2025-26 Department of Mathematics and Statistics Events |
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July 2026 |
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Wed. |
MS Defense Committee/Co-advisors: Dr. Francis Motta and Dr. Papiya Bhattacharjee; committee members: Dr. Parkers Edwards, Dr. Veronika Kuchta Title: Persistent homology, persistence diagrams, and an elementary proof of their stability. Abstract: During this talk, we will explore the foundations of homology needed to build up to persistent homology and its use in Topological Data Analysis (TDA). We will look at the implications of a keystone paper in the field, "Vines and Vineyards by Updating Persistence in Linear Time", by Cohen-Steiner, Edelsbrunner, and Morozov (2006), which provides a conceptually simple proof of diagram stability using boundary matrices and transposition points. My presentation follows "Notes on an Elementary Proof for the Stability of Persistence Diagrams", by Skraba and Turner (2021), which takes the result from Cohen-Steiner, et al. and explores the proof of stability between filtrations. Filtrations can be thought of as functions on a fixed simplicial complex, which may change under perturbations of the underlying data. We track these functions on a simplicial complex noting where there are possible “collisions” of simplices (i.e., non-injectivity of a filtration) as persistence is computed over an interval. I will present on a straight-line homotopy of filtrations and go through the scheme to show that the bottleneck distance between two persistence diagrams can be controlled by the usual infinity norm on filtration values of simplices at the beginning and end of the homotopy interval. |
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Thurs. |
Ph.D. Dissertation Defense Speaker: Matthew Trang, Ph.D. candidate Committee/Co-advisors: Dr. Zvi Rosen (advisor), Dr. Timothy Ford, Dr. Francis Motta, Dr. Stephen Locke Title: Algebraic and Topological Methods in Computational Neuroscience Abstract: Neural data is incredibly rich in combinatorial, topological, and geometrical information, reflecting the intricate shape and connectivity of neural firing patterns. To decipher these structures, neuroscience increasingly relies on advanced mathematical tools to analyze neural activity. Here we study (1) neural codes within the poset PCode of neural codes and (2) the connectivity of neural population activity within the insular cortex when responding to interoceptive information. In (1), we establish combinatorial constructions for all upward covering relations based on what we call "isolated subsets" with supporting theorems and give a slight modification of the existing downward covering relations. We then provide an enumeration algorithm to exhaust all codes covering a given neural code, followed by some computational results. In (2), we infer topological and geometrical differences between the neural connectivity observed around eating events and the activity observed around drinking events of food- and water-deprived mice observed from various analyses via statistical and topological data analysis tools, indicating the use of different neural mechanisms for these two behaviors. We conclude with a discussion of future directions and open problems related to these two areas. |
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Friday |
MS Defense Speaker: Andreus Brammer, Masters candidate Title: A Dynamics Viewpoint on the Approximation Capabilities of Infinite-Depth Neural Networks Abstract: Neural ordinary differential equations (Neural ODEs) provide a framework for modeling continuous-time dynamical systems by parameterizing differential equations with neural networks. This work explores a spectral approach to Neural ODEs in which the solution trajectories and learned dynamics are represented using polynomial basis expansions. By incorporating ideas from numerical approximation theory, the proposed framework replaces traditional time-stepping approaches with global polynomial representations, enabling high-order approximation and improved analysis of the learned dynamics. Chebyshev and Taylor polynomial bases are investigated for representing nonlinear neural differential systems, with efficient spectral differentiation and polynomial multiplication techniques used to evaluate the governing equations. Optimization strategies including resolution continuation, coefficient regularization, and hybrid gradient/Newton methods are examined to improve stability and convergence. Numerical experiments on several target dynamical behaviors demonstrate the ability of the proposed approach to accurately approximate continuous trajectories while providing insight into the relationship between neural parameterization, approximation error, and numerical stability. |
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August 2026 |
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August 3-7 |
CryptoTeens in South Florida Summer Camp Shaping the Future of Cybersecurity The "CryptoTeens in South Florida" is an exciting program tailored for talented high school students - boys and girls - who are passionate about mathematics, computer science, and cybersecurity. Open to all schools in the South Florida region (and beyond), the camp attracts a diverse group of participants and emphasizes a unique focus on cryptography, particularly the cutting-edge field of post-quantum cryptography. There is a participation fee of $90; however, scholarships will be available to fully reimburse students who successfully attend all five days of the program. If you have any questions, please feel free to contact the summer camp organizers at cryptoteens@fau.edu. |
