Automatic transversality in contact homology I: Regularity

This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological invariant in the presence of multiply covered curves. We then introduce a large subclass of dynamically convex contact forms in dimension 3, termed dynamically separated, and demonstrate automatic transversality holds, therby allowing us to define the desired chain complex. The Reeb orbits of dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder index under iteration, typically up to large action; see Definition 1.15 These contact forms arise naturally as perturbations of Morse-Bott contact forms such as those associated to $S^1$-bundles. In subsequent work, we give a direct proof of invariance for this subclass and, when further proportionality holds between the index and action, powerful geometric computations in a wide variety of examples.Comment: 68 pages, added more information about bad Reeb orbits, added a proof of a beloved folk theorem concerning the factorization of multiply covered curves, contains expository revisions helpfully suggested by the refere

Radio Project in the Land of the Midnight Sun

Getting bogged down with the daily routine of broadcasting? Or weary of entertaining foreign visitors? Then go off on a short-term professional foreign visit yourself

LOCAL GOVERNMENT LAND USE RESTRICTIONS AND SELECTED FIRST AMENDMENT ISSUES

A local government\u27s power to enact zoning regulations falls within the general power to provide for the health, safety, and welfare of its citizenry. This thesis addresses a few selected First Amendment issues as they apply to zoning and land use restrictions in Georgia. Free speech review of zoning ordinances applies to zoning for adult sex businesses, such as adult book stores and cinemas. The First Amendment balancing test that is applicable to adult entertainment ordinances is discussed in Chapter One. The free speech impact of restrictions on signs and billboards is discussed in Chapter Two. Finally, in Chapter Three, zoning and the free exercise clause of the First Amendment is discussed as it applies to the regulation of religious uses of land. The United States Congress enacted legislation to protect religious institutions from government regulation. However, the legislation was unconstitutional. Georgia courts do not recognize First Amendment protection for religious uses in the face of zoning regulations. A traditional zoning analysis is applied with regulations being invalidated only if they are unreasonable or arbitrary

Torus knot filtered embedded contact homology of the tight contact 3-sphere

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n-1) via a quotient by Weiler. While straightforward toric constructions can be used to understand the ECH chain complexes of open books along the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3-sphere in terms of their presentation as open books and as Seifert fibered spaces. We provide Morse-Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg-Witten theory developed by Hutchings and Taubes, and use them to compute the T(2,q) knot filtered embedded contact homology, for q odd and positive. In the sequel we complete the computation for positive T(p,q) knots (where there is a nonvanishing differential) and use our results to deduce quantitative existence results for torus knotted Reeb dynamics on the tight 3-sphere and the mean action of area preserving diffeomorphisms of once punctured surfaces of arbitrary genus arising as Seifert surfaces of positive torus knots.Comment: 85 pages, arXiv insists the primary is GT rather than S

Embedded contact homology of prequantization bundles

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism. We fill in some technical details, including the Morse-Bott direct limit argument and some writhe bounds. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings--Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang. In a subsequent paper, we plan to extend these methods for Seifert fiber spaces and relate the ECH U-map to orbifold Gromov-Witten invariants of the base.Comment: 85 page

Co-delivery of human cancer-testis antigens with adjuvant in protein nanoparticles induces higher cell-mediated immune responses.

Nanoparticles have attracted considerable interest as cancer vaccine delivery vehicles for inducing sufficient CD8+ T cell-mediated immune responses to overcome the low immunogenicity of the tumor microenvironment. Our studies described here are the first to examine the effects of clinically-tested human cancer-testis (CT) peptide epitopes within a synthetic nanoparticle. Specifically, we focused on two significant clinical CT targets, the HLA-A2 restricted epitopes of NY-ESO-1 and MAGE-A3, using a viral-mimetic packaging strategy. Our data shows that simultaneous delivery of a NY-ESO-1 epitope (SLLMWITQV) and CpG using the E2 subunit assembly of pyruvate dehydrogenase (E2 nanoparticle), resulted in a 25-fold increase in specific IFN-γ secretion in HLA-A2 transgenic mice. This translated to a 15-fold increase in lytic activity toward target cancer cells expressing the antigen. Immunization with a MAGE-A3 epitope (FLWGPRALV) delivered with CpG in E2 nanoparticles yielded an increase in specific IFN-γ secretion and cell lysis by 6-fold and 9-fold, respectively. Furthermore, combined delivery of NY-ESO-1 and MAGE-A3 antigens in E2 nanoparticles yielded an additive effect that increased lytic activity towards cells bearing NY-ESO-1+ and MAGE-A3+. Our investigations demonstrate that formulation of CT antigens within a nanoparticle can significantly enhance antigen-specific cell-mediated responses, and the combination of the two antigens in a vaccine can preserve the increased individual responses that are observed for each antigen alone