Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions

We introduce a notion of a weak Poisson structure on a manifold $M$ modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra \cA \subeq C^\infty(M) which has to satisfy a non-degeneracy condition (the differentials of elements of \cA separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form $\kappa$ on a Lie algebra \g and a $\kappa$-skew-symmetric derivation $D$ a weak affine Poisson structure on \g itself. This leads naturally to a concept of a Hamiltonian $G$-action on a weak Poisson manifold with a \g-valued momentum map and hence to a generalization of quasi-hamiltonian group actions

Unitary Representations of Unitary Groups

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group \U(\cH) of a real, complex or quaternionic separable Hilbert space and the subgroup \U_\infty(\cH), consisting of those unitary operators $g$ for which g - \1 is compact. The Kirillov--Olshanski theorem on the continuous unitary representations of the identity component \U_\infty(\cH)_0 asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell's Theorem, asserting that the separable unitary representations of \U(\cH), for a separable Hilbert space \cH, are uniquely determined by their restriction to \U_\infty(\cH)_0. For the $10$ classical infinite rank symmetric pairs $(G,K)$ of non-unitary type, such as (\GL(\cH),\U(\cH)), we also show that all separable unitary representations are trivial.Comment: 42 page

Large N limit of SO(N) scalar gauge theory

In this paper we study the large $N_c$ limit of SO(N_c) gauge theory coupled to a real scalar field following ideas of Rajeev. We see that the phase space of this resulting classical theory is Sp_1(H)/U(H_+) which is the analog of the Siegel disc in infinite dimensions. The linearized equations of motion give us a version of the well-known 't Hooft equation of two dimensional QCD.Comment: 16 pages, no figure

Localization via Automorphisms of the CARs. Local gauge invariance

The classical matter fields are sections of a vector bundle E with base manifold M. The space L^2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^\infty(M) on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra A, of the canonical anticommutation relations on L^2(E), with which we can perform the analogous localization. That is, the net structure of the CAR, A, w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps. As a corollary, we prove a well-known "folk theorem," that the algebra A contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E.Comment: 15 page

Closed forms and multi-moment maps

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.Comment: 36 page

Invariants of Lie algebras extended over commutative algebras without unit

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.Comment: v3: added a footnote on p.10 about a wrong derivation of the correct statemen

Prostate-specific Membrane Antigen Heterogeneity and DNA Repair Defects in Prostate Cancer.

BackgroundProstate-specific membrane antigen (PSMA; folate hydrolase) prostate cancer (PC) expression has theranostic utility.ObjectiveTo elucidate PC PSMA expression and associate this with defective DNA damage repair (DDR).Design, setting, and participantsMembranous PSMA (mPSMA) expression was scored immunohistochemically from metastatic castration-resistant PC (mCRPC) and matching, same-patient, diagnostic biopsies, and correlated with next-generation sequencing (NGS) and clinical outcome data.Outcome measurements and statistical analysisExpression of mPSMA was quantitated by modified H-score. Patient DNA was tested by NGS. Gene expression and activity scores were determined from mCRPC transcriptomes. Statistical correlations utilised Wilcoxon signed rank tests, survival was estimated by Kaplan-Meier test, and sample heterogeneity was quantified by Shannon's diversity index.Results and limitationsExpression of mPSMA at diagnosis was associated with higher Gleason grade (p=0.04) and worse overall survival (p=0.006). Overall, mPSMA expression levels increased at mCRPC (median H-score [interquartile range]: castration-sensitive prostate cancer [CSPC] 17.5 [0.0-60.0] vs mCRPC 55.0 [2.8-117.5]). Surprisingly, 42% (n=16) of CSPC and 27% (n=16) of mCRPC tissues sampled had no detectable mPSMA (H-score ConclusionsMembranous PSMA expression is upregulated in some but not all PCs, with mPSMA expression demonstrating marked inter- and intrapatient heterogeneity. DDR aberrations are associated with higher mPSMA expression and merit further evaluation as predictive biomarkers of response for PSMA-targeted therapies in larger, prospective cohorts.Patient summaryThrough analysis of prostate cancer samples, we report that the presence of prostate-specific membrane antigen (PSMA) is extremely variable both within one patient and between different patients. This may limit the usefulness of PSMA scans and PSMA-targeted therapies. We show for the first time that prostate cancers with defective DNA repair produce more PSMA and so may respond better to PSMA-targeting treatments

