@phdthesis{20551, abstract = {The space of codimension-2 shapes, such as curves in 3D and surfaces in 4D, is an infinite-dimensional manifold. This thesis explores geometric structures and dynamics on this space, with emphasis on their implications for physics, particularly hydrodynamics. Our investigation ranges from theoretical studies of infinite-dimensional symplectic and prequantum geometry to numerical computation of the time evolution of shapes. The thesis presents four main contributions. In the first part, we introduce implicit representations of codimension-2 shapes using a class of complex-valued functions, and prove that the space of these implicit representations forms a prequantum bundle over the codimension-2 shape space. This reveals a new geometric interpretation of the canonical symplectic structure on the codimension-2 shape space. In the second part, we use implicit representations to develop a simulation method for the dynamics of space curves. To handle chaotic systems such as vortex filaments in hydrodynamics, we exploit the infinite degrees of freedom, hidden in both the configuration and dynamics of implicit representations. In the third part, we introduce new symplectic structures on the space of space curves, which generalize the only previously known symplectic structure on this space, allowing for new Hamiltonian dynamics of space curves. In the fourth part, we apply a symplectic viewpoint to a differential geometric problem with practical applications. We derive a new area formula for spherical polygons via prequantization. }, author = {Ishida, Sadashige}, isbn = {978-3-99078-070-1}, issn = {2663-337X}, pages = {141}, publisher = {Institute of Science and Technology Austria}, title = {{Symplectic-prequantum structures and dynamics on the codimension-2 shape space}}, doi = {10.15479/AT-ISTA-20551}, year = {2025}, } @phdthesis{20357, author = {Ruzickova, Natalia}, isbn = {978-3-99078-066-4}, issn = {2663-337X}, keywords = {gene regulation, networks, omnigenic model, pancreas, collective behaviour}, pages = {160}, publisher = {Institute of Science and Technology Austria}, title = {{Effect propagation in biological networks}}, doi = {10.15479/AT-ISTA-20357}, year = {2025}, } @phdthesis{20147, abstract = {Quantitative properties offer a framework for specifying and verifying system behaviors beyond the traditional boolean perspective. For example, while a boolean property may specify whether a server eventually grants every request it receives, a quantitative one may map each server execution to its average response time. This quantitative view is relatively well-studied in the context of static verification. However, although such properties often appear in practice as performance or robustness measures in a dynamic verification context, a general theoretical framework for their analysis and classification from a monitoring perspective is still missing. In this thesis, we aim to develop such a framework that takes resource-precision tradeoffs of monitors as a central consideration. We present the first theory of monitorability for quantitative properties where monitors can be naturally approximate and compared regarding their precision and resource use. In particular, we show that additional monitor resources such as registers or states lead to strictly better approximations for some properties. To enable such analyses in a machine-model independent way, we describe an abstract notion of monitors that can be instantiated with concrete models of monitors. Within this framework, we study how abstract monitors behave and identify classes of properties amenable to approximate monitoring with resource-precision considerations. We then extend the boolean safety-liveness dichotomy and safety-progress hierarchy to the quantitative setting with a monitoring perspective. In particular, we prove that every property is the pointwise minimum of a safety property and a liveness property, and properties that are both safe and co-safe can be approximately monitored arbitrarily precisely using only finitely many states. We also study the classes of quantitative properties definable by finite-state quantitative automata and provide algorithms for deciding their safety or liveness as well as their safety-liveness decompositions. Finally, we present the first general-purpose tool for automating the analysis, verification, and monitoring of quantitative automata. -------------------------------------------------------------------------------------------------------------------------------------------------------------- In reference to IEEE copyrighted material which is used with permission in this thesis, the IEEE does not endorse any of ISTA's products or services. Internal or personal use of this material is permitted. If interested in reprinting/republishing IEEE copyrighted material for advertising or promotional purposes or for creating new collective works for resale or redistribution, please go to http://www.ieee.org/publications_standards/publications/rights/rights_link.html to learn how to obtain a License from RightsLink. }, author = {Sarac, Naci E}, issn = {2663-337X}, pages = {149}, publisher = {Institute of Science and Technology Austria}, title = {{A monitoring-oriented theory and classification of quantitative specifications}}, doi = {10.15479/AT-ISTA-20147}, year = {2025}, } @phdthesis{20741, abstract = {Life on Earth emerged when biomacromolecules were membrane-enclosed in a confined space where many essential chemical reactions were more likely to happen and thereby accelerate evolution. These kinds of membranes separated internal reactions from the outside chaos while staying flexible so that those primordial cells can move, adopt their shape and, most importantly, propagate. Such membrane plasticity still remains a defining feature of all modern cell types. This remarkable ability to change their shape is most prominently observed during their propagation (i.e., cell division). Throughout division, a cell undergoes drastic change in its shape, usually at the middle of the cell, pulling the two opposite membrane sides inward, closer to each other, and, finally, culminating in pinching off to separate the cell into two daughter cells. To achieve this, a cell needs to employ a protein machinery, usually termed divisome, that can coordinate all necessary intracellular processes with membrane remodelling and synthesis of other extracellular structures that decorate a cell. The focus of this dissertation is a membrane-remodelling FtsZ system that is present across all domains of life. FtsZ forms filaments that further self-organize into ring-like structures at the cell septum and together with other division proteins perform cell envelope synthesis and constriction. However, there are still knowledge gaps in our mechanistic understanding of division in both archaea and bacteria. My work presented in this dissertation centres around a simple yet not well understood question: How is the divisome positioned correctly at the mid-cell? To achieve the proper positioning, the divisome needs to (i) be recruited to the mid-cell and (ii) localized orthogonally to the long cell axis. I tackle these processes in two different systems by applying an in vitro biochemical bottom-up reconstitution approach. I use purified components of Haloferax volcanii and Escherichia coli divisome to explore how divisome is recruited to the mid-cell in archaea and how the Z-ring positions orthogonally to the long cell axis in bacteria, respectively. Firstly, I collaborate with archaeal cell and structural biologists to explore the assembly of early division proteins in two FtsZ-containing archaeon H. volcanii, a standard model system for understudied archaeal organisms. I particularly address the hierarchy of interactions that allow a tripartite complex formation (SepF-CdpB1-CdpB2) and how the hierarchy of interactions ultimately leads to the recruitment of FtsZ filaments to the septum. This part of work has been published in (Nußbaum et al., 2024). In collaboration with evolutionary biologists, I shed light on ancient features that archaeal divisome has retained to this day and also speculate on a property that it might have lost during the course of evolution. Next, I switch my attention to E. coli divisome. Particularly, I address the FtsZ’s intrinsic biophysical property that drives the Z-ring diameter, and thereby the perpendicular orientation of the Z-ring to the long cell axis based on suggested membrane curvature sensing mechanism (Vanhille-Campos et al., 2024). This property allows formation of different Z-ring diameters that match the variety of cell diameters present in prokaryotes. The results showcase that the distribution of charged amino acids in the intrinsically disordered linker at the C-terminus (CTL) of FtsZ is the major determining factor of Z-ring diameter with inter-CTL interactions as an underlying mechanism. Finally, I thoroughly explain the methodology I used to address the abovementioned projects, and I finish with a discussion on how early archaeal divisome assembly and curvature sensing mechanism in bacteria, at first sight unrelated topics, are interconnected and important groundwork for both fundamental and translational research. }, author = {Kojic, Marko}, isbn = {978-3-99078-073-2}, issn = {2663-337X}, publisher = {Institute of Science and Technology Austria}, title = {{Towards understanding the assembly mechanisms of the Z-ring in Archaea and Bacteria}}, doi = {10.15479/AT-ISTA-20741}, year = {2025}, } @phdthesis{19903, abstract = {Cooperation, that is, one person paying a cost for another's benefit, is a fundamental principle without which no form of society could exist. The extent to which humans cooperate with each other is also an essential feature that differentiates them from other animals. Cooperation occurs even in the absence of altruistic motivations, when it is selfishly incentivised by the expectation of a future reward. For example, many economic interactions are well described that way. This kind of cooperation requires that people exhibit reciprocal behaviour that acts as a mechanism that rewards cooperation. With game-theoretic models, it is possible to formally study potential such mechanisms and under what conditions they can exist. This thesis contributes to