Bachelor Thesis at Maastricht University : "Duologues in Social Networks: Quick and Low-Variance Paths"
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Abstract: A social network is a set of connections between individuals. We modeled such a network as a graph and investigated how information spreads through isolated paths in this graph. We applied a labeling algorithm by Tung and Chew to find multicriteria optimal paths that minimize the expected arrival time and variance in the arrival time of information at vertices in the graph. We also presented three speed-up heuristics to reduce the practical computational complexity of the labeling algorithm. Lastly, we looked at some real-world implications of our findings. We explained that two ways of reducing the expected arrival time and variance in the arrival time of information at vertices, are to add edges if the graph is sparse, or to increase information flow over specific edges if the graph is dense.
Bachelor Thesis at Radboud University : "Zooming in on the Computational Complexity of the Functional Renormalization Group"
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Abstract: The extraction of physics predictions from an asymptotically safe QFT requires the construction of the UV-critical hypersurface of the underlying renormalization group fixed point. This thesis investigates the computational complexity of the boundary value method as a scalable alternative to the traditional shooting method. The boundary value method reformulates this construction of the UV-critical surface $S_{UV}$ as an optimization problem for generating functions $F_\mu$. Here we solve this problem using a genetic algorithm that has been optimized using modern principles from computer science. To provide a clear example of the functioning of this new method, it has been applied to a scalar field theory in two Euclidean dimensions. A complexity analysis demonstrates the genetic algorithm's superiority for high-dimensional surfaces. While successfully predicting the $k=0$ endpoint, the genetic algorithm has trouble producing connected $S_{UV}$ trajectories. The work introduces systematic meta-optimization through testing suites and Pareto analysis. These algorithmic treatment of the very general mathematical problem offers a broad applicability to asymptotic safety, dynamical systems, fluid dynamics, and more. We also outline future research directions, including adaptive basis functions, neural network based implementations, and applications to theories that offer ties to real world observables, e.g. in the context of asymptotically safe quantum gravity.
"The Poincaré Group and its Conserved Quantities in Physics"
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Abstract: In this exposition we derive from basic symmetry principles the conserved quantities that are important in physics: total energy, linear momentum, angular momentum, center of momentum, and parity. To do so we will construct the Poincaré group of spacetime symmetries step-by-step. We will use mathematical rigor to really convince the reader of the validity and fundamental nature of the results, but will choose forming good intuition over leaning into heavy group theory in order to keep the text accessible.
"The Standard Model as a Geometric Theory" [Work In Progress]
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Introduction: Quantum Field Theory (QFT) is one of the two most successful theories describing our reality, the other being General Relativity (GR). While GR is treated as a geometrical theory first and foremost, this is not the case for most introductions to QFT. This choice is easily understood, as one most naturally arrives at QFT from single-particle Quantum Mechanics (QM), so it is only natural to primarily treat the "quantum" and "field" aspect of QFT. As arguably the most important product of QFT, the Standard Model of Particle Physics (SM) is treated much the same way. Dirac fermions are introduced as matter particles and Yang-Mills (YM) gauge fields host gauge bosons as force carriers. In doing so, the SM is built primarily as a theory arising from local unitary gauge symmetries. The first peek behind the curtain comes in the form of the covariant derivative. This object is in most cases introduced as a replacement to the normal derivative, with the purpose of magically fixing the Lagrangian. The Lagrangian had lost its invariance under gauge transformations now that these transformations have been made local, and the covariant derivative neatly fixes this. However, this poses the question: What is the covariant derivative really doing to undo the mess that the local gauge transformation caused? To answer this question neatly, one must zoom out and treat the SM as a geometric theory, just as GR. Doing so exposes the beautiful mathematical structure underlying the fundamental forces, as well as exposing some of the many difficulties of treating gravity as a theory of quantum fields.