REU Project Summaries

Large cardinal axioms postulate the existence of combinatorial properties of infinity. These axioms are far beyond the usual axioms of mathematics and expand the burden of the mathematical universe. A well-studied consequence of large cardinal axioms is compactness. Compactness denotes the extent to which mathematical structures are determined by their local behavior. In this project we continued the study of compactness properties for algebraic structures. We proved two compactness theorems in the context of almost projective modules from lambda-strongly compact and weakly compact, respectively.

Social learning studies how people combine their own private information with the choices of others. Although observing others can improve decision-making, it can also create information cascades, where early mistakes lead an entire population away from the truth. We investigate three aspects of this problem. First, we show that predicting the behavior and accuracy of social-learning networks is computationally difficult, even in several restricted settings. Second, we introduce a multipass model in which agents may repeatedly revise their decisions. While repeated updates can improve learning on some networks, we construct examples where they make learning strictly worse. Third, we study incentives that reward agents differently depending on how many others have already made the same correct prediction. We show that slowly decreasing rewards can eliminate permanent herding and lead almost everyone to learn the truth, whereas rewards that decrease too quickly can prevent learning.

Gauge theory gives useful tools for studying the geometry and topology of four-dimensional manifolds. We develop the differential-geometric background needed to set up the Yang-Mills and anti-self-duality equations, including connections, curvature, differential forms, and the Hodge star operator. We then study the gluing method, which constructs new anti-self-dual connections by combining known solutions and correcting the resulting approximate solution.

There is a growing effort to formalise mathematical knowledge in modern proof assistants, with Lean and the mathlib library playing a central role. Most areas of undergraduate mathematics are already covered well. However, graph theory is a subject which is still relatively underdeveloped in mathlib. We survey what in the field has been done and how and give examples of the various difficulties the formalisation of graphs poses in an attempt to understand this disparity. Then we present our own contribution, developing a formal environment for planar simple graphs in Lean with the aim of keeping its use intuitive and minimising the artifacts caused by the strict type-checking nature of this language. The description of planarity we use is combinatorial, via rotation systems. We demonstrate the practicality of our work by proving that the disjoint union of planar graphs is planar with the tools we develop. The proof includes a description of the connected components of a disjoint union, which is also a previously unformalised result.

Graph-based Multi-Robot Path Planning (MRPP), also known as Multi-Agent Path Finding (MAPF), seeks to quickly find high-quality paths for routing a large number of robots. MRPP/MAPF has numerous important applications, such as enabling the coordination of thousands of robots at Amazon warehouses. In this project, we leverage graph connectivity to design better MAPF algorithms. In particular, we focus on 3-regular graphs, and note the sharp structural transition that occurs between 2 and 3-regular graphs. We integrate the routing-via-matchings theorem of Alon, Chung, and Graham and give a transfer argument to simulate each matching as a MAPF instance.

Repeaters are key infrastructure in the building of long-range quantum networks, enabling long-distance entanglement between qubits. However, their performance may be limited by an asymmetric error model that is incompatible with their common symmetric error correction practices. This imbalance in the errors experienced by the control and the target motivates the use of distinct quantum codes in neighboring repeater stations. We desire a CX gate that has some notion of fault-tolerance, but all fully transversal gates between distinct code pairs have the code distance in the opposite configuration from the error distribution. We loosen the fault-tolerance constraint and show that even with partial transversality, logical CX with the desired asymmetrical protection is not constructable.

A patient who requires a kidney transplant may have a proxy donor: a person who wishes to donate a kidney to the patient, but cannot due to medical incompatibility. However, the patient may be compatible with another patient's proxy donor. Kidney exchange programs allow patients to swap proxy donors, so that each swapping patient receives a compatible transplant. The program determines which swaps occur, with the objective of maximizing the number of transplants (or a related objective representing the total health benefit to patients from transplants). At the same time, the program must account for constraints such as simultaneity of surgeries, cancellations, participation incentives and more. Mathematically, the underlying problem is what's known as a packing problem on a directed graph: finding a set of vertex-disjoint directed cycles and paths which maximize an objective function under constraints. In this project, we will designed algorithms that solve this problem for new combinations of objectives and constraints, motivated by real-world challenges faced by kidney exchange programs.

Large language models, like Llama-4 and GPT-5, provide the foundation upon which they can be adapted to a specialized domain through fine-tuning methods. Existing methods in post-training, like full-parameter fine tuning, can be computationally expensive and memory exhaustive, even with some parameter reduction found in commonly used low-rank adaptation. We first examine existing tensor-based adaptation methods, and their reported impact on parameter count and accuracy based on varying compression techniques. Furthermore, we attempt to develop a unified framework for low tensor rank adaptations that remains agnostic to non-uniform model weights, reflecting developments in recent architecture. This framework will also generalize different tensor compression methods found, highlighting the tradeoffs and benefits to accuracy and parameter count associated with each factorization.

We study the problem of online fair allocation of indivisible goods when sharing goods among a limited number of agents is permitted. In this problem, the goods arrive sequentially one by one and must be allocated irrevocably before the future good sequence is revealed. Existing literature shows that even under normalized additive valuations (the value of all items for a particular agent is known beforehand), no deterministic online algorithm can avoid an arbