On a problem of A. Weil

A topological invariant of the geodesic laminations on a modular surface is constructed. The invariant has a continuous part (the tail of a continued fraction) and a combinatorial part (the singularity data). It is shown, that the invariant is complete, i.e. the geodesic lamination can be recovered from the invariant. The continuous part of the invariant has geometric meaning of a slope of lamination on the surface.Comment: to appear Beitr\"age zur Algebra und Geometri

A glimpse into Thurston's work

We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book "In the tradition of Thurston: Geometry and topology", edited by K. Ohshika and A. Papadopoulos (Springer, 2020)

The <i>CHEK2</i> Variant C.349A>G Is Associated with Prostate Cancer Risk and Carriers Share a Common Ancestor.

The identification of recurrent founder variants in cancer predisposing genes may have important implications for implementing cost-effective targeted genetic screening strategies. In this study, we evaluated the prevalence and relative risk of the CHEK2 recurrent variant c.349A>G in a series of 462 Portuguese patients with early-onset and/or familial/hereditary prostate cancer (PrCa), as well as in the large multicentre PRACTICAL case-control study comprising 55,162 prostate cancer cases and 36,147 controls. Additionally, we investigated the potential shared ancestry of the carriers by performing identity-by-descent, haplotype and age estimation analyses using high-density SNP data from 70 variant carriers belonging to 11 different populations included in the PRACTICAL consortium. The CHEK2 missense variant c.349A>G was found significantly associated with an increased risk for PrCa (OR 1.9; 95% CI: 1.1-3.2). A shared haplotype flanking the variant in all carriers was identified, strongly suggesting a common founder of European origin. Additionally, using two independent statistical algorithms, implemented by DMLE+2.3 and ESTIAGE, we were able to estimate the age of the variant between 2300 and 3125 years. By extending the haplotype analysis to 14 additional carrier families, a shared core haplotype was revealed among all carriers matching the conserved region previously identified in the high-density SNP analysis. These findings are consistent with CHEK2 c.349A>G being a founder variant associated with increased PrCa risk, suggesting its potential usefulness for cost-effective targeted genetic screening in PrCa families

Additional SNPs improve risk stratification of a polygenic hazard score for prostate cancer.

BackgroundPolygenic hazard scores (PHS) can identify individuals with increased risk of prostate cancer. We estimated the benefit of additional SNPs on performance of a previously validated PHS (PHS46).Materials and method180 SNPs, shown to be previously associated with prostate cancer, were used to develop a PHS model in men with European ancestry. A machine-learning approach, LASSO-regularized Cox regression, was used to select SNPs and to estimate their coefficients in the training set (75,596 men). Performance of the resulting model was evaluated in the testing/validation set (6,411 men) with two metrics: (1) hazard ratios (HRs) and (2) positive predictive value (PPV) of prostate-specific antigen (PSA) testing. HRs were estimated between individuals with PHS in the top 5% to those in the middle 40% (HR95/50), top 20% to bottom 20% (HR80/20), and bottom 20% to middle 40% (HR20/50). PPV was calculated for the top 20% (PPV80) and top 5% (PPV95) of PHS as the fraction of individuals with elevated PSA that were diagnosed with clinically significant prostate cancer on biopsy.Results166 SNPs had non-zero coefficients in the Cox model (PHS166). All HR metrics showed significant improvements for PHS166 compared to PHS46: HR95/50 increased from 3.72 to 5.09, HR80/20 increased from 6.12 to 9.45, and HR20/50 decreased from 0.41 to 0.34. By contrast, no significant differences were observed in PPV of PSA testing for clinically significant prostate cancer.ConclusionsIncorporating 120 additional SNPs (PHS166 vs PHS46) significantly improved HRs for prostate cancer, while PPV of PSA testing remained the same

Prostate cancer risk stratification improvement across multiple ancestries with new polygenic hazard score.

BackgroundProstate cancer risk stratification using single-nucleotide polymorphisms (SNPs) demonstrates considerable promise in men of European, Asian, and African genetic ancestries, but there is still need for increased accuracy. We evaluated whether including additional SNPs in a prostate cancer polygenic hazard score (PHS) would improve associations with clinically significant prostate cancer in multi-ancestry datasets.MethodsIn total, 299 SNPs previously associated with prostate cancer were evaluated for inclusion in a new PHS, using a LASSO-regularized Cox proportional hazards model in a training dataset of 72,181 men from the PRACTICAL Consortium. The PHS model was evaluated in four testing datasets: African ancestry, Asian ancestry, and two of European Ancestry-the Cohort of Swedish Men (COSM) and the ProtecT study. Hazard ratios (HRs) were estimated to compare men with high versus low PHS for association with clinically significant, with any, and with fatal prostate cancer. The impact of genetic risk stratification on the positive predictive value (PPV) of PSA testing for clinically significant prostate cancer was also measured.ResultsThe final model (PHS290) had 290 SNPs with non-zero coefficients. Comparing, for example, the highest and lowest quintiles of PHS290, the hazard ratios (HRs) for clinically significant prostate cancer were 13.73 [95% CI: 12.43-15.16] in ProtecT, 7.07 [6.58-7.60] in African ancestry, 10.31 [9.58-11.11] in Asian ancestry, and 11.18 [10.34-12.09] in COSM. Similar results were seen for association with any and fatal prostate cancer. Without PHS stratification, the PPV of PSA testing for clinically significant prostate cancer in ProtecT was 0.12 (0.11-0.14). For the top 20% and top 5% of PHS290, the PPV of PSA testing was 0.19 (0.15-0.22) and 0.26 (0.19-0.33), respectively.ConclusionsWe demonstrate better genetic risk stratification for clinically significant prostate cancer than prior versions of PHS in multi-ancestry datasets. This is promising for implementing precision-medicine approaches to prostate cancer screening decisions in diverse populations