Homological Algebra and Its Application: A Descriptive Study

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory

Clustering Trajectories to Study Diabetic Kidney Disease

Diabetic kidney disease (DKD) is a serious complication of type-2 diabetes, defined prominently by a reduction in estimated glomerular filtration rate (eGFR), a measure of renal waste excretion capacity. However DKD patients present high heterogeneity in disease trajectory and response to treatment, making the one-model-fits-all pro- tocol for estimating prognosis and expected response to therapy as proposed by guidelines obsolete. As a solution, precision or stratified medicine aims to define subgroups of patients with similar pathophysi- ology and response to the therapy, allowing to select the best drug com- binations for each subgroup. We focus on eGFR when aiming to identify eGFR decline trends by clustering patients according to their eGFR tra- jectory shape-similarity. The study involved 256 DKD patients observed annually for four years. Using the Fr ́echet distance, we built clusters of patients according to the similarity of their eGFR trajectories to identify distinct clusters. We formalized the trajectory-clustering approach through category the- ory. Characteristics of patients within different progression clusters were compared at the baseline and over time. We identified five clusters of eGFR progression over time. We noticed a bifurcation of eGFR mean trajectories and a switch between two other mean trajectories. This particular clustering approach identified different mean eGFR trajectories. Our findings suggest the existence of distinct dynamical behaviors in the disease progression