@unpublished{petrov2024essence,
  title         = {The Essence of de Rham Cohomology},
  author        = {Petrov, Alice},
  year          = {2024},
  note          = {arXiv preprint},
  archivePrefix = {arXiv},
  eprint        = {2411.06296},
  url_paper     = {https://alicepetrov.github.io/assets/pdf/papers/2024/petrov2024essence.pdf},
  abstract      = {The study of differential forms that are closed but not exact reveals important information about the global topology of a manifold, encoded in the de Rham cohomology groups H^k(M), named after Georges de Rham (1903-1990). This expository paper provides an explanation and exploration of de Rham cohomology and its equivalence to singular cohomology. We present an intuitive introduction to de Rham cohomology and discuss four associated computational tools: the Mayer-Vietoris theorem, homotopy invariance, Poincaré duality, and the Künneth formula. We conclude with a statement and proof of de Rham's theorem, which asserts that de Rham cohomology is equivalent to singular cohomology.},
  keywords      = {Algebraic Topology}
}

The study of differential forms that are closed but not exact reveals important information about the global topology of a manifold, encoded in the de Rham cohomology groups H^k(M), named after Georges de Rham (1903-1990). This expository paper provides an explanation and exploration of de Rham cohomology and its equivalence to singular cohomology. We present an intuitive introduction to de Rham cohomology and discuss four associated computational tools: the Mayer-Vietoris theorem, homotopy invariance, Poincaré duality, and the Künneth formula. We conclude with a statement and proof of de Rham's theorem, which asserts that de Rham cohomology is equivalent to singular cohomology.

@mastersthesis{petrov2024dissertation,
  title      = {Persistent Homology for the Analysis of Stratified Spaces},
  author     = {Petrov, Alice},
  school     = {University of Oxford},
  year       = {2024},
  url_paper  = {https://alicepetrov.github.io/assets/pdf/papers/2024/petrov2024dissertation.pdf},
  abstract   = {This dissertation investigates the application of persistent homology to the analysis of stratified spaces, focusing on word embeddings as a case study. Stratified spaces present unique challenges for traditional topological data analysis techniques, particularly in identifying and analyzing singularities. To address these challenges, we discuss persistent intersection homology and extend the concepts of kernel, image, and cokernel persistence. We also present a novel approach to analyzing singularities using bifiltrations and sliding window (co)kernel diagrams. In addition, we discuss multiparameter persistence. These tools allow for a more nuanced analysis, enabling the differentiation of various types of singularities, such as those associated with polysemous words in word embeddings.
We demonstrate the practical utility of these theoretical advancements through computational pipelines and experiments on word embeddings, specifically analyzing how dimensionality reduction techniques influence the topology of local neighborhoods. The results indicate that our methods can reveal significant structural differences in word embeddings, offering new insights into their topological properties.
This work advances the theoretical understanding of persistent homology in stratified spaces and opens new avenues for applying these techniques to other complex, high-dimensional datasets. We conclude with a discussion of potential future research directions, including extending these methods to multifiltrations, exploring different invariants, and broader applications beyond word embeddings.},
  keywords   = {Computational Topology}
}

This dissertation investigates the application of persistent homology to the analysis of stratified spaces, focusing on word embeddings as a case study. Stratified spaces present unique challenges for traditional topological data analysis techniques, particularly in identifying and analyzing singularities. To address these challenges, we discuss persistent intersection homology and extend the concepts of kernel, image, and cokernel persistence. We also present a novel approach to analyzing singularities using bifiltrations and sliding window (co)kernel diagrams. In addition, we discuss multiparameter persistence. These tools allow for a more nuanced analysis, enabling the differentiation of various types of singularities, such as those associated with polysemous words in word embeddings. We demonstrate the practical utility of these theoretical advancements through computational pipelines and experiments on word embeddings, specifically analyzing how dimensionality reduction techniques influence the topology of local neighborhoods. The results indicate that our methods can reveal significant structural differences in word embeddings, offering new insights into their topological properties. This work advances the theoretical understanding of persistent homology in stratified spaces and opens new avenues for applying these techniques to other complex, high-dimensional datasets. We conclude with a discussion of potential future research directions, including extending these methods to multifiltrations, exploring different invariants, and broader applications beyond word embeddings.

