A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

Bounded derived categories of very simple manifolds

An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical terms. This is a partial generalization of an impressive result due to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2) and many typos are correcte

Worldline Superfield Actions for N=2 Superparticles

We propose doubly supersymmetric actions in terms of n=2(D-2) worldline superfields for N=2 superparticles in D=3,4 and Type IIA D=6 superspaces. These actions are obtained by dimensional reduction of superfield actions for N=1 superparticles in D=4,6 and 10, respectively. We show that in all these models geometrodynamical constraints on target superspace coordinates do not put the theory on the mass shell, so the actions constructed consistently describe the dynamics of the corresponding N=2 superparticles. We also find that in contrast to the IIA D=6 superparticle a chiral IIB D=6 superparticle, which is not obtainable by dimensional reduction from N=1, D=10, is described by superfield constraints which produce dynamical equations. This implies that for the IIB D=6 superparticle the doubly supersymmetric action does not exist in the conventional form.Comment: Latex, 20 pp. Minor corrections, acknowledgements adde

Outcomes of haematology/oncology patients admitted to intensive care unit at The Canberra Hospital

BACKGROUND: Outcomes for haematology/oncology patients have improved; however, determining their suitability for intensive care unit (ICU) admission remains challenging and controversial. AIM: Examine outcomes of patients admitted to an Australian tertiary hospital ICU and explore potential prognostic factors. METHODS: A retrospective review of patients with haematological and solid tumour malignancies non-electively admitted to The Canberra Hospital (TCH) ICU, between January 2008 and December 2012. Patient demographics, cancer details, reasons for ICU admission and Acute Physiologic and Chronic Health Evaluation (APACHE) II scores were collected, and survival rates calculated and correlated with potential prognostic factors. RESULTS Of 205 patients, 113 (55%) had haematological malignancies, and 92 (45%) had solid tumours: 58% male and mean age 60.3 years (standard deviation (SD) 13.4). Eighty-two per cent of solid tumour patients had metastatic disease and 55% received palliative chemotherapy. Primary reasons for ICU admission included sepsis (59%), respiratory distress (37%) and hypotension/shock (18%). Mean APACHE II score was 20.1(SD 0.55); mean length of stay in ICU, 4 days (SD 5.2); ICU survival was 76% with 62% and 41% alive at 30 days and 6 months respectively. Overall 1-year survival was 36%. High APACHE II scores and ≥2 organs failing were significant risk factors for 30-day mortality. CONCLUSION: Short-term outcomes were similar to contemporary studies from a general tertiary hospital setting and better than historical data. Sixty-two per cent of patients were alive 30 days post-ICU admission, with a significant minority alive at 12 months, confirming some patients achieved worthwhile outcomes. Further research is needed to ensure appropriate patient selection and to explore quality of life post ICU

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

The Topology of Parabolic Character Varieties of Free Groups

Let G be a complex affine algebraic reductive group, and let K be a maximal compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or equal to 0, let X (respectively, Y) be the space of equivalence classes of representations of the free group of m+n generators in G (respectively, K) such that for each i between 1 and m, the image of the i-th free generator is conjugate to h_i. These spaces are parabolic analogues of character varieties of free groups. We prove that Y is a strong deformation retraction of X. In particular, X and Y are homotopy equivalent. We also describe explicit examples relating X to relative character varieties.Comment: 16 pages, version 2 includes minor revisions and some modified proofs, accepted for publication in Geometriae Dedicat

Brown representability does not come for free

"Vegeu el resum a l'inici del document del fitxer adjunt"

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.Comment: 28 page

Coherent analogues of matrix factorizations and relative singularity categories

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.Comment: LaTeX 2e with pb-diagram and xy-pic; 114 pages, 13 commutative diagrams. v.8: new sections 2.10, 3.1 and 3.7 inserted; v.9: appendix B added, remarks inserted in sections 2.10 and 2.7, section 1.8 expanded; v.10: new section 3.3 inserted, the whole paper has two authors now; v.11: small corrections, additions, and improvements -- this is intended as the final versio