This thesis presents a general background on discrete Morse theory, as developed by Robin Forman, as well as an introduction to computability and computational complexity. Since general point-set data equipped with a smooth structure can admit a triangulation, discrete Morse theory finds numerous applications in data analysis which can range from traffic control to geographical interpretation. Currently, there are various methods which convert point-set data to simplicial complexes or piecewise-smooth manifolds; however, this is not the focus of the thesis. Instead, this thesis will show that the Morse homology of such data is computable in the classical sense of Turing decidability, bound the complexity of finding the Morse homology of a given simplicial complex, and provide a measure for when this is more efficient than simplicial homology. / Master of Science / With the growing prevalence of data in the technological world, there is an emerging need to identify geometric properties (such as holes and boundaries) to data sets. However, it is often fruitless to employ an algorithm if it is known to be too computationally expensive (or even worse, not computable in the traditional sense). However, discrete Morse theory was originally formulated to provide a simplified manner of calculating these geometric properties on discrete sets. Therefore, this thesis outlines the general background of Discrete Morse theory and formulates the computational cost of computing specific geometric algorithms from the Discrete Morse perspective.

Thesis advisor: John A. Baldwin / Khovanov homology and Heegaard Floer homology have opened new horizons in knot theory and three-manifold topology, respectively. The two invariants have distinct origins, but the Khovanov homology of a link is related to the Heegaard Floer homology of its branched double cover by a spectral sequence constructed by Ozsváth and Szabó. In this thesis, we construct an equivalent spectral sequence with a much more transparent connection to Khovanov homology. This is the first step towards proving Seed and Szabó's conjecture that Szabó's geometric spectral sequence is isomorphic to Ozsváth and Szabó's spectral sequence. These spectral sequences connect information about contact structures contained in each invariant. We construct a braid conjugacy class invariant κ from Khovanov homology by adapting Floer-theoretic tools. There is a related transverse invariant which we conjecture to be effective. The conjugacy class invariant solves the word problem in the braid group among other applications. We have written a computer program to compute the invariant. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis (Ph. D.)--Duke University, 2008.

This thesis proposes an improved notion of coefficient system for Lagrangian Floer Homology which allows one to produce nontrivial invariants away from characteristic 2, even when coherent orientations of moduli spaces of Floer trajectories do not exist. This explains a suggestion of Witten. The invariant can be computed in examples, and the method explained below should be extensible to other Floer-theoretic invariants. The basic idea is that the moduli spaces of curves admit fundamental classes in homology with coefficients in the orientation lines of the moduli spaces, and the usual construction of coherent orientations actually shows that these fundamental classes naturally map to spaces of paths twisted with appropriate coefficient systems. These twisted path spaces admit enough algebraic structure to make sense of Floer homology with coefficients in these path spaces.

No description available.

published_or_final_version / Mathematics / Master / Master of Science

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

No description available.

Recent studies have identified the Class B g-protein coupled receptor (GPCR) pituitary adenylate cyclase activating polypeptide type 1 (PAC1R) as a key component in physiological stress management. Over-activity of neurological stress response systems due to prolonged or extreme exposure to traumatic events has led researchers to investigate PAC1R inhibition as a possible treatment for anxiety disorders such as post-traumatic stress disorder (PTSD). In 2008, Beebe and coworkers identified two such small molecule hydrazide antagonists and a general pharmacaphore for PAC1R inhibition. However, a relative dearth of information about Class B GPCRs in general, and PAC1R in specific, has significantly hindered progress toward the development of small molecule antagonists of PAC1R. The recent crystallization of the homologically similar glucagon receptor (GCGR) by Siu and coworkers in 2013, also a Class B receptor, has provided an experimentally resolved template from which to base computationally derived models of PAC1R. Initially, this research was focused towards synthesizing small molecule antagonists for PAC1R which were to be biologically screened via a qualitative western blot assay followed by a radioisotope binding assay for those hydrazides exhibiting down-stream signaling inhibitory capabilities. However, the resolution of the GCGR crystal structure shifted research objectives towards developing a homology model of PAC1R and evaluating that computationally created model with Beebe's known small molecule antagonists. Created using academic versions of on-line resources including UniProtKB, Swiss-Model and Maestro, a homology model for PAC1R is presented here. The model is validated and evaluated for the presence of conserved Class B GPCR residues and motifs, including expected disulfide bridges, a conserved tyrosine residue, a GWGxP motif, a conserved glutamic acid residue and the extension of the transmembrane helix 1 (TM1) into the extra-cellular domain. Having determined this virtual PAC1R an acceptable model, ligand docking studies of known antagonists to the receptor were undertaken using AutoDock Vina in conjunction with AutoDock Tools and PyMol. Computational docking results were evaluated via comparison of theoretical binding affinity results to Beebe's experimental data. Based on hydrogen bonding capabilities, several residues possibly key to the ligand-receptor binding complex are identified and include ASN 240, TYR 241 and HIST 365. Although the docking software does not identify non-bonding interactions other than hydrogen-bonding, the roles of additional proposed binding pocket residues are discussed in terms of hydrophobic interactions, π-π interactions and halogen bonding. These residues include TYR 161, PHE 196, VAL 203, PHE 204, ILE 209, LEU 210, VAL 237, TRP 297, PHE 362 and LEU 386. Although theoretical in nature, this reported homology modeling and docking exercise details a proposed binding site that may potentially further the development of drugs designed for the treatment of PTSD.

Grid homology is a combinatorial version of knot Floer homology. In a previous thesis, the author established an unoriented skein exact triangle for grid homology, giving a combinatorial proof of Manolescu’s unoriented skein exact triangle for knot Floer homology, and extending Manolescu’s result from Z/2Z coefficients to coefficients in any commutative ring. In Part II of this dissertation, after recalling the combinatorial proof mentioned above, we track the delta-gradings of the maps involved in the skein exact triangle, and use them to establish the Floer-homological sigma-thinness of quasi-alternating links over any commutative ring. Tangle Floer homology is a combinatorial extension of knot Floer homology to tangles, introduced by Petkova–Vertesi; it assigns an A-infinity-(bi)module to each tangle, so that the knot Floer homology of a link L obtained by gluing together tangles T_1, ..., T_n can be recovered from a tensor product of the A-infinity-(bi)modules assigned to the tangles T_i. Currently, tangle Floer homology has only been defined over Z/2Z. Part III of this dissertation presents a joint result with Ina Petkova, establishing an analogous unoriented skein relation for tangle Floer homology over Z/2Z, and tracking the delta-gradings involved.