Research
My work is in applied topology and applications to biology with machine learning algorithms. I believe many classical mathematical theories have yet to find their applications in data science problems. Below is a more detailed description of my projects past and current.
Neuroscience Projects
Currently I am working with Dr. Itskov in applied topology and theoretical neuroscience on the following projects:
The combinatorial code and the graph rules of Dale recurrent networks
A preprint of this work is available here. We describe the combinatorics of equilibria and steady states of neurons in threshold-linear networks that satisfy the Dale’s law. The combinatorial code of a Dale network is characterized in terms of two conditions: (i) a condition on the network connectivity graph, and (ii) aspectral condition on the synaptic matrix. We find that in the weak coupling regime the combinatorial code depends only on the connectivity graph, and not on the particulars of the synaptic strengths. Moreover, we prove that the combinatorial code of a weakly coupled network is a sublattice, and we provide a learning rule for encoding a sublattice in a weakly coupled excitatory network. In the strong coupling regime we prove that the combinatorial code of a generic Dale network is intersection-complete and is therefore a convex code, as is common in some sensory systems in the brain.
Shapley fields: interpretable chemical maps on the oflactory bulb of mice
This work is in preparation. We build a machine learning model to predict glomeruli activation in the olfactory bulb of mice, given a set of stimuli. The features of the model are chemical features provided by experts. We provide machine learning interpretability to the experts by computing Shapley values of the model. Shapley values provide both local and global ``fair" distribution of feature contributions to a model . We introduce Shapley fields; one field per chemical feature. These are akin to a probability distribution on the olfactory bulb. For a given feature, the field represents a distribution of its contribution for the model prediction for each glomerulus across all stimuli. We show the Shapley fields are stereotypical across animals and are stable with respect to hyperparameter changes. This finding is consistent across different feature sets as well. Using the Shapley fields, we perform hierarchical clustering on the olfactory bulb, revealing a mosaic distribution. This distribution is statistically real and furthermore some clusters correspond to a class of well identified glomeruli. There is a small gap present between the performances of our model and the baseline peer-prediction model. This suggests that there might be a better set of chemical features to use for predicting glomerular activation. In a future project we plan on learning the relevant chemistry using machine learning, as opposed to relying on features suggested by experts.
Social Networks Analysis Projects
I am also working on two projects on social networks analysis with Dr. Nina Otter being the PI. These projects were initiated at the 2022 MRC in Applied Category Theory. Talks about both of these projects will be given at the JMM in January 2023.
Positional analysis for social simplicial complexes
Role analysis for social simplicial complexes
Applied Topology Projects
Singular homology of roots of unity
A preprint of this work is available here. I proved analogues of fundamental results from algebraic topology in the setting of Čech's closure spaces. For a singular homology theory of closure spaces, I proved analogues of the excision and Mayer-Vietoris theorems. Furthermore, I proved a Hurewicz theorem in dimension one. I used these results to calculate examples of singular homology groups of spaces that are not topological, but one often encounters in applied topology, such as simple undirected graphs. I mainly focused on singular homology of roots of unity with closure structures arising from considering nearest neighbors. These ''simpler'' closure spaces can then serve as building blocks along with Mayer-Vietoris and excision theorems for calculating the singular homology of more complex closure spaces.
Eilenberg-Steenrod homology and cohomology theories for Čech's closure spaces
A preprint of this work is available here. We generalize some of the fundamental results of algebraic topology from topological spaces to Čech's closure spaces, also known as pretopological spaces. Using simplicial sets and cubical sets with connections, we define three distinct singular (relative) simplicial and six distinct singular (relative) cubical (co)homology groups of closure spaces. Using acyclic models we show that the three simplicial groups have isomorphic cubical analogues among the six cubical groups. Thus, we obtain a total of six distinct singular (co)homology groups of closure spaces. Each of these is shown to have a compatible homotopy theory that depends on the choice of a product operation and an interval object. We give axioms for an Eilenberg-Steenrod (co)homology theory with respect to a product operation and an interval object. We verify these axioms for three of our six (co)homology groups. For the other three, we verify all of the axioms except excision, which remains open for future work. We also show the existence of long exact sequences of (co)homology groups coming from Künneth theorems and short exact (co)chain complex sequences of pairs of closure spaces.