IER2-induced senescence drives melanoma invasion through osteopontin

Expression of the immediate-early response gene IER2 has been associated with the progression of several types of cancer, but its functional role is poorly understood. We found that increased IER2 expression in human melanoma is associated with shorter overall survival, and subsequently investigated the mechanisms through which IER2 exerts this effect. In experimental melanoma models, sustained expression of IER2 induced senescence in a subset of melanoma cells in a p53/MAPK/AKT-dependent manner. The senescent cells produced a characteristic secretome that included high levels of the extracellular phosphoglycoprotein osteopontin. Nuclear localization of the IER2 protein was critical for both the induction of senescence and osteopontin secretion. Osteopontin secreted by IER2-expressing senescent cells strongly stimulated the migration and invasion of non-senescent melanoma cells. Consistently, we observed coordinate expression of IER2, p53/p21, and osteopontin in primary human melanomas and metastases, highlighting the pathophysiological relevance of IER2-mediated senescence in melanoma progression. Together, our study reveals that sustained IER2 expression drives melanoma invasion and progression through stimulating osteopontin secretion via the stochastic induction of senescence

Search for the exotic Ξ−−(1860)\Xi^{--}(1860) Resonance in 340GeV/c Σ−\Sigma^--Nucleus Interactions

We report on a high statistics search for the $\Xi^{--}(1860)$ resonance in $\Sigma^-$-nucleus collisions at 340GeV/c. No evidence for this resonance is found in our data sample which contains 676000 $\Xi^-$ candidates above background. For the decay channel $\Xi^{--}(1860) \to \Xi^-\pi^-$ and the kinematic range 0.15$<x_F<$0.9 we find a 3$\sigma$ upper limit for the production cross section of 3.1 and 3.5 $\mu$b per nucleon for reactions with carbon and copper, respectively.Comment: 5 pages, 4 figures, modification of ref. 43 and 4

Identification of an Imidazopyridine-based Compound as an Oral Selective Estrogen Receptor Degrader for Breast Cancer Therapy.

UNLABELLED: The pro-oncogenic activities of estrogen receptor alpha (ERα) drive breast cancer pathogenesis. Endocrine therapies that impair the production of estrogen or the action of the ERα are therefore used to prevent primary disease metastasis. Although recent successes with ERα degraders have been reported, there is still the need to develop further ERα antagonists with additional properties for breast cancer therapy. We have previously described a benzothiazole compound A4B17 that inhibits the proliferation of androgen receptor-positive prostate cancer cells by disrupting the interaction of the cochaperone BAG1 with the AR. A4B17 was also found to inhibit the proliferation of estrogen receptor-positive (ER+) breast cancer cells. Using a scaffold hopping approach, we report here a group of small molecules with imidazopyridine scaffolds that are more potent and efficacious than A4B17. The prototype molecule X15695 efficiently degraded ERα and attenuated estrogen-mediated target gene expression as well as transactivation by the AR. X15695 also disrupted key cellular protein-protein interactions such as BAG1-mortalin (GRP75) interaction as well as wild-type p53-mortalin or mutant p53-BAG2 interactions. These activities together reactivated p53 and resulted in cell-cycle block and the induction of apoptosis. When administered orally to in vivo tumor xenograft models, X15695 potently inhibited the growth of breast tumor cells but less efficiently the growth of prostate tumor cells. We therefore identify X15695 as an oral selective ER degrader and propose further development of this compound for therapy of ER+ breast cancers. SIGNIFICANCE: An imidazopyridine that selectively degrades ERα and is orally bioavailable has been identified for the development of ER+ breast cancer therapeutics. This compound also activates wild-type p53 and disrupts the gain-of-function tumorigenic activity of mutant p53, resulting in cell-cycle arrest and the induction of apoptosis