A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page

A matroid extension result

Adding elements to matroids can be fraught with difficulty. In the V\'amos matroid $V_8$, there are four independent sets $X_1,X_2, X_3,$ and $X_4$ such that $(X_1 \cup X_2,X_3 \cup X_4)$ is a $3$-separation while exactly three of the local connectivities $\sqcap(X_1,X_{3})$, $\sqcap(X_1,X_{4})$, $\sqcap(X_2,X_{3})$, and $\sqcap(X_2,X_{4})$ are one, with the fourth being zero. As is well known, there is no extension of $V_8$ by a non-loop element $p$ such that $X_j \cup p$ is a circuit for all $j$. This paper proves that a matroid can be extended by a fixed element in the guts of a $3$-separation provided no V\'amos-like structure is present

Extremal problems on shadows and hypercuts in simplicial complexes

Let $F$ be an $n$-vertex forest. We say that an edge $e\notin F$ is in the shadow of $F$ if $F\cup\{e\}$ contains a cycle. It is easy to see that if $F$ is "almost a tree", that is, it has $n-2$ edges, then at least $\lfloor\frac{n^2}{4}\rfloor$ edges are in its shadow and this is tight. Equivalently, the largest number of edges an $n$-vertex cut can have is $\lfloor\frac{n^2}{4}\rfloor$. These notions have natural analogs in higher $d$-dimensional simplicial complexes, graphs being the case $d=1$. The results in dimension $d>1$ turn out to be remarkably different from the case in graphs. In particular the corresponding bounds depend on the underlying field of coefficients. We find the (tight) analogous theorems for $d=2$. We construct $2$-dimensional "$\mathbb Q$-almost-hypertrees" (defined below) with an empty shadow. We also show that the shadow of an "$\mathbb F_2$-almost-hypertree" cannot be empty, and its least possible density is $\Theta(\frac{1}{n})$. In addition we construct very large hyperforests with a shadow that is empty over every field. For $d\ge 4$ even, we construct $d$-dimensional $\mathbb{F} _2$-almost-hypertree whose shadow has density $o_n(1)$. Finally, we mention several intriguing open questions

Algebraic recognizability of regular tree languages

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem \`{a} la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization

Effects on the transcriptome upon deletion of a distal element cannot be predicted by the size of the H3K27Ac peak in human cells.

Genome-wide association studies (GWAS) have identified single nucleotide polymorphisms (SNPs) associated with increased risk for colorectal cancer (CRC). A molecular understanding of the functional consequences of this genetic variation is complicated because most GWAS SNPs are located in non-coding regions. We used epigenomic information to identify H3K27Ac peaks in HCT116 colon cancer cells that harbor SNPs associated with an increased risk for CRC. Employing CRISPR/Cas9 nucleases, we deleted a CRC risk-associated H3K27Ac peak from HCT116 cells and observed large-scale changes in gene expression, resulting in decreased expression of many nearby genes. As a comparison, we showed that deletion of a robust H3K27Ac peak not associated with CRC had minimal effects on the transcriptome. Interestingly, although there is no H3K27Ac peak in HEK293 cells in the E7 region, deletion of this region in HEK293 cells decreased expression of several of the same genes that were downregulated in HCT116 cells, including the MYC oncogene. Accordingly, deletion of E7 causes changes in cell culture assays in HCT116 and HEK293 cells. In summary, we show that effects on the transcriptome upon deletion of a distal regulatory element cannot be predicted by the size or presence of an H3K27Ac peak