Research

For a comprehensive list of my papers click here .

Bounding the interleaving distance on concrete categories using a loss function

Olave, Astrid A. and Munch, Elizabeth
arXiv: 2601.09034, 2026

We generalized the loss function defined in Chambers et al., 2025 , to bound the interleaving distance between generalized persistence modules on concrete categories. Our result opens up the potential for an algorithmic approach to bound from above the interleaving distance on commonly used constructions such as multiparameter persistence modules.
arXiv Version
```
@misc{Olave2026bounding,
  title = {Bounding the interleaving distance on concrete categories using a loss function},
  author = {Olave, Astrid A. and Munch, Elizabeth},
  year = {2026},
  eprint = {2601.09034},
  archiveprefix = {arXiv},
  primaryclass = {math.AT},
  arxiv = {2601.09034},
  img = {../images/functor_diagrams.gif},
  comment = {We generalized the loss function defined in  Chambers et al., 2025 , to bound the interleaving distance between generalized persistence modules on concrete categories. Our result opens up the potential for an algorithmic approach to bound from above the interleaving distance on commonly used constructions such as multiparameter persistence modules.}
}
```
The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and zigzag modules. However, computation of the interleaving distance in the vast majority of this settings is known to be NP-hard, limiting its use in practical settings. Inspired by the work of Chambers et al. on the interleaving distance for mapper graphs, we solve a more general problem bounding the interleaving distance between generalized persistence modules on concrete categories via a loss function. This loss function measures how far an assignment, which can be thought of as an interleaving that might not commute, is from defining a true interleaving. We give settings for which the loss can be computed in polynomial time, including for certain assumptions on -parameter persistence modules.
Generalizing the interleaving distance using categorical flows

Olave, Astrid A.
Interleaving distance is a central concept in algebraic topology and topological data analysis. Through a guiding example, this video introduces categorical flows, a functorial operation, that extends the interleaving distance into a measure between objects of virtually any category. This video is part of the tutorial-a-thon 2025 Spring by the Applied Algebraic Topology Research Network (AATRN)
```
@misc{Olavevideo,
  title = {Generalizing the interleaving distance using categorical flows},
  author = {Olave, Astrid A.},
  year = {2025},
  youtube = {https://www.youtube.com/embed/Yxar6UuoGSk?si=Z7UR5bL9u7z4CKiT},
  comment = {Interleaving distance is a central concept in algebraic topology and topological data analysis. Through a guiding example, this video introduces categorical flows, a functorial operation, that extends the interleaving distance into a measure between objects of virtually any category. This video is part of the  tutorial-a-thon 2025 Spring  by the Applied Algebraic Topology Research Network (AATRN)}
}
```
Revealing brain network dynamics during the emotional state of suspense using TDA

Olave, Astrid A. and Perea, Jose A. and Gómez, Francisco
Network Neuroscience, 2025

We applied Mapper, a tool from Topological Data Analysis (TDA), to find the relation of brain dynamics and suspense. As result, we found changes in the functional connectivity within and between large-scale networks associated with the level of suspense. If you want check all the details in my master’s thesis .
Published Version arXiv Version
```
@article{Olave2025,
  title = {Revealing brain network dynamics during the emotional state of suspense using TDA},
  issn = {2472-1751},
  url = {http://dx.doi.org/10.1162/netn.a.34},
  biorxiv = {2024.01.29.577820v2},
  doi = {10.1162/netn.a.34},
  journal = {Network Neuroscience},
  publisher = {MIT Press},
  author = {Olave, Astrid A. and Perea, Jose A. and Gómez, Francisco},
  year = {2025},
  month = aug,
  pages = {1–25},
  img = {../images/chosen_mapper.png},
  comment = {We applied Mapper, a tool from Topological Data Analysis (TDA), to find the relation of brain dynamics and suspense. As result, we found changes in the functional connectivity within and between large-scale networks associated with the level of suspense.  
  If you want check all the details in my  master's thesis . }
}
```
Suspense is an affective state that is ubiquitous in human life, from art to quotidian events. However, little is known about the behavior of large-scale brain networks during suspenseful experiences. To address this question, we examined the continuous brain responses of participants watching a suspenseful movie, along with reported levels of suspense from an independent set of viewers. We employ sliding window analysis and Pearson correlation to measure functional connectivity states over time. Then, we use Mapper, a topological data analysis tool, to obtain a graphical representation that captures the dynamical transitions of the brain across states; this representation enables the anchoring of the topological characteristics of the combinatorial object with the measured suspense. Our analysis revealed changes in functional connectivity within and between the salience, fronto-parietal, and default networks associated with suspense. In particular, the functional connectivity between the salience and fronto-parietal networks increased with the level of suspense. In contrast, the connections of both networks with the default network decreased. Together, our findings reveal specific dynamical changes in functional connectivity at the network level associated with variation in suspense, and suggest topological data analysis as a potentially powerful tool for studying dynamic brain networks.
Lift & project systems performing on the partial-vertex-cover polytope