@inproceedings{petrov2023state,
  title        = {From State Spaces to Semigroups: Leveraging Algebraic Formalism for Automated Planning},
  author       = {Petrov, Alice and Muise, Christian},
  booktitle    = {Workshop on Heuristics and Search for Domain-Independent Planning (HSDIP)},
  organization = {ICAPS},
  year         = {2023},
  url_paper    = {https://alicepetrov.github.io/assets/pdf/papers/2023/petrov2023state.pdf},
  abstract     = {This paper introduces an algebraic formalism linking transformation semigroups and the state transition systems induced by classical planning problems. We investigate some basic planning problems with interesting properties and establish fundamental characteristics of the corresponding semigroups, such as their ideals and Green's relations. Furthermore, we leverage semigroup theory to propose new approaches to existing concepts in automated planning, including the identification of landmark actions and the study of dead ends. We demonstrate that algebraic results can be applied to facilitate an understanding of a planning problem's state space and explore its solutions, thus verifying the relevance and effectiveness of such formal modeling.},
  keywords     = {AI Planning & Reasoning}
}

This paper introduces an algebraic formalism linking transformation semigroups and the state transition systems induced by classical planning problems. We investigate some basic planning problems with interesting properties and establish fundamental characteristics of the corresponding semigroups, such as their ideals and Green's relations. Furthermore, we leverage semigroup theory to propose new approaches to existing concepts in automated planning, including the identification of landmark actions and the study of dead ends. We demonstrate that algebraic results can be applied to facilitate an understanding of a planning problem's state space and explore its solutions, thus verifying the relevance and effectiveness of such formal modeling.

@inproceedings{christen2023paris,
  title        = {{PARIS}: Planning Algorithms for Reconfiguring Independent Sets},
  author       = {Christen, Remo and Eriksson, Salom{\'e} and Katz, Michael and Muise, Christian and Petrov, Alice and Pommerening, Florian and Seipp, Jendrik and Sievers, Silvan and Speck, David},
  booktitle    = {Proceedings of the 30th European Conference on Artificial Intelligence (ECAI 2023)},
  year         = {2023},
  url_paper    = {https://alicepetrov.github.io/assets/pdf/papers/2023/christen2023paris.pdf},
  abstract     = {Combinatorial reconfiguration is the problem of transforming one solution of a combinatorial problem into another, where each transformation may only apply small changes to a solution and may not leave the solution space. An important example is the independent set reconfiguration (ISR) problem, where an independent set of a graph (a subset of its vertices without edges between them) has to be transformed into another by a sequence of transformations that can replace a vertex in the current subset such that the new subset is still an independent set. The 1st Combinatorial Reconfiguration Challenge (CoRe Challenge 2022) was a competition focused on the ISR problem. The PARIS team successfully participated with two solvers that model the ISR problem as a planning task and employ different planning techniques for solving it. In this work, we describe these models and solvers. For a fair comparison to competing ISR approaches, we re-run the entire competition under equal computational conditions. Besides showcasing the success of planning technology, we hope that this work will create a cross-fertilization of the two research fields. },
  keywords     = {AI Planning & Reasoning}
}

Combinatorial reconfiguration is the problem of transforming one solution of a combinatorial problem into another, where each transformation may only apply small changes to a solution and may not leave the solution space. An important example is the independent set reconfiguration (ISR) problem, where an independent set of a graph (a subset of its vertices without edges between them) has to be transformed into another by a sequence of transformations that can replace a vertex in the current subset such that the new subset is still an independent set. The 1st Combinatorial Reconfiguration Challenge (CoRe Challenge 2022) was a competition focused on the ISR problem. The PARIS team successfully participated with two solvers that model the ISR problem as a planning task and employ different planning techniques for solving it. In this work, we describe these models and solvers. For a fair comparison to competing ISR approaches, we re-run the entire competition under equal computational conditions. Besides showcasing the success of planning technology, we hope that this work will create a cross-fertilization of the two research fields.

@inproceedings{petrov2023automated,
  title        = {Automated Planning Techniques for Elementary Proofs in Abstract Algebra},
  author       = {Petrov, Alice and Muise, Christian},
  booktitle    = {SPARK: Scheduling and Planning Applications Workshop},
  organization = {ICAPS},
  year         = {2023},
  url_paper    = {https://alicepetrov.github.io/assets/pdf/papers/2023/petrov2023automated.pdf},
  abstract     = {This paper explores the application of automated planning to automated theorem proving, which is a branch of automated reasoning concerned with the development of algorithms and computer programs to construct mathematical proofs. In particular, we investigate the use of planning to construct elementary proofs in abstract algebra, which provides a rigorous and axiomatic framework for studying algebraic structures such as groups, rings, fields, and modules. We implement basic implications, equalities, and rules in both deterministic and non-deterministic domains to model commutative rings and deduce elementary results about them. The success of this initial implementation suggests that the well-established techniques seen in automated planning are applicable to the relatively newer field of automated theorem proving. Likewise, automated theorem proving provides a new, challenging domain for automated planning.},
  keywords     = {AI Planning & Reasoning}
}