this effort by analysing recently introduced models of cooperation that advance on previous work by taking into account the potential for pre-existing inequality among cooperating individuals as well as the different forms that reciprocity can take. Individuals may differ both intrinsically, in their abilities, as well as extrinsically, in the amount of resources they have available. Allowing for such differences in a model of cooperation helps to understand how inequality affects the potential for, and outcomes of, cooperation among unequals. In this thesis, it is shown that in the presence of intrinsic inequality, a similar unequal distribution of resources can increase the potential for cooperation. This effect is stronger the smaller the group is in which cooperation takes place. It is also shown that under particular assumptions, if the unequal members of a group vary the size of their contributions to a cooperative effort over time, they can thereby increase their efficiency and improve the collective outcome. Cooperative behaviour in a two-person interaction can be rewarded either by direct reciprocation whenever the same two people interact again, or indirectly by a third party who observed the interaction. In the latter case of indirect reciprocity, individuals are proximally rewarded by a good reputation, which ultimately translates to being rewarded with cooperative behaviour by others. This mechanism can enable selfishly motivated cooperation even in circumstances where individuals are unlikely to meet again, akin to how money facilitates trade. While these two forms of reciprocity have mostly been studied in isolation, this thesis analyses both direct and indirect reciprocity in a general model in order to compare their relative effectiveness under different circumstances. The contribution of this thesis is an extension of previous work regarding a specific kind of interaction, whose parameters allow for convenient mathematical analysis, to the most general set of possible interactions.}, author = {Hübner, Valentin}, issn = {2663-337X}, pages = {157}, publisher = {Institute of Science and Technology Austria}, title = {{Reciprocity and inequality in social dilemmas}}, doi = {10.15479/AT-ISTA-19903}, year = {2025}, } @phdthesis{20607, author = {Mondal, Soumyadip}, isbn = {978-3-99078-071-8}, issn = {2663-337X}, pages = {71}, publisher = {Institute of Science and Technology Austria}, title = {{Oxygen and sulfur redox : Conversion kinetics and phase equilibria}}, doi = {10.15479/AT-ISTA-20607}, year = {2025}, } @phdthesis{20449, author = {Bett, Vincent K}, issn = {2663-337X}, pages = {114}, publisher = {Institute of Science and Technology Austria}, title = {{Evolution and regulation of the Z chromosome}}, doi = {10.15479/AT-ISTA-20449}, year = {2025}, } @phdthesis{20777, author = {Zivadinovic, Predrag}, issn = {2663-337X}, pages = {104}, publisher = {Institute of Science and Technology Austria}, title = {{Scale-free activity as a basis for spatial learning and memory in the brain}}, doi = {10.15479/AT-ISTA-20777}, year = {2025}, } @phdthesis{20234, abstract = {Game Theory is the mathematical formalization of social dynamics - systems where agents interact over time and the evolution of the state of the system depends on the decisions of every player. This thesis takes the perspective of a single player and focuses on what they can guarantee in the worst case over the behavior of other players. In other words, we consider that the objective of every other player in the game is exactly the opposite to the player. We focus on sustained interactions over time, where the players repeatedly obtain quantitative rewards over time, and they are interested in maximizing their long-term performance. Formally, this thesis focuses on zero-sum games with the liminf average objective. Two fundamental questions that Game Theory aims to answer are the following. 1. How much can a player guarantee to obtain after the interaction? 2. How to act in order to obtain the previously mentioned guarantee? These questions are formalized by the concepts of "value" and "optimal strategies". We study their properties on games that exhibit one or more of the following properties. 1. Partial Observation: the players can not perfectly observe the current state of the system during the game. We consider the model of (finite) Partially Observable Markov Decision Processes and prove that finite-memory strategies are sufficient to approximately guarantee the value. 2. Perturbed Description: the formal description of the game is perturbed by a small parameter. We consider the model of (finite) Perturbed Matrix Games, and provide algorithms to check various robustness properties and to compute the parameterized value and optimal strategies. 3. Stochastic Transitions: the actions of the players determine the behavior of the evolution of the system, described as a probability distribution over the next state. We consider the model of (finite) Perturbed Stochastic Games and provide formulas for the marginal value. 4. Infinite States: the system can be in infinitely many states. We consider the model of Random Dynamic Games on a class of infinite graphs, prove the existence of the value, and quantify the concentration of finite-horizon values.