DIMACS REU 2025
Name: Nadya Belova
Project Mentor: Facundo Mémoli
Email: nadyabelova271@gmail.com
Office Location: CoRE, Room 434
University: Rutgers University
I would like to thank Facundo Mémoli and Sriram Raghunath for their guidance and support throughout this project. I'd also like to thank the Rutgers Math Department and the NSF for funding my participation in DIMACS.
I am an undergraduate at Rutgers University studying mathematics. I particularly enjoy algebra and feel an affinity for classifying finite groups. The kinds of arguments in Section 6.2 Applications to Groups of Medium Order of Dummit and Foote's algebra book are a good motivating example of what I like to do. I'm also interested more sophisticated machinery in group classification, especially those used in the famous classification of finite simple groups (shout out to Daniel Gorenstein for this and also founding DIMACS!). I also enjoy category theory, homological algebra, and 'abstract nonsense' in general. Mathematicians whose names I've heard a lot and wish to know about their work include Galois, Burnside, Jordan, Serre, Cartan, Grothendieck, Mac Lane, and Eilenberg (this list is also intended to give a vibe for my interests). I am also interested in algebraic topology and commutative algebra. In the future I've been thinking of perusing low dimensional topology or algebraic geometry, but would love to stay in pure algebra.
Math is honestly the main thing I've got going on, but outside of that, I really enjoy spending time with friends and others generally. I like to watch movies, particular movies intended for women in the USSR from 1955-1985. I love the music of Adrianne Lenker, and I casually play tennis.
This project aims to explore how persistent homology—a key tool from topological data analysis—can be used to investigate the time-dependent structure and the formation of "memories" in biological networks. In particular, we will focus on networks that arise in computational neuroscience, such as those modeling the hippocampus, a brain region known to play a central role in spatial navigation and memory formation.
The project will begin with an introduction to the mathematical foundations of persistent homology, including simplicial complexes, filtrations, and persistence diagrams. Students will also be introduced to basic concepts in computational neuroscience relevant to network structure and function, with an emphasis on how activity and connectivity in the brain can be represented as dynamic graphs or simplicial complexes.
A significant component of the project will involve learning how to implement computational pipelines for analyzing such networks using persistent homology. Depending on the student's background and interest, this may include hands-on work with simulated data, numerical experiments, and an exploration of how persistent features correlate with biologically meaningful patterns—such as the representation of spatial environments or the formation of stable memory traces.
I moved into the DIMACS apartments and met my roommates. I also had my first meeting with my mentor and his research group. My research partner Andrew and I were given some papers to read and started thinking about research directions for our project.
Andrew and I presented to our fellow DIMACS participants about my project. We presented to our research group members about the paper we read. We also settled on a concrete direction to study for our project and started thinking about some basic questions.
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