Ximena Fernández

Ximena Fernández

Lecturer in Mathematics

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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

Pages

Posts

Future Blog Post

less than 1 minute read

Published:

This post will show up by default. To disable scheduling of future posts, edit config.yml and set future: false.

Blog Post number 4

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 3

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 2

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 1

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

code

GAP Package Posets

Published:

Authors: Ximena L. Fernández, Kevin I. Piterman and Iván Sadofschi Costa.

A GAP package for posets and its applications to problems in Combinatorial Group Theory.

outreach

portfolio

publications

Homotopy colimits of diagrams over posets and variations on a theorem of Thomason Permalink

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.

The cylinder of a relation and generalized versions of the Nerve Theorem Permalink

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.

Topological biomarkers for real-time detection of epileptic seizures. Permalink

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.

Intrinsic persistent homology via density-based metric learning. Permalink

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.

Modeled grid cells aligned by a flexible attractor. Permalink

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.

Topological Fingerprints for Audio Identification Permalink

Published in SIAM Journal on Mathematics of Data Science, Vol. 6, Iss. 3 (2024), 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.

Morse theory for group presentations Permalink

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.

talks

teaching

Teaching experience 1

Undergraduate course, University 1, Department, 2014

This is a description of a teaching experience. You can use markdown like any other post.

Teaching experience 2

Workshop, University 1, Department, 2015

This is a description of a teaching experience. You can use markdown like any other post.

work-experience

CAMMESA Permalink

2018. Specialist in Mathematical Modelling.

I was part of the project of optimization of energetic resources, developing mixed linear programming models and stochastic dynamical programming models. I also designed a model for prediction of demand of electric energy in Argentina. I used machine learning, combinatorial optimization and statistical tools. I developed software in Python, Fortran and VisualBasic. I used as input the historical data of electric production and demand, as well as external meteorological and economical features that had latent predictive qualities for this problem.

Properati-OLX Permalink

2019. Data Scientist.

I developed models for: prediction of the price of properties, recommender systems, statistical reports and data analysis. I used machine learning, deep learning and statistical tools, using as input for the models the historical data of advertisements published in the web page, and the interaction of users with the web platform. I developed software in Python and I deployed the models using Google Cloud tools.