Advanced maths in cosmology

In this talk, I will present recent advancements in mathematical and cosmological frameworks, drawing from my key studies. I will introduce functorial methods in Functors of Actions Theories (FATs), bridging mathematical rigor with cosmological phenomena, by predicting actions. In response to the need for advanced math, I will introduce the exploration of novel tensor-based approaches from Advancing Tensor Theories (ATTs), and Advanced Manifold–Metric Pairs (AMMPs), offering new tools for modeling. ATTs merge novel advanced indices with corresponding tensors which are defined using standard calculus operators and fractional calculus operators. AMMPs introduce more abstract Manifold–Metric Pairs in which the line elements are constructured through a functional tensor of (U,L)-rank, rather than a (0,2)-rank, which opens a new road to introduce more sophisticated mathematical structure to physics problems. Finally, I will discuss the dynamics of the quintessence in the ϕCDM, simple ϕΛCDM, and polyΛCDM models, providing insights into scalar field-driven cosmology and its implications for dark energy and modified gravity. The analytical polyΛCDM model incorporates scalar–vector–tensor modified gravity theories into a lagrangian, and under appropriate assumptions, it results in a polynomial depenence on energy density ratios that can be measured by observational data, and distinguishes which modified gravity model is more apparent from the background cosmology data. If time permits, I will talk about a novel test of fundamental principles of gravity in extra dimensions given large-scale structure surveys with contaminated samples [8]. These works collectively highlight innovative mathematical methods to advance our understanding of the universe’s evolution and structure.