This paper explores the application of automated planning to automated theorem proving, which is a branch of automated reasoning concerned with the development of algorithms and computer programs to construct mathematical proofs. In particular, we investigate the use of planning to construct elementary proofs in abstract algebra, which provides a rigorous and axiomatic framework for studying algebraic structures such as groups, rings, fields, and modules. We implement basic implications, equalities, and rules in both deterministic and non-deterministic domains to model commutative rings and deduce elementary results about them. The success of this initial implementation suggests that the well-established techniques seen in automated planning are applicable to the relatively newer field of automated theorem proving. Likewise, automated theorem proving provides a new, challenging domain for automated planning.

@unpublished{sha2022analyzing,
  title     = {Analyzing the Robustness of Open Source Software Ecosystems to the Loss of Contributors: A Case Study},
  author    = {Sha, Zhendong and Petrov, Alice and Tian, Yuan and Hu, Ting},
  year      = {2022},
  note      = {SSRN Electronic Journal, Available at SSRN: https://ssrn.com/abstract=4082801},
  url_paper = {https://alicepetrov.github.io/assets/pdf/papers/2022/sha2022analyzing.pdf},
  abstract  = {The health and sustainability of an open-source software (OSS) ecosystem depends on its contributors' initiative. Unfortunately, studies have shown that OSS projects often suffer from high contributor turnover rates due to their open-source nature. High contributor turnover rates can have negative impacts on the community involved in individual projects and the ecosystem that relies on such projects. We propose a computational model to quantify the robustness of a software ecosystem to contributor loss and perform a case study on two OSS ecosystems, i.e., Ruby and PyPI. We utilize a simulation method to analyze project extinction risk due to contributor loss at the ecosystem-level. We apply the proposed method on 12,267,933 and 1,459,322 commits scraped from 102,447 Ruby projects and 69,311 PyPI libraries respectively, hosted on GitHub. To identify the factors that influence the robustness of ecosystems, we propose an ecosystem simulation method to generate artificial ecosystems with different control parameters.We find that contributor turnover and project abandonment frequently happen in Ruby and PyPI ecosystems. Moreover, the preference of developers to contribute to large projects in an OSS ecosystem negatively affects the robustness of the ecosystem. Both studied ecosystems are less robust to the loss of active contributors who either intensively or extensively contribute to the ecosystem than the random loss of contributors.Our proposed methods can be leveraged to analyze the robustness of a software ecosystem to contributor loss over time. Moreover, we provide a ecosystem simulation method to analyze how various factors determine the robustness of an OSS ecosystem, demonstrating the potential of testing a hypothesis without empirical data for ecosystem robustness analysis.},
  keywords  = {Software Engineering}
}

The health and sustainability of an open-source software (OSS) ecosystem depends on its contributors' initiative. Unfortunately, studies have shown that OSS projects often suffer from high contributor turnover rates due to their open-source nature. High contributor turnover rates can have negative impacts on the community involved in individual projects and the ecosystem that relies on such projects. We propose a computational model to quantify the robustness of a software ecosystem to contributor loss and perform a case study on two OSS ecosystems, i.e., Ruby and PyPI. We utilize a simulation method to analyze project extinction risk due to contributor loss at the ecosystem-level. We apply the proposed method on 12,267,933 and 1,459,322 commits scraped from 102,447 Ruby projects and 69,311 PyPI libraries respectively, hosted on GitHub. To identify the factors that influence the robustness of ecosystems, we propose an ecosystem simulation method to generate artificial ecosystems with different control parameters.We find that contributor turnover and project abandonment frequently happen in Ruby and PyPI ecosystems. Moreover, the preference of developers to contribute to large projects in an OSS ecosystem negatively affects the robustness of the ecosystem. Both studied ecosystems are less robust to the loss of active contributors who either intensively or extensively contribute to the ecosystem than the random loss of contributors.Our proposed methods can be leveraged to analyze the robustness of a software ecosystem to contributor loss over time. Moreover, we provide a ecosystem simulation method to analyze how various factors determine the robustness of an OSS ecosystem, demonstrating the potential of testing a hypothesis without empirical data for ecosystem robustness analysis.