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.

Cette thèse contient quatre articles qui étudient les phénomènes de rigidité des transforma- tions hamiltoniennes des variétés symplectiques. Le premier article, rédigé en collaboration avec Egor Shelukhin, examine les obstructions à l’existence de symétries hamiltoniennes d’ordre fini sur une variété symplectique fermée (M,ω); c’est-à-dire de torsion hamiltonienne. En d’autres termes, nous étudions les sous- groupes finis du groupe des difféomorphismes hamiltoniens Ham(M,ω). Nous identifions trois sources principales d’obstructions: Contraintes topologiques. Inspirés par un résultat de Polterovich montrant que les variétés symplectiques asphériques n’admettent pas de torsion hamiltonienne, nous établissons que la présence d’un sous-groupe fini non trivial de Ham(M, ω) implique l’existence d’une sphère A ∈ π2(M) avec ⟨[ω],A⟩ > 0 et ⟨c1(M),A⟩ > 0. En particulier, les variétés symplectiques négativement monotones et les variétés symplectiques Calabi-Yau n’admettent pas de torsion hamiltonienne. Présence de courbes J-holomorphes. De manière générale, il y a de nombreux exemples de torsion hamiltonienne, par exemple toute rotation de la sphère de dimension deux par une fraction irrationnelle de π. Lorsque (M,ω) est positivement monotone, nous montrons que l’existence de torsion hamiltonienne impose une condition géométrique qui implique que les sphères J-holomorphes non constantes sont présentes partout. Ce phénomène était prédit dans une liste de problèmes contenue dans la monographie d’introduction de McDuff et de Salamon. Rigidité métrique spectrale. Notre analyse révèle que, pour les variétés symplectiques posi- tivement monotones, il existe un voisinage de l’identité dans Ham(M,ω) dans la topologie induite par la métrique spectrale qui ne contient aucun sous-groupe fini non trivial. Le principal résultat du deuxième article établit que, pour une large classe de variétés sym- plectiques, le flux d’un lacet de difféomorphismes symplectiques est entièrement déterminé par la classe d’homotopie de ses orbites. Comme application, nous obtenons de nouveaux exemples où l’existence d’un point fixe d’une action symplectique du cercle implique qu’elle est hamiltonienne et de nouvelles conditions assurant que le groupe de flux est trivial. De plus, nous obtenons des obstructions à l’existence d’éléments non triviaux de Symp0(M,ω) ayant un ordre fini. Le troisième article, rédigé en collaboration avec Han Lou, démontre une version de la conjecture de Hofer-Zehnder pour les variétés symplectiques fermées semi-positives dont l’homologie quantique est semi-simple; ce résultat généralise le travail révolutionnaire de Shelukhin sur les variétés symplectiques monotones. Le résultat montre qu’un difféomor- phisme hamiltonien possédant plus de points fixes contractiles, comptés homologiquement, que le nombre total de Betti de la variété doit avoir une infinité de points périodiques. La composante clé de la preuve est une nouvelle étude de l’effet de la réduction modulo p, un nombre premier, sur les bornes de l’homologie de Floer filtrée qui proviennent de la semi- simplicité. Cette étude repose sur la théorie des extensions algébriques des corps équipés d’une norme non-archimédienne. Le quatrième article, écrit en collaboration avec Habib Alizadeh et Dylan Cant, examine la déplaçabilité d’une sous-variété lagrangienne fermée L d’une variété symplectique convexe á l’infini par un difféomorphisme hamiltonien à support compact. Nous concluons qu’un difféomorphisme hamiltonien φ dont la norme spectrale est plus petite qu’un ħ(L) > 0 ne dépendant que de L ⊆ W ne peut pas déplacer L. De plus, nous établissons une estimation du nombre de valeurs d’action en terme de la longueur du cup-produit pour le nombre de valeurs d’action; lorsque L est rationnelle, cela implique une estimation du nombre de points d’intersection L ∩ φ(L) en terme de la longueur du cup-produit. Ainsi, nous montrons que le nombre de points fixes d’un difféomorphisme hamiltonien d’une variété symplectique fermée rationnelle (M, ω) dont la norme spectrale est plus petite que la constante de rationalité est au moins de 1 plus la longueur du cup-produit de M. / This thesis comprises four articles that study rigidity phenomena of Hamiltonian transfor- mations of symplectic manifolds. The first article, co-authored with Egor Shelukhin, examines obstructions to the existence of Hamiltonian symmetries of finite order on a closed symplectic manifold (M,ω); Hamil- tonian torsion. In other words, we study the finite subgroups of the group of Hamiltonian diffeomorphisms Ham(M, ω). We identify three primary sources of obstructions: Topological constraints. Inspired by a result of Polterovich showing that symplectically aspherical symplectic manifolds do not admit Hamiltonian torsion, we establish that the presence of a non-trivial finite subgroup of Ham(M,ω) implies that there exists a sphere A ∈ π2(M) with ⟨[ω],A⟩ > 0 and ⟨c1(M),A⟩ > 0. In particular, symplectically Calabi-Yau, and spherically negative-monotone symplectic manifolds do not admit Hamiltonian torsion. The presence of J-holomorphic curves. For general closed symplectic manifolds, there are plenty of examples of Hamiltonian torsion, for instance, any rotation of the two-sphere by an irrational fraction of π. When (M, ω) is spherically positive-monotone, we show the existence of Hamiltonian torsion imposes geometrical uniruledness, which implies that non-constant J-holomorphic spheres are ubiquitous. This phenomenon was predicted in a list of problems contained in the introductory monograph of McDuff and Salamon. The spectral metric rigidity. Our study reveals that for spherically positive-monotone (M, ω), there exists a neighbourhood of the identity in Ham(M,ω), in the topology induced by the spectral metric, that does not contain any non-trivial finite subgroup. The main result of the second article establishes that for a broad class of symplectic manifolds the flux of a loop of symplectic diffeomorphisms is completely determined by the homotopy class of its orbits. As an application, we obtain a new vanishing result for the flux group and new instances where the existence of a fixed point of a symplectic circle action implies that it is Hamiltonian. Moreover, we obtain obstructions to the existence of non-trivial elements of Symp0(M,ω) that have finite order. The third article, co-authored with Han Lou, proves a version of the Hofer-Zehnder conjec- ture for closed semipositive symplectic manifolds whose quantum homology is semisimple; this result generalizes the groundbreaking work of Shelukhin in the spherically positive- monotone setting. The result shows that a Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the mani- fold, must have infinitely many periodic points. The key component of the proof is a new study of the effect of reduction modulo a prime on the bounds on filtered Floer homology that arise from semisimplicity. It relies on the theory of algebraic extensions of non-Archimedean normed fields. The fourth article, co-authored with Habib Alizadeh and Dylan Cant, investigates the dis- placeability of a closed Lagrangian submanifold L of a convex-at-infinity symplectic manifold by a compactly supported Hamiltonian diffeomorphism. We conclude that a Hamiltonian diffeomorphism φ whose spectral norm is smaller than some ħ(L) > 0, depending only on L ⊂ W , cannot displace L. Furthermore, we establish a cup-length estimate for the number of action values; when L is rational, this implies a cup-length estimate on the number of intersection points L ∩ φ(L). As a corollary, we demonstrate that the number of fixed points of a Hamiltonian diffeomorphism of a closed rational symplectic manifold (M,ω), whose spectral norm is smaller than the rationality constant, is bounded below by one plus the cup-length of M.