}, author = {Saona Urmeneta, Raimundo J}, issn = {2663-337X}, pages = {125}, publisher = {Institute of Science and Technology Austria}, title = {{Robustness of solutions in game theory : Values and strategies in partially observable, perturbed, stochastic, and infinite games}}, doi = {10.15479/AT-ISTA-20234}, year = {2025}, } @phdthesis{20575, abstract = {This thesis deals with eigenvalue and eigenvector universality results for random matrix ensembles equipped with non-trivial spatial structure. We consider both mean-field models with a general variance profile (Wigner-type matrices) and correlation structure (correlated matrices) among the entries, as well as non-mean-field random band matrices with bandwidth W >> N^(1/2). To extract the universal properties of random matrix spectra and eigenvectors, we obtain concentration estimates for their resolvent, the local laws, which generalize the celebrated Wigner semicircle law for a broad class of random matrices to much finer spectral scales. The local laws hold for both a single resolvent as well as for products of multiple resolvents, known as resolvent chains, and express the remarkable approximately-deterministic behavior of these objects down to the microscopic scale. Our primary tool for establishing the local laws is the dynamical Zigzag strategy, which we develop in the setting of spatially-inhomogeneous random matrices. Our proof method systematically addresses the challenges arising from non-trivial spatial structures and is robust to all types of singularities in the spectrum, as we demonstrate in the correlated setting. Furthermore, we incorporate the analysis of the deterministic resolvent chain approximations into the dynamical framework of the Zigzag strategy, synthesizing a unified toolkit for establishing multi-resolvent local laws. Using these methods, we prove complete eigenvector delocalization, the Eigenstate Thermalization Hypothesis, and Wigner-Dyson universality in the bulk for random band matrices down to the optimal bandwidth W >> N^(1/2). For mean-field ensembles, we establish universality of local eigenvalue statistics at the cups for random matrices with correlated entries, and the Eigenstate Thermalization Hypothesis for Wigner-type matrices in the bulk of the spectrum. Finally, this thesis also contains other applications of the multi-resolvent local laws to spatially-inhomogeneous random matrices, obtained prior to the development of the Zigzag strategy. In particular, we provide a complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-type matrices in all spectral regimes, including the novel cusps, and rigorously establish the prethermalization phenomenon for deformed Wigner matrices. The main body of this thesis consists of seven research papers (listed on page xi), each presented in a separate chapter with its own introduction and all relevant context, suitable to be read independently. We ask the reader’s indulgence for the repetitions in the historical overviews and other minor redundancies that remain among the chapters as a result. The overall Introduction, preceding the chapters, provides a condensed, informal summary of the main ideas and concepts at the core of these works. }, author = {Riabov, Volodymyr}, isbn = {978-3-99078-064-0}, issn = {2663-337X}, pages = {436}, publisher = {Institute of Science and Technology Austria}, title = {{Universality in random matrices with spatial structure}}, doi = {10.15479/AT-ISTA-20575}, year = {2025}, } @phdthesis{20811, abstract = { This thesis is organized into two parts, each comprising two chapters: Chapter 1 and 2 offer models for the evolution of vaccine resistance in response to diverse vaccination strategies. Chapter 3 and 4 review the statistics of records, their connection to models of innovation and an application to the cultural evolution of sports. In chapter 1 we present a modelling study from 2021 on the evolution of SARS-CoV-2. At that time the vaccine-resistant Omicron variant had not yet evolved. In our model we consider a population that is becoming vaccinated over time, while a pathogen is spreading in the population and eventually becoming resistant to the vaccine. We explore effective pharmaceutical and non-pharmaceutical interventions to prevent the emergence of vaccine resistance. In chapter 2 we model a particular set of complex vaccination strategies, mosaic and pyramid vaccination, where an immunologically diverse portfolio of vaccines is considered. We find that a bet-hatching strategy, in which vaccine types are distributed in the population, is effective at hindering the evolution of vaccine resistance if mutation rates are high. In chapter 3 we switch gears and present a review on the statistics of records. We highlight similarities and analogies to other models in the fields of statistical physics, evolution and innovation. This offers interesting complimentary perspectives on well-known models. In chapter 4 we apply models of record statistics and innovation to study cultural evolution in sport. We propose a model of sport evolution