Thesis advisor: Julia Elisenda Grigsby / Thesis advisor: David Treumann / In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar geometric techniques to prove analogous rank inequalities for the knot Floer homology of 2-periodic links. We use combinatorial and space-level techniques to prove analogous Smith-type inequalities for various flavors of Khovanov homology for periodic links in the 3-sphere of any prime periodicity. First, we prove a graded rank inequality for the annular Khovanov homology of 2-periodic links by showing grading obstructions to longer differentials in a localization spectral sequence. We remark that the same method can be extended to p-periodic links. Second, in joint work with Matthew Stoffregen, we construct a Z/p-equivariant stable homotopy type for odd and even, annular and non-annular Khovanov homologies, using Lawson, Lipshitz, and Sarkar's Burnside functor construction of a Khovanov stable homotopy type. Then, we identify the fixed-point sets and apply a version of the classical Smith inequality to obtain spectral sequences and rank inequalities relating the Khovanov homology of a periodic link with the annular Khovanov homology of the quotient link. As a corollary, we recover a rank inequality for Khovanov homology conjectured by Seidel and Smith's work on localization and symplectic Khovanov homology. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Critical velocity is defined as projectile striking velocity that causes instantaneous rupture of the specimen under transverse impact. The main goal of this dissertation was to determine the critical velocities of a Twaron^® 2040 warp yarn impacted by different round indenters. Special attention was placed to develop models to predict the critical velocities when transversely impacted by the indenters. An MTS 810 load frame was utilized to perform quasi-static transverse and uniaxial tension experiments to examine the stress concentration and the constitutive mechanical properties of the yarn which were used as an input to the models. A gas/powder gun was utilized to perform ballistic experiments to evaluate the critical velocities of a Twaron^® 2040 warp yarn impacted by four different type of round projectiles. These projectiles possessed a radius of curvature of 2 μm, 20 μm, 200 μm and 2 mm. The results showed that as the projectile radius of curvature increased, the critical velocity also increased. However, these experimental critical velocities showed a demonstrated reduction as compared to the classical theory. Post-mortem analysis via scanning electron microscopy on the recovered specimens revealed that the fibers failure surfaces changed from shear to fibrillation as the radius of curvature of the projectile increased. To improve the prediction capability, two additional models, Euler-Bernoulli beam and Hertzian contact, were developed to predict the critical velocity. For the Euler–Bernoulli beam model, the critical velocity was obtained by assuming the specimen ruptured instantaneously when the maximum flexural strain reached the ultimate tensile strain of the yarn upon impact. On the other hand, for the Hertzian contact model, the yarn was assumed to fail when the indentation depth was equivalent to the diameter of the yarn. Unlike Smith theory, the Euler-Bernoulli beam model underestimated the critical velocity for all cases. The Hertzian model was capable of predicting the critical velocities of a Twaron^® 2040 yarn transversely impacted by 2 μm and 20 μm round projectiles.