Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

Prognostic factors and risk groups in T1G3 non-muscle-invasive bladder cancer patients initially treated with Bacillus Calmette-Guérin: results of a retrospective multicenter study of 2451 patients

BACKGROUND The impact of prognostic factors in T1G3 non-muscle-invasive bladder cancer (BCa) patients is critical for proper treatment decision making. OBJECTIVE To assess prognostic factors in patients who received bacillus Calmette-Guérin (BCG) as initial intravesical treatment of T1G3 tumors and to identify a subgroup of high-risk patients who should be considered for more aggressive treatment. DESIGN, SETTING, AND PARTICIPANTS Individual patient data were collected for 2451 T1G3 patients from 23 centers who received BCG between 1990 and 2011. OUTCOME MEASUREMENTS AND STATISTICAL ANALYSIS Using Cox multivariable regression, the prognostic importance of several clinical variables was assessed for time to recurrence, progression, BCa-specific survival, and overall survival (OS). RESULTS AND LIMITATIONS With a median follow-up of 5.2 yr, 465 patients (19%) progressed, 509 (21%) underwent cystectomy, and 221 (9%) died because of BCa. In multivariable analyses, the most important prognostic factors for progression were age, tumor size, and concomitant carcinoma in situ (CIS); the most important prognostic factors for BCa-specific survival and OS were age and tumor size. Patients were divided into four risk groups for progression according to the number of adverse factors among age ≥ 70 yr, size ≥ 3 cm, and presence of CIS. Progression rates at 10 yr ranged from 17% to 52%. BCa-specific death rates at 10 yr were 32% in patients ≥ 70 yr with tumor size ≥ 3 cm and 13% otherwise. CONCLUSIONS T1G3 patients ≥ 70 yr with tumors ≥ 3 cm and concomitant CIS should be treated more aggressively because of the high risk of progression. PATIENT SUMMARY Although the majority of T1G3 patients can be safely treated with intravesical bacillus Calmette-Guérin, there is a subgroup of T1G3 patients with age ≥ 70 yr, tumor size ≥ 3 cm, and concomitant CIS who have a high risk of progression and thus require aggressive treatment.

Molecular Classification of Ependymal Tumors across All CNS Compartments, Histopathological Grades, and Age Groups

Ependymal tumors across age groups are currently classified and graded solely by histopathology. It is, however, commonly accepted that this classification scheme has limited clinical utility based on its lack of reproducibility in predicting patients' outcome. We aimed at establishing a uniform molecular classification using DNA methylation profiling. Nine molecular subgroups were identified in a large cohort of 500 tumors, 3 in each anatomical compartment of the CNS, spine, posterior fossa, supratentorial. Two supratentorial subgroups are characterized by prototypic fusion genes involving RELA and YAP1, respectively. Regarding clinical associations, the molecular classification proposed herein outperforms the current histopathological classification and thus might serve as a basis for the next World Health Organization classification of CNS tumors.