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Let X be a compact oriented Riemannian manifold and let φ : X → S 1 be a circle-valued Morse function. Under some mild assumptions on φ, we prove a formula relating: (a) the number of closed orbits of the gradient flow of φ of any given degree; (b) the torsion of a " Morse complex " , which counts gradient flow lines between critical points of φ; and (c) a(More)
The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance—the Reidemeister torsion τ F of the Floer complex of (possibly non-hamiltonian) symplectomorphisms. τ F turns out not to be invariant under hamiltonian isotopies, but we(More)
This is an expansion on my talk at the Geometry and Topol-ogy conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an intermediate version of Floer theory, which(More)
Protein arginine methyltransferase (PRMT) 1 is the most conserved and widely distributed PRMT in eukaryotes. PRMT8 is a vertebrate-restricted paralogue of PRMT1 with an extra N-terminal sequence and brain-specific expression. We use zebrafish (Danio rerio) as a vertebrate model to study PRMT8 function and putative redundancy with PRMT1. The transcripts of(More)
Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface auto-morphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer cohomology and the corresponding(More)
The availability of adequate cancer stem cells or cancer stem-like cell (CSC) is important in cancer study. From ovarian cancer cell lines, SKOV3 and OVCAR3, we induced peritoneal ascites tumors in immunodeficient mice. Among the cells (SKOV3.PX1 and OVCAR3.PX1) from those tumors, we sorted both CD44 and CD133 positive cells (SKOV3.PX1_133+44+,(More)
This is the second part of an article in two parts , which builds the foundation of a Floer-theoretic invariant, I F. (See [Pt1] for part I). Having constructed I F and outlined a proof of its invariance based on bi-furcation analysis in part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in part I. These(More)
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