REU and Star abstracts summer 2025
9:30 AM to 9:55 AM
Tiling: Even Harder to Decide Than a Restaurant
Keynote speaker, Dr. Casey Mann
Abstract: In this talk, we will discuss various problems related to the complicated kinds of tilings (and non-tilings) that can formed by copies of a single tile and how these problems are linked to the Tiling Problem for Monotiles, which asks if there can exist an algorithm that takes as input a single tile decides whether or not a tiling of the plane can be formed from congruent copies of the input tile; if no such algorithm can exist, the Tiling Problem for Monotiles is said to be undecidable. The Tiling Problem for Monotiles is a central unsolved problem in the theory of tilings.
10:00 AM to 10:25 AM
Topological data analysis techniques in genetic stock identification of Chinook Salmon
Rosemary Hartless, Abigail Nigro and Alison Watson
Research mentor: Dr. Elizabeth Field.
Abstract: The Southern Resident Killer Whale (SRKW) is an endangered species of orca living off the coast of the Pacific Northwest and has evolved to eat primarily Chinook salmon. The survival of the SRKW is contingent on ensuring the continued availability of this food source. As Chinook salmon are important culturally, ecologically, and economically, conservation efforts depend on precise identification of what geographic and temporal genetic stocks of salmon are being consumed by the SRKW. Scientists can only use non-invasive methods to study the SRKW, which means that the individual genetic stocks of salmon must be identified through fecal samples which contain a mixture of individuals. We utilized the tools of topological data analysis to identify viable techniques to differentiate between these genetic stocks. The techniques we explored include computing persistent homology and utilizing the Mapper algorithm. Several of our approaches worked well on certain mock datasets. However, we found the assumptions used to create those datasets do not always hold for real-world data. Better understanding of salmon genomics is required to assess whether our techniques generalize to data drawn from actual salmon.
11:35 AM to 12:00 PM
Delayed Weighted Subgradient Method with Spectral Steps
Grace Ko, Apol Medrano, and Tim Scheldt.
Dr. Milagros Loreto
Abstract: The Delayed Weighted Subgradient Method with Spectral Steps (DWSM) is proposed for addressing non-smooth unconstrained optimization problems. This innovative method expands on the Delayed Weighted Gradient Method by incorporating subgradients, spectral steps, and inexact line search techniques. The theoretical results for the DWSM are discussed, and numerical findings are presented based on standard non-smooth optimization problems. Performance profiles are utilized to identify the optimal stepsize for the algorithm and to assess its effectiveness in comparison to other non-smooth optimization solvers.
1:00 PM to 1:25 PM
CT Image Reconstruction with Unrolled Iterative Networks
Alexander Huang, Zachary Linder, Dana Pantoja, NYU
Research mentor: Dr. Thomas Humphries.
Abstract: In recent years, there has been a rising interest in reducing the radiation dose associated with computed tomography (CT) scans; however, dose reduction introduces additional image noise and artifacts. This motivates the use of neural networks as a part of the image reconstruction process. One method for doing this is unrolling, a process which incorporates neural networks by fixing the total length of a traditional iterative algorithm and training it as a neural network. We primarily work with the It-Net (Genzel, et al. 2022), which unrolls a gradient descent algorithm, and utilizes a U-Net between iterations to remove artifacts and noise. In our exploration, we emphasize balancing model-driven and data-driven regularization. Notably, we are able to significantly improve the performance of It-Net by including total variation regularization in the algorithm. Additionally, we study adding fixed convolutional layers to the network and replacing gradient descent with the limited memory BFGS update, but find that these do not significantly improve performance. We use the LION library to implement our approaches, providing a standardized framework to compare different algorithms. Finally, we contribute to the LION library by implementing the DU-GAN algorithm (Huang et al. 2021), which provides an additional point of comparison for our approach and is the first implementation of an adversarially trained algorithm in LION.
1:30 PM to 1:55 PM
A computerized search for Heesch numbers of polykites
Aiden Kreeger
Research mentor: Dr. Casey Mann
Abstract: The Heesch number of a tile is the maximum number of layers that can be formed from copies of the tile, without gaps or overlaps, around a centrally placed copy of the tile. A kite is a special quadrilateral related to the regular hexagon, and a polykite is a shape formed by fusing several kites together along their edges. In this presentation, an algorithm for determining the Heesch number of an edge-marked polykite will be described and demonstrated. Further, we discuss how we are implementing this algorithm on a high performance computing cluster to find the Heesch numbers of a large class of edge-marked polykites.
2:05 PM to 2:25 PM
A dance of squares and symmetries: Unilateral and equitransitive tilings by squares of consecutive integer sizes
Layan Arrabi and Rowan Surkan
Research mentor: Dr. Casey Mann.
Abstract: A tiling by squares is unilateral if no two squares of the same size share a whole edge and is equitransitive if for any two same size squares in the tiling there is a symmetry of the tiling mapping one to the other. In this presentation, we will give a progress report on our efforts to (1) find all unilateral and equitransitive tilings by squares of sizes 1, 2, 3, and 4, and (2) discuss ideas for algorithms that can be used to find all unilateral and equiltransitive tilings by squares of sizes 1, 2, 3, …, n for arbitrary positive integers n