Georgiou, Konstantinos and Jiang, Andy (Jia) and Lee, Edward and Olave, Astrid A. and Seong, Ian and Upadhyaya, Twesh
Theoretical Computer Science, 2020

In the 2017 Fields Undergraduate Summer Research Program we investigated integrality gaps of linear programming and semidefinite programming derived by lift-and-project systems for the NP-hard t-Partial-Vertex-Cover problem.
Published Version arXiv Version
```
@article{Georgiou2020,
  doi = {10.1016/j.tcs.2020.03.004},
  url = {https://doi.org/10.1016/j.tcs.2020.03.004},
  arxiv = {1409.6365},
  year = {2020},
  month = jun,
  publisher = {Elsevier {BV}},
  volume = {820},
  pages = {1--16},
  author = {Georgiou, Konstantinos and Jiang, Andy (Jia) and Lee, Edward and Olave, Astrid A. and Seong, Ian and Upadhyaya, Twesh},
  title = {Lift {\&} project systems performing on the partial-vertex-cover polytope},
  journal = {Theoretical Computer Science},
  img = {../images/serre-matrix.png},
  comment = {In the 2017 Fields Undergraduate Summer Research Program we investigated integrality gaps of linear programming and semidefinite programming derived by lift-and-project systems for the NP-hard t-Partial-Vertex-Cover problem.}
}
```
We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovász-Schrijver-SDP (LS+), Sherali-Adams-SDP (SA+), and Lasserre-SDP (La) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. With starting point this LP relaxation, our main results assert that for every , level- LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least , where n is the number of vertices of the input graph. Our lower bounds are nearly tight, in that level-n relaxations, even of the weakest systems, have integrality gap 1. Additionally, we give a integrality gap for the Level-1 Lasserre system and a superconstant general integrality gap for all Level- Lasserre derived SDPs. As lift-and-project systems have given the best algorithms known for numerous combinatorial optimization problems, our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for . As further motivation for our results, we show that the SDP that has given the best algorithm known for t-PVC has integrality gap on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, we believe our results are of independent interest as they are among the very few known integrality gap lower bounds for LP and SDP 0-1 relaxations in which not all variables possess the same semantics in the underlying combinatorial optimization problem. To achieve our results, we utilize a common methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system). The latter sheds some light as to why La tightenings seem strictly stronger than LS+ or SA+ tightenings.
Persistent Homology for clusterization of RNA secondary structures

Olave, Astrid A.
As part of my bachelor’s thesis I worked with Professors Gustavo Rubiano and Clara Bermudez applying persistent homology to cluster RNA secondary structures. We interpreted the 0-homology classes (connected-components) as clusters, resulting a method comparable to hierarchical clustering.
```
@thesis{Olavepershom,
  title = {Persistent Homology for clusterization of RNA secondary structures},
  author = {Olave, Astrid A.},
  year = {2018},
  school = {Universidad Nacional de Colombia},
  img = {../images/comp_bar20.gif},
  comment = {As part of my  bachelor's thesis  I worked with Professors Gustavo Rubiano and Clara Bermudez applying persistent homology to cluster RNA secondary structures. We interpreted the 0-homology classes (connected-components) as clusters, resulting a method comparable to hierarchical clustering.}
}
```

Research

Bounding the interleaving distance on concrete categories using a loss function

Olave, Astrid A. and Munch, Elizabeth

Generalizing the interleaving distance using categorical flows

Olave, Astrid A.

Revealing brain network dynamics during the emotional state of suspense using TDA

Olave, Astrid A. and Perea, Jose A. and Gómez, Francisco

Lift & project systems performing on the partial-vertex-cover polytope

Georgiou, Konstantinos and Jiang, Andy (Jia) and Lee, Edward and Olave, Astrid A. and Seong, Ian and Upadhyaya, Twesh

Persistent Homology for clusterization of RNA secondary structures

Olave, Astrid A.