2025-26 Department of Mathematics and Statistics Events |
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July 2026 |
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Wed. |
MS Presentation 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 Presentation 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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Wed. |
MS Presentation Speaker: Sanat Acharya, Masters Candidate Title: Consistency and Asymptotic Normality of Hill's Estimator Abstract: Hill estimator is one of the most widely used estimators for the study of the tail index of heavy-tailed distributions. It has a number of applications such as in finance, banking, insurance, public Health, epidemiology, telecommunication, and information technology. We try to cover the construction of Hill estimator by using conditional likelihood function and giving two special cases- lower tail and upper tail. Furthermore, we will introduce first and second order regular variation to study the consistency and asymptotic normality of the Hill estimator. All are invited to attend. |
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Monday |
MS Presentation Speaker: James Klinkenberg, Masters Candidate Title: Maximal d-elements and a Lemma on Ultrafilters Abstract: For a Tychonoff space X, its ring of continuous functions, C(X), is widely studied. It has been shown that, for C(X), its space of maximal d-ideals is in one-to-one correspondence with the space of ultrafilters on the closures of interiors of zero-sets. In an algebraic frame with the finite intersection property (FIP), say, L, the notion of d-ideals and maximal d-ideals correspond with d-elements and maximal d-elements. In this talk, we will explore a set of ultrafilters, the ultrafilters on the compact polars of L, and prove that the maximal d-elements of L are in one-to-one correspondence with these ultrafilters. In this way, we will have generalized the result above from C(X) to general algebraic frames with FIP. For those who cannot attend in person, James has provided a Zoom link. Join Zoom Meeting https://fau-edu.zoom.us/j/6909396009?omn=82395948308 Meeting ID: 690 939 6009 Find your local number: https://fau-edu.zoom.us/u/kzacsUwex |
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Monday |
MS Presentation Speaker: Morgan Williams, MS Candidate Title: Exploring Gradient Methods in Neural Networks Abstract: This presentation explores the mathematical foundations of gradient-based optimization in neural networks. Beginning with classical optimization, we examine the strengths and limitations of gradient descent before investigating why it remains so successful in training modern neural networks. A bottom-up approach using simple neural network architectures motivates several directions for future research. |
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Friday |
MS Presentation Speaker: Aysha Mohamed Nuhuman, MS Candidate Title: A Dual Representation of Minimal Primes in M-Frames: From Ultrafilters to Maximal Ideals Abstract: The study of minimal prime ideal spaces originated in commutative rings and lattice-ordered groups, then generalized to algebraic frames. In an M-frame (an algebraic frame satisfying the finite intersection property), the Lemma on Ultrafilters establishes a bijective correspondence between the minimal prime elements of the frame and the ultrafilters on its join semi-lattice of compact elements. This talk explores a dual perspective, shifting our focus from the compact elements to their polars. By examining the bounded distributive sub-lattice formed by these compact polars, we establish a dual correspondence paralleling the Lemma on Ultrafilters. Ultimately, we prove that the minimal prime elements of an M-frame can be constructed from the maximal ideals of the polar sub-lattice of compact elements. For those who cannot attend in person, Aysha has provided a Zoom link. Join Zoom Meeting: Zoom Link: https://fau-edu.zoom.us/j/81109050739?pwd=akVZ89aCVar7YeMFawJRoOTWApN5us.1 Meeting ID: 811 0905 0739 Passcode: zhN4hn One tap mobile +1 305 224 1968 US Meeting ID: 811 0905 0739 Find your local number: https://fau-edu.zoom.us/u/kcAdnVMmYf All are invited to attend! |
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Friday |
MS Presentation Speaker: Andrew McCartney, MS Candidate Title: The Priestley Duality Abstract: Stone duality—the relationship between Boolean algebras and compact, zero-dimensional Hausdorff spaces—is a driving force behind the study of ordered algebra. We extend Stone's framework to a duality between bounded distributive lattices and Priestley spaces. Chapter 1 explores the prime ideal space of a bounded distributive lattice as a topological space. Building on this foundation, Chapter 2 constructs the Priestley duality. Finally, Chapter 3 examines the special case of Boolean algebras, demonstrating how the Priestley duality naturally extends Stone's original work. For those who cannot attend in person, Andrew has provided a Zoom link. Join Zoom Meeting: https://fau-edu.zoom.us/j/88665840678?pwd=F2QKwbYsLzReCHhakatatJE1HRr7x0.1 Topic: MS EXAM - ANDREW MCCARTNEY Time: Jul 31, 2026 02:00 PM Eastern Time (US and Canada) Meeting ID: 886 6584 0678 Passcode: a9gUJN One tap mobile +1 305 224 1968 US Meeting ID: 886 6584 0678 Passcode: 326356 All are invited to attend. |
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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. |