Past Projects
I worked in the field of Topological Data Analysis (TDA) as part of Dr. Bubenik’s research group in the Department of Mathematics at the University of Florida. TDA is a new branch of mathematics and some theoretical foundational work is still required. I developed theoretical aspects of TDA. In particular, I used applied sheaf theory in advancing ideas in TDA. I also used old ideas by Eduard Čech about Čech closure spaces, a generalization of topological spaces. I also collaborated with the Vitriol Lab and using TDA and machine learning algorithms to fluorescent microscopy images of cells that have undergone some perturbations.
I have also mentored undergraduate students in Peter Bubenik’s research group:
David Freeman, analysis of microscopic cell images using TDA.
Gianfranco Cortes, characterizing types of singularities in algebraic varieties via machine learning.
More detailed descriptions of the above mentioned projects are given below:
Homological Algebra for Persistence Modules
A published version of this work is available here. There is a video of my co-author Peter Bubenik talking about the project at the AATRN seminar here. A short tutorial discussing some ideas of this work, for the AATRN and WinCompTop Tutorial-a-thon is available here.
In TDA, we often start with data that was processed to obtain a diagram of topological spaces, such as a filtered cubical or simplicial complex. Applying a homology functor with field coefficients we obtain a diagram of vector spaces.Such a diagram is called a persistence modules and in applications this object decomposes as a direct sum of indecomposables thanks to a theorem by Gabriel.In this project, together with Peter Bubenik, I developed some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. I showed the two theories are different. I considered the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and gave explicit computations for interval modules. I gave a classification of injective, projective, and flat interval modules. I stated Kunneth theorems and universal coefficient theorems for the homology and cohomology of complexes of persistence modules in both the sheaf and graded modules settings and showed how these theorems can be applied to persistence modules arising from filtered cell complexes. I also gave a Gabriel-Popescu theorem for persistence modules. Finally, I examined categories enriched over persistence modules. I showed that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.
TDAExplore: quantitative analysis of fluorescence microscopy images through topology based machine learning
This work is published here. A preprint is also available here. We presented a new set of features for image analysis based on two topological data analysis (TDA) methods: persistent homology and persistence landscapes. We then exhibit an image analysis pipeline, TDAExplore, that uses these features for image classification and image segmentation. In addition to being highly accurate, with minimal training, in assigning images to their correct group, TDAExplore quantifies how much images resemble the training data and identifies which parts are different, an improvement over other machine learning models that do not permit insight into how classification tasks were made. This work represents progress into a future where machine learning identifies and describes nuanced image features in ways that allow researchers to answer important biological questions and generate new hypotheses for future studies.
Convolution of persistence modules
A preprint of this work is available here.
Sheaves and cosheaves have found many applications in data science problems of the local-to-global character. A common perspective in applications is to study sheaves and cosheaves on partially ordered sets often valued in vector spaces over a field. The thesis work of Justin Curry showed that functors from a partially ordered set into a “nice” category are equivalent to sheaves and cosheaves on open and closed sets of the Alexandrov topology on the partially ordered set, respectively, valued in said category. An example of this are cellular sheaves and cosheaves. On an arbitrary topological space sheaf cohomology is well defined and studied in the derived setting for any sheaf. On the other hand, cosheaf homology is only defined for constant or locally constant cosheaves. However, on finite partially ordered sets one can construct a rich sheaf cohomology andcosheaf homology theory in the framework of derived functors for any sheaf and cosheaf. One can even study entropy and information theory from this point of view.
This project is analogous in spirit to the work by Kashiwara and Schapira for bounded derived complexes of constructible sheaves. It turns out that persistence modules themselves are both sheaves and cosheaves on R with the Alexandrov topology. I define sheaf and cosheaf convolution operations for bounded derived complexes of persistence modules. I define a convolution distance on the bounded derived category of (multiparameter) persistence modules. I show this distance extends the classical interleaving distance of persistence modules. I prove stability results for this distance. I show that the convolution of cosheaves is canonically isomorphic to the derived graded module tensor product operation. I also show the cosheaf convolution has a right adjoint. This adjunction is analogous to the Tamarkin-Kashiwara-Schapira adjunction for constructible sheaves.
Homotopy, homology and persistent homology using closure spaces and filtered closure spaces.
A preprint of this work is available here. A video of my talk on this work is available here.
We use Čech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain nine homology and six homotopy theories of closure spaces. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Čech complexes to closure spaces and prove that their persistent homology is stable.
