Gauge Theoretic Invariants of, Dehn Surgeries on Knots

New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) connections on homology 3-spheres obtained by 1/k Dehn surgery on (2,q) torus knots. The methods are then applied to compute the SU(3) gauge theoretic Casson invariant (introduced in [H U Boden and C M Herald, The SU(3) Casson invariant for integral homology 3--spheres, J. Diff. Geom. 50 (1998) 147-206]) for Dehn surgeries on (2,q) torus knots for q=3,5,7 and 9.Comment: Version 3: minor corrections from version 2. Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper6.abs.htm

Further solutions of critical ABF RSOS models

The restricted SOS model of Andrews, Baxter and Forrester has been studied. The finite size corrections to the eigenvalue spectra of the transfer matrix of the model with a more general crossing parameter have been calculated. Therefore the conformal weights and the central charges of the non-unitary or unitary minimal conformal field have been extracted from the finite size corrections.Comment: Pages 11; revised versio

Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces

We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.Comment: 15 pages, Introduction is modifie

Jacobi fields in optimal control: one-dimensional variations

We study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have much more singularities than the control. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighborhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps

Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type

Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization

General Spectral Flow Formula for Fixed Maximal Domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.Comment: 22 page

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.Comment: 34 pages, 13 figures, 45 references, to appear in Ann Glob Anal Geom. The final publication will be available at http://www.springerlink.com. arXiv admin note: substantial text overlap with arXiv:math/040613

Multiple Cancer Testis Antigens Function To Support Tumor Cell Mitotic Fidelity

While the expression of genes that are normally involved in spermatogenesis is frequently detected in tumors, the extent to which these gene products are required for neoplastic behaviors is unclear. To begin to address their functional relevance to tumorigenesis, we identified a cohort of proteins which display synthetic lethality with paclitaxel in non-small-cell lung cancer and whose expression is biased toward testes and tumors. Remarkably, these testis proteins, FMR1NB, NXF2, MAGEA5, FSIP1, and STARD6, are required for accurate chromosome segregation in tumor cells. Their individual depletion enhances the generation of multipolar spindles, increases mitotic transit time, and induces micronucleation in response to an otherwise innocuous dose of paclitaxel. The underlying basis for abnormal mitosis is an alteration in microtubule function, as their depletion increases microtubule cytaster formation and disrupts microtubule stability. Given these observations, we hypothesize that reactivated testis proteins may represent unique tumor cell vulnerabilities which, if targeted, could enhance responsiveness to antimitotic therapy. Indeed, we demonstrate that combining paclitaxel with a small-molecule inhibitor of the gametogenic and tumor cell mitotic protein TACC3 leads to enhanced centrosomal abnormalities, activation of death programs, and loss of anchorage-independent growth

A rare cause of chronic mesenteric ischemia from fibromuscular dysplasia: a case report

<p>Abstract</p> <p>Introduction</p> <p>Chronic mesenteric ischemia is a condition that is classically associated with significant atherosclerosis of the abdominal arteries, causing postprandial abdominal pain out of proportion to physical examination. The abdominal pain is exacerbated after meals due to the shunting of blood away from the intestines to the stomach, causing relative ischemia. More than 95% of chronic mesenteric ischemia cases are due to atherosclerosis. We report the first known case of chronic mesenteric ischemia from fibromuscular dysplasia. To the best of our knowledge, this is also the first known case in the literature where postprandial abdominal pain was the presenting symptom of fibromuscular dysplasia.</p> <p>Case presentation</p> <p>A 44-year-old Caucasian woman with a history of hypertension and preeclampsia, who had taken oral contraceptive pills for 15 years, presented with an intractable, colicky abdominal pain of two weeks duration. This abdominal pain worsened with oral intake. It was also associated with diarrhea and vomiting. Physical examination revealed stage III hypertension out of proportion to her risk factors and diffuse abdominal pain without peritoneal signs. An abdominal computed tomography scan, completed in the emergency room, revealed nonspecific colitis. Laboratory work revealed leukocytosis with a left shift, an erythrocyte sedimentation rate of 79 and a C-reactive protein level of 100. She was started on intravenous flagyl and intravenous ciprofloxacin. However, all microbial cultures were negative including three cultures for clostridium difficile. Urine analysis revealed nephritic range proteinuria. The laboratory profile was within normal limits for perinuclear-anti-neutrophil cytoplasmic antibody, cytoplasmic-anti-neutrophil cytoplasmic antibody, anti-saccharomyces cerevisiae antibody, antinuclear antibody test, celiac profile, lactate, carbohydrate antigen-125 and thyroid stimulating hormone. A colonoscopy was completed, which revealed diffuse colonic lymphoid reactive hyperplasia. A small bowel series was negative for any inflammation. An indium scan, pan-computed tomography scan and transvaginal ultrasound were also negative. Magnetic resonance angiography of her abdomen revealed proximal superior mesenteric artery stenosis, which was confirmed by computed tomography angiogram findings of severe proximal and distal superior mesenteric artery stenosis, consistent with the appearance of fibromuscular dysplasia on angiography in the absence of vasculitis or atherosclerotic disease. The patient's superior mesenteric artery stenosis was subsequently angioplastied suboptimally and had to be stented with an Angioplus stent. One month after she was admitted, her abdominal pain and tolerance to oral feeds improved tremendously.</p> <p>Conclusion</p> <p>Fibromuscular dysplasia most commonly presents with renal artery stenosis, which rarely causes abdominal pain. This case illustrates how fibromuscular dysplasia can present as a rare cause of chronic mesenteric ischemia, similar to chronic mesenteric ischemia from atherosclerosis.</p