We use molecular dynamics simulation data of the CXCR7 protein. This protein is a G protein-coupled receptor (GPCRs), it scavenges chemokines and opioids, and recruit β-arrestins, so it is an atypical GPCR. For the protein helices, we extract the spatial coordinates of atoms of alpha carbon (“CA”) and monitor how their positions evolve over time using static and dynamic graphical analysis. In parallel, we compute pairwise contact distances between protein residues and analyze how these distances fluctuate throughout the simulation trajectory.

In this project we expanded previous work related to discrete abelian gauge theories through the group SU(2). We examined the SU(2) Hopf bundle on S⁷ through the Hurwitz group, then conducted several simulations involving Wilson loops in two dimensions for both U(1) and SU(2) and studied their different behaviour on both usual lattices and hypergraphs.

I identified four kinds of symmetric fractal trees in 3D that are determined by the type of expressions found in their boundary equations; these are trees with number of branches b=4n-1, b=4n, b=4n+1 and b=4n+2 where n takes the integer values from 1 to ∞. Several animations were produced when one walks around the critical boundaries of the parameter space, showing interesting dynamics and topological critical changes for certain angles.

In adaptive evolution, a system is iteratively modified in order to optimize a given fitness function. This biologically inspired technique is commonly used to search for solutions in large, complex spaces. In this project, we investigate the adaptive evolution of hypergraph rewriting systems. Our goal is to discover rewrite rules that evolve a hypergraph for a finite number of iterations and then halt. As an extension, we explore the adaptive evolution of hypergraph multiway systems, aiming to identify rules that produce halting multiway graphs with the maximal possible number of nodes.

In this work, a transformer neural network was trained on periodic and aperiodic graph rewriting systems to predict the next (or previous) step in their evolution, and to iteratively predict the entire forwards (or backwards) evolution of the system by using the previous prediction as the basis for the next. A simple transformer was able to predict the bidirectional evolution of a periodic case with perfect accuracy, but prediction of an aperiodic case had almost no success. These findings may suggest that artificial intelligences based on current neural networks architectures are generally limited only to tasks also capable by humans by tapping into the same pockets of computational reducibility accessible to human brains, but not tasks completely beyond human capability.

This report defines a hole for the proof space arising in multiway system graphs generated by string-substitution rules. A proof is a path from an initial string to a target string (the initial and final string is a theorem) in the graph. By contracting each proof to a single vertex and connecting two proofs when they differ in exactly one step (when they form a single diamond sub-graph), at the end it constructs a proof graph whose connectedness tells the fact of continuously deforming one proof into another. A hole is defined as follows: #Holes = #ConnectedComponents - 1. Adding new symbols or asymmetric rules quickly introduces holes.

The aim of this project is to identify the manifolds corresponding to networks that are generated by simple substitution rules from Stephen Wolfram’s physics project. At first glance, some of the networks resemble a cell complex of a known surface. The idea is to provide techniques to substantiate this impression and figure out the geometric structure that might underlie the purely combinatorially given network.