Physical Sciences and Mathematics Commons^™

Computing Intersection Multiplicity Via Triangular Decomposition, Paul Vrbik

Electronic Thesis and Dissertation Repository

Fulton’s algorithm is used to calculate the intersection multiplicity of two plane curves about a rational point. This work extends Fulton’s algorithm first to algebraic points (encoded by triangular sets) and then, with some generic assumptions, to l many hypersurfaces.

Out of necessity, we give a standard-basis free method (i.e. practically efficient method) for calculating tangent cones at points on curves.

Cohomology Of Absolute Galois Groups, Claudio Quadrelli

Electronic Thesis and Dissertation Repository

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-p case, i.e., one would like to know which pro-p groups occur as maximal pro-p Galois groups, i.e., maximal pro-p quotients of absolute Galois groups. Indeed, pro-p groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group.

We define a new class of pro-p groups, called Bloch-Kato …

Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, Youlong Yan

Electronic Thesis and Dissertation Repository

The derived category of coherent sheaves on a smooth projective variety is an important object of study in algebraic geometry. One important device relevant for this study is the notion of tilting sheaf.

This thesis is concerned with the existence of tilting sheaves on some smooth projective varieties. The main technique we use in this thesis is Galois descent theory. We first construct tilting bundles on general Brauer-Severi varieties. Our main result shows the existence of tilting bundles on some Brauer-Severi schemes. As an application, we prove that there are tilting bundles on an arithmetic toric variety whose toric variety …

Polynomial Identities On Algebras With Actions, Chris Plyley

Electronic Thesis and Dissertation Repository

When an algebra is endowed with the additional structure of an action or a grading, one can often make striking conclusions about the algebra based on the properties of the structure-induced subspaces. For example, if A is an associative G-graded algebra such that the homogeneous component A₁ satisfies an identity of degree d, then Bergen and Cohen showed that A is itself a PI-algebra. Bahturin, Giambruno and Riley later used combinatorial methods to show that the degree of the identity satisfied by A is bounded above by a function of d and |G|. Utilizing a …

Ghost Number Of Group Algebras, Gaohong Wang

Electronic Thesis and Dissertation Repository

The generating hypothesis for the stable module category of a finite group is the statement that if a map in the thick subcategory generated by the trivial representation induces the zero map in Tate cohomology, then it is stably trivial. It is known that the generating hypothesis fails for most groups. Generalizing work done for p-groups, we define the ghost number of a group algebra, which is a natural number that measures the degree to which the generating hypothesis fails. We describe a close relationship between ghost numbers and Auslander-Reiten triangles, with many results stated for a general projective class …

Optimizing The Analysis Of Electroencephalographic Data By Dynamic Graphs, Mehrsasadat Golestaneh

Electronic Thesis and Dissertation Repository

The brain’s underlying functional connectivity has been recently studied using tools offered by graph theory and network theory. Although the primary research focus in this area has so far been mostly on static graphs, the complex and dynamic nature of the brain’s underlying mechanism has initiated the usage of dynamic graphs, providing groundwork for time sensi- tive and finer investigations. Studying the topological reconfiguration of these dynamic graphs is done by exploiting a pool of graph metrics, which describe the network’s characteristics at different scales. However, considering the vast amount of data generated by neuroimaging tools, heavy computation load and …

Classification Of W-Groups Of Pythagorean Formally Real Fields, Fatemeh Bagherzadeh Golmakani

Electronic Thesis and Dissertation Repository

In this work we consider the Galois point of view in determining the structure of
a space of orderings of fields via considering small Galois quotients of absolute Galois
groups G _F of Pythagorean formally real fields. Galois theoretic, group theoretic and
combinatorial arguments are used to reduce the structure of W-groups.

Extracting Vessel Structure From 3d Image Data, Yuchen Zhong

Electronic Thesis and Dissertation Repository

This thesis is focused on extracting the structure of vessels from 3D cardiac images. In many biomedical applications it is important to segment the vessels preserving their anatomically-correct topological structure. That is, the final result should form a tree. There are many technical challenges when solving this image analysis problem: noise, outliers, partial volume. In particular, standard segmentation methods are known to have problems with extracting thin structures and with enforcing topological constraints. All these issues explain why vessel segmentation remains an unsolved problem despite years of research.

Our new efforts combine recent advances in optimization-based methods for image analysis …