that combines deterministic improvements in performance and stochastic bursts of improvements due to innovation. }, author = {Rella, Simon}, issn = {2663-337X}, pages = {95}, publisher = {Institute of Science and Technology Austria}, title = {{Adaptive processes in biology and culture : Models of evolving vaccine resistance and the record statistics of innovation}}, doi = {10.15479/AT-ISTA-20811}, year = {2025}, } @phdthesis{20798, author = {Wald, Sebastian}, isbn = {978-3-99078-075-6}, issn = {2663-337X}, keywords = {entanglement-enhanced atom interferometry, cavity QED, spin-squeezing, dipole trap, quantum optics}, pages = {152}, publisher = {Institute of Science and Technology Austria}, title = {{Atoms in a propagating-wave cavity for squeezed Mach-Zehnder atom interferometry}}, doi = {10.15479/AT-ISTA-20798}, year = {2025}, } @phdthesis{20339, abstract = {This thesis investigates the interplay between algebraic and topological methods and combinatorial problems, focusing on approximate graph colourings and mass partitioning. The unifying theme throughout the dissertation is the use of continuous maps and symmetry constraints to extract combinatorial insights. We first explore approximate graph colouring problems and more generally promise constraint satisfaction problems. Using tools from equivariant topology in combination with the general theory of polymorphism of a promise constraint satisfaction problem, we establish hardness for specific types of approximations. In the second part, we address mass partitioning problems, where one seeks to divide geometric objects or measures in Euclidean space into parts of equal size using hyperplanes. Employing techniques from topological combinatorics (configuration space/test map setup and Borsuk–Ulam type theorems), we both obtain a new equipartitioning result in the and provide a fast algorithm for computing equipartitioning of point sets in 3D. }, author = {Tasinato, Gianluca}, issn = {2663-337X}, pages = {106}, publisher = {Institute of Science and Technology Austria}, title = {{Topological methods in discrete geometry and theoretical computer science : Measure partitioning and constraint satisfaction problems}}, doi = {10.15479/AT-ISTA-20339}, year = {2025}, } @phdthesis{19540, abstract = {This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. In Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices. In order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. In Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. In Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. Additionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs). }, author = {Henheik, Sven Joscha}, isbn = {978-3-99078-057-2}, issn = {2663-337X}, pages = {720}, publisher = {Institute of Science and Technology Austria}, title = {{Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems}}, doi = {10.15479/AT-ISTA-19540}, year = {2025}, } @phdthesis{20212, author = {Miranda, Osvaldo}, isbn = {978-3-99078-063-3}, issn = {2663-337X}, keywords = {Pten, mtor, cortical development, MADM, Mapk}, pages = {119}, publisher = {Institute of Science and Technology Austria}, title = {{Unraveling the role of Pten in cortical stem cell lineage progression using MADM}}, doi = {10.15479/AT-ISTA-20212}, year = {2025}, } @phdthesis{20737, author = {Casado Polanco, Raquel}, isbn = {978-3-99078-072-5}, issn = {2663-337X}, keywords = {NOTCH, radial glial progenitor, lineage progression, cortical development}, pages = {133}, publisher = {Institute of Science and Technology Austria}, title = {{Role of NOTCH signaling in radial glial progenitor lineage progression}}, doi = {10.15479/AT-ISTA-20737}, year = {2025}, } @phdthesis{19557, author = {Schwarz, Lena A}, issn = {2663-337X}, pages = {124}, publisher = {Institute of Science and Technology Austria}, title = {{Mapping developmental dynamics of autism spectrum disorder mouse models at single-cell resolution}}, doi = {10.15479/AT-ISTA-19557}, year = {2025}, } @phdthesis{20393, author = {Kishi, Kasumi}, issn = {2663-337X}, pages = {102}, publisher = {Institute of Science and Technology Austria}, title = {{Regulation of notochord and floor plate size during mouse development}}, doi = {10.15479/AT-ISTA-20393}, year = {2025}, } @phdthesis{20371, abstract = {Quantum mechanics reveals a world that defies classical determinism, where uncertainty, superposition, and fluctuations are fundamental aspects. Engineering devices that harness these quantum features requires not only precision, but also a deep understanding of how they interact with their surrounding environment. Superconducting circuits, which exploit macroscopic quantum coherence in low-loss superconducting materials, provide a scalable platform for implementing such systems. Among the critical elements in these circuits, superinductors—high-impedance, dissipation-free inductive components—play a central role by suppressing charge fluctuations. They allow quantum states to be delocalized in phase space, protect qubits from environmental noise, and facilitate access to phenomena such as dual Josephson physics and ultra-strong coupling regimes. This thesis explores two complementary implementations