On radical square zero rings

Let Λ be a connected left artinian ring with radical square zero and with n simple modules. If Λ is not self-injective, then we show that any module M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of the artin algebras with radical square zero and n simple modules which have a non-projective module M such that Exti(M, Λ) = 0 for 1 ≤ i ≤ n

Stability conditions and Stokes factors

Let A be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on A parametrises an isomonodromic family of irregular connections on P^1 with values in the Hall algebra of A. The residues of these connections are given by the holomorphic generating function for counting invariants in A constructed by D. Joyce.Comment: Very minor changes. Final version. To appear in Inventione

Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's comments. To appear in Geometriae Dedicat

Tilted algebras and short chains of modules

We provide an affirmative answer for the question raised almost twenty years ago concerning the characterization of tilted artin algebras by the existence of a sincere finitely generated module which is not the middle of a short chain

Cluster structures on quantum coordinate rings

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C_w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.Comment: v2: minor corrections. v3: references updated, final version to appear in Selecta Mathematic

C^2/Z_n Fractional branes and Monodromy

We construct geometric representatives for the C^2/Z_n fractional branes in terms of branes wrapping certain exceptional cycles of the resolution. In the process we use large radius and conifold-type monodromies, and also check some of the orbifold quantum symmetries. We find the explicit Seiberg-duality which connects our fractional branes to the ones given by the McKay correspondence. We also comment on the Harvey-Moore BPS algebras.Comment: 34 pages, v1 identical to v2, v3: typos fixed, discussion of Harvey-Moore BPS algebras update

Development and external validation of a clinical prediction model for functional impairment after intracranial tumor surgery

OBJECTIVE Decision-making for intracranial tumor surgery requires balancing the oncological benefit against the risk for resection-related impairment. Risk estimates are commonly based on subjective experience and generalized num-bers from the literature, but even experienced surgeons overestimate functional outcome after surgery. Today, there is no reliable and objective way to preoperatively predict an individual patient's risk of experiencing any functional impair-ment. METHODS The authors developed a prediction model for functional impairment at 3 to 6 months after microsurgical resection, defined as a decrease in Karnofsky Performance Status of >= 10 points. Two prospective registries in Swit- zerland and Italy were used for development. External validation was performed in 7 cohorts from Sweden, Norway, Germany, Austria, and the Netherlands. Age, sex, prior surgery, tumor histology and maximum diameter, expected major brain vessel or cranial nerve manipulation, resection in eloquent areas and the posterior fossa, and surgical approach were recorded. Discrimination and calibration metrics were evaluated. RESULTS In the development (2437 patients, 48.2% male; mean age +/- SD: 55 +/- 15 years) and external validation (2427 patients, 42.4% male; mean age +/- SD: 58 +/- 13 years) cohorts, functional impairment rates were 21.5% and 28.5%, respectively. In the development cohort, area under the curve (AUC) values of 0.72 (95% CI 0.69-0.74) were observed. In the pooled external validation cohort, the AUC was 0.72 (95% CI 0.69-0.74), confirming generalizability. Calibration plots indicated fair calibration in both cohorts. The tool has been incorporated into a web-based application available at https://neurosurgery.shinyapps.io/impairment/. CONCLUSIONS Functional impairment after intracranial tumor surgery remains extraordinarily difficult to predict, al- though machine learning can help quantify risk. This externally validated prediction tool can serve as the basis for case by-case discussions and risk-to-benefit estimation of surgical treatment in the individual patient.Scientific Assessment and Innovation in Neurosurgical Treatment Strategie