Narrow Results

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Given a principal $G$-bundle $P\to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The main result in this paper is that if $G$ is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, $C^1$ homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.

In this paper, relations between topological entropy, volume growth and the growth in topological complexity from homotopical and homological point of view are discussed for random dynamical systems. It is shown that, under certain conditions, the volume growth, the growth in fundamental group and the growth in homological group are all bounded from above by the topological entropy.

Revolutionary in scope and application, the CRISPR Cas9 endonuclease system can be guided by 20-nt single guide RNA (sgRNA) to any complementary loci on the double-stranded DNA. Once the target site is located, Cas9 can then cleave the DNA and introduce mutations. Despite the power of this system, sgRNA is highly susceptible to off-target homologous attachment and can consequently cause Cas9 to cleave DNA at off-target sites. In order to better understand this flaw in the system, the human genome and Streptococcus pyogenes Cas9 (SpCas9) were used in a mathematical and computational study to analyze the probabilities of potential sgRNA off-target homologies. It has been concluded that off-target sites are nearly unavoidable for large-size genomes, such as the human genome. Backed by mathematical analysis, a viable solution is the double-nicking method which has the promise for genome editing specificity. Also applied in this study was a computational algorithm for off-target homology search that was implemented in Java to confirm the mathematical analysis.