Research
My PhD thesis is about:
My current research interests fall into two main areas:
- Applied Algebraic Topology and Geometry,
- Computational and Combinatorial Topology.
I am specially interested in Topological Data Analysis, Geometric Inference and Homotopy Theory.
Publications & preprints
Published in Trans. Amer. Math. Soc. 377 (2024), 2495-2523, 2024
We introduce a novel combinatorial method to study Q∗∗-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews-Curtis conjecture do satisfy the conjecture.
Code. Tutorial.
Published in SIAM Journal on Mathematics of Data Science (accepted), 2024
(with W. Reise, M. Dominguez, H.A. Harrington and M. Beguerisse-Diaz) We present a topological audio fingerprinting approach for robustly identifying duplicate audio tracks. Our method applies persistent homology on local spectral decompositions of audio signals, using filtered cubical complexes computed from mel-spectrograms. By encoding the audio content in terms of local Betti curves, our topological audio fingerprints enable accurate detection of time-aligned audio matchings. Experimental results demonstrate the accuracy of our algorithm in the detection of tracks with the same audio content, even when subjected to various obfuscations. Our approach outperforms existing methods in scenarios involving topological distortions, such as time stretching and pitch shifting.
Published in eLife12:RP89851 , 2023
(with S. Benas and E. Kropff) Grid cells are a key component of the neural system for mapping the position of an individual within a physical environment. These entorhinal neurons fire in a characteristic hexagonal pattern of locations, and are organized in modules that collectively form a population code for the animal’s allocentric position. Whereas the population activity of grid cells in the same module lies in a toroidal manifold, the connectivity pattern of grid cells in the brain is still unknown. It is conjectured that the neural network of grid cells has also a toroidal architecture, inherited by the geometry of the attractor. In this work, we give a negative answer to this question for modeled grid cells.
Poster.
Published in Journal of Machine Learning Research 24(75):1−42, 2023
(with E. Borghini, P. Groisman and G. Mindlin) We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this method outperforms more standard approaches based on Euclidean norm with theoretical results and computational experiments.
Code. Tutorial.
Published in arXiv:2211.02523, 2022
(with D. Mateos) Automated seizure detection is a fundamental problem in computational neuroscience towards diagnosis and treatment's improvement of epileptic disease. We propose a real-time computational method for automated tracking and detection of epileptic seizures from raw neurophysiological recordings. Our mechanism is based on the topological analysis of the sliding-window embedding of the time series derived from simultaneously recorded channels. We extract topological biomarkers from the signals via the computation of the persistent homology of time-evolving topological spaces. Remarkably, the proposed biomarkers robustly captures the change in the brain dynamics during the ictal state. We apply our methods in different types of signals including scalp and intracranial EEG and MEG, in patients during interictal and ictal states, showing high accuracy in a range of clinical situations.
Slides.
Published in Discrete and Computational Geometry 63, 549–559, 2020
(with E.G. Minian) We introduce the notion of cylinder of a relation in the context of posets, extending the construction of the mapping cylinder. We establish a local-to-global result for relations, generalizing Quillen's Theorem A for order preserving maps, and derive novel formulations of the classical Nerve Theorem for posets and simplicial complexes, suitable for covers with not necessarily contractible intersections.
Published in Homology, Homotopy and Applications vol. 18 issue 2., 2016
(with E.G. Minian) We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason’s theorem on homotopy colimits over posets. In particular, this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen’s Theorem A for posets.
