness of continuous splines (or lack thereof).
From the perspective of approximation theory, it is important to have a basis for the vector space C^{r}_{d}(\PC) of splines of degree at most d which is `locally supported' in some reasonable sense. For simplicial complexes, such a basis consists of splines which are…
*Advisors/Committee Members: Schenck, Henry (advisor), Dutta, Sankar (Committee Chair), Yong, Alexander (committee member), Nevins, Thomas A. (committee member).*

Subjects/Keywords: Algebraic Splines; Commutative Algebra

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APA (6^{th} Edition):

DiPasquale, M. R. (2015). Splines on polytopal complexes. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/87949

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

DiPasquale, Michael Robert. “Splines on polytopal complexes.” 2015. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed July 02, 2020. http://hdl.handle.net/2142/87949.

MLA Handbook (7^{th} Edition):

DiPasquale, Michael Robert. “Splines on polytopal complexes.” 2015. Web. 02 Jul 2020.

Vancouver:

DiPasquale MR. Splines on polytopal complexes. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2015. [cited 2020 Jul 02]. Available from: http://hdl.handle.net/2142/87949.

Council of Science Editors:

DiPasquale MR. Splines on polytopal complexes. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2015. Available from: http://hdl.handle.net/2142/87949

University of Illinois – Urbana-Champaign

2. Seceleanu, Alexandra. The syzygy theorem and the weak Lefschetz Property.

Degree: PhD, 0439, 2011, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/26091

This thesis consists of two research topics in commutative algebra.
In the first chapter, a comprehensive analysis is given of the Weak Lefschetz property in the case of ideals generated by powers of linear forms in a standard graded polynomial ring of characteristic zero. The main point to take away from these developments is that, via the inverse system dictionary,
one is able to relate the failure of the Weak Lefschetz property to the geometry of the fat point scheme associated to the powers of linear forms. As a natural outcome of this research we describe conjectures on the asymptotical
behavior of the family of ideals that is being studied.
In the second chapter, we solve some relevant cases of the Evans-Griffith syzygy conjecture in the case of (regular) local rings of unramif ed mixed characteristic p, with the case of syzygies of prime ideals of Cohen-Macaulay
local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by p^s, s > 0, that have finite projective dimension over a hypersurface ring. Our main results are
obtained as a byproduct of two theorems that establish a weak order ideal property for kth syzygy modules under conditions allowing for comparison ofsyzygies over the original ring versus the hypersurface ring.
*Advisors/Committee Members: Schenck, Henry K. (advisor), Griffith, Phillip A. (Committee Chair), Schenck, Henry K. (committee member), Dutta, Sankar P. (committee member), Evans, Graham (committee member).*

Subjects/Keywords: syzygy; syzygy theorem; weak Lefschetz Property; fat points; homological conjectures

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

APA (6^{th} Edition):

Seceleanu, A. (2011). The syzygy theorem and the weak Lefschetz Property. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/26091

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

Seceleanu, Alexandra. “The syzygy theorem and the weak Lefschetz Property.” 2011. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed July 02, 2020. http://hdl.handle.net/2142/26091.

MLA Handbook (7^{th} Edition):

Seceleanu, Alexandra. “The syzygy theorem and the weak Lefschetz Property.” 2011. Web. 02 Jul 2020.

Vancouver:

Seceleanu A. The syzygy theorem and the weak Lefschetz Property. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2011. [cited 2020 Jul 02]. Available from: http://hdl.handle.net/2142/26091.

Council of Science Editors:

Seceleanu A. The syzygy theorem and the weak Lefschetz Property. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2011. Available from: http://hdl.handle.net/2142/26091

3. Beder, Jesse. The Grade Conjecture and asymptotic intersection multiplicity.

Degree: PhD, 0439, 2013, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/42274

In this thesis, we study Peskine and Szpiro's Grade Conjecture and its connection with asymptotic intersection multiplicity χ_∞. Given an A-module M of finite projective dimension and a system of parameters x_{1}, …, x_{r} for M, we show, under certain assumptions on M, that χ_∞(M, A/\underline{x}) > 0. We also give a necessary and sufficient condition on M for the existence of a system of parameters \underline{x} with χ_∞(M, A/\underline{x}) > 0.
We then prove that if the Grade Conjecture holds for a given module M, then there is a system of parameters \underline{x} such that χ_∞(M, A/\underline{x}) > 0. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic.
*Advisors/Committee Members: Dutta, Sankar P. (advisor), Griffith, Phillip A. (Committee Chair), Dutta, Sankar P. (committee member), Schenck, Henry K. (committee member), Haboush, William J. (committee member).*

Subjects/Keywords: commutative algebra; grade conjecture; characteristic p; frobenius; intersection multiplicity

Record Details Similar Records

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

APA (6^{th} Edition):

Beder, J. (2013). The Grade Conjecture and asymptotic intersection multiplicity. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/42274

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

Beder, Jesse. “The Grade Conjecture and asymptotic intersection multiplicity.” 2013. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed July 02, 2020. http://hdl.handle.net/2142/42274.

MLA Handbook (7^{th} Edition):

Beder, Jesse. “The Grade Conjecture and asymptotic intersection multiplicity.” 2013. Web. 02 Jul 2020.

Vancouver:

Beder J. The Grade Conjecture and asymptotic intersection multiplicity. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2013. [cited 2020 Jul 02]. Available from: http://hdl.handle.net/2142/42274.

Council of Science Editors:

Beder J. The Grade Conjecture and asymptotic intersection multiplicity. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2013. Available from: http://hdl.handle.net/2142/42274