of high-impedance circuits: geometric superinductors, demonstrating that high impedance can be achieved beyond kinetic inductance, and Josephson junction chains, used to investigate both microwave mode properties and DC transport across the superconductor-to-insulator transition. Part I addresses geometric superinductors. Contrary to the common belief that high-impedance superconducting circuits require kinetic inductance, we demonstrate that purely geometric designs can achieve characteristic impedance exceeding the resistance quantum. By exploiting mutual coupling between adjacent turns, coil-based inductors achieve enhanced self-inductance, creating a reliable platform for qubits and resonators. Modeling, simulation, fabrication, and characterization confirm that these elements behave as superinductor. With low loss, high linearity, and minimal stray capacitance, these elements are reproducible, free of uncontrolled tunneling events, and capable of strong magnetic coupling. This establishes geometric superinductors as robust, single-wave-function superconducting devices suitable for hardware protected qubits and hybrid systems. Part II presents classical numerical simulations of a Quantum Phase Slip circuit to study dual Shapiro steps. The circuit consists of an ideal Quantum Phase Slip element embedded in a resistive-inductive environment with a parasitic capacitance. Part III extends the investigation of high characteristic-impedance circuit elements to one-dimensional Josephson junction chains, which act as a quantum simulator for many-body physics and the superconductor–insulator transition. Different devices are realized on both sides of the DC phase transition, showing either a supercurrent branch or Coulomb blockade at zero bias. The effect of the crossover on microwave modes, however, remains insufficiently investigated. Studying these modes provides insight into the interplay between disorder and phase-slip events. Small differences in circuit component sizes determine which side of the transition a device falls on, making these results relevant not only for fundamental understanding but also for the design of quantum devices, emphasizing the crucial role of the electromagnetic environment in stabilizing and controlling fragile quantum states. Together, these results illustrate how carefully engineered high characteristic-impedance elements provide a link between macroscopic circuits and the inherently uncertain quantum world, enabling experiments that probe, control, and ultimately exploit quantum fluctuations for applications in quantum information, metrology, solid state physics and beyond. }, author = {Trioni, Andrea}, isbn = {978-3-99078-067-1}, issn = {2663-337X}, pages = {202}, publisher = {Institute of Science and Technology Austria}, title = {{High-impedance quantum circuits for mesoscopic physics : Geometric superinductors and insulating Josephson Chains}}, doi = {10.15479/AT-ISTA-20371}, year = {2025}, } @phdthesis{19630, abstract = {This thesis consists of three chapters, each corresponding to one publication. While each of these projects tackles a topic in a different area of research, they all share a common thread in the type of topological structure they handle - a partition of space into volumes separated by interfaces that meet in non-manifold junctions. In Chapter 2, we study clusters of soap bubbles from a simulation perspective. In particular, we develop a surface-only algorithm that couples large scale motion and shape deformation of soap bubble clusters with the small scale evolution of the thin film's thickness, which is responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. We model film thickness as a reduced degree of freedom in the Navier-Stokes equations and from them derive three sets of equations governing normal and tangential motion of the soap film surface, as well as the evolution of the thin film thickness. We discretize these equations on a non-manifold triangle mesh, extending and adapting operators to handle complex topology. We also present an incompressible fluid solver for 2.5D films and an advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film. In Chapter 3, we introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts mesh defects, such as overlaps, self-intersections, and inversions into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations, such as those presented in Chapter 2, but with an order of magnitude more interacting bubbles than what we could achieve before, and Boolean unions of non-manifold meshes consisting of millions of triangles. Lastly, in Chapter 4, we utilize developments in the theory of random geometric complexes facilitated by observations from Discrete Morse theory. We survey the methods and results obtained with this new approach, and discuss some of its shortcomings. We use simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.}, author = {Synak, Peter}, issn = {2663-337X}, pages = {106}, publisher = {Institute of Science and Technology Austria}, title = {{Methods for fluid simulation, surface tracking, and statistics of non-manifold structures}}, doi = {10.15479/AT-ISTA-19630}, year = {2025}, }