The bounded negativity conjecture predicts that on a smooth complex surface X, there is a bound b such that for every (reduced) curve C on X, the self-intersection of C satisfies C^2 >b. It was stated one century ago by the Italian geometers but it is still quite open. Recently people introduced new tools -called Harbourne constants- in order to study that conjecture. In these lectures, we will explain these tools and give an overview of the present knowledge on that conjecture.

The bounded negativity conjecture predicts that on a smooth complex surface X, there is a bound b such that for every (reduced) curve C on X, the self-intersection of C satisfies C^2 >b. It was stated one century ago by the Italian geometers but it is still quite open. Recently people introduced new tools -called Harbourne constants- in order to study that conjecture. In these lectures, we will explain these tools and give an overview of the present knowledge on that conjecture.

The bounded negativity conjecture predicts that on a smooth complex surface X, there is a bound b such that for every (reduced) curve C on X, the self-intersection of C satisfies C^2 >b. It was stated one century ago by the Italian geometers but it is still quite open. Recently people introduced new tools -called Harbourne constants- in order to study that conjecture. In these lectures, we will explain these tools and give an overview of the present knowledge on that conjecture.

issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci. We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity. Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes. An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html. 

Abstract: issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci. We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity. Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes. An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html.

Abstract: issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci. We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity. Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes. An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html.

Issue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single nucleotide mutations at a selected set of loci

We carry out such inference on the basis of a Bayesian feature allocation model. We jointly model subclonal copy numbers and the corresponding allele sequences for the same loci, using three random matrices, L, Z and w to represent subclonal copy numbers (L), the number of sub- clonal variant alleles (Z) and the cellular fractions (w) of subclones in one or more tumor samples, respectively. The unknown number of subclones implies a random number of columns. More than one subclone indicates tumor heterogeneity.

Using simulation studies and a real data analysis with next-generation sequencing data, we demonstrate how posterior inference on the subclonal structure is enhanced with the joint modeling of both structure and sequencing variants on subclonal genomes

An R package is available at http://cran.r-project.org/web/packages/ BayClone2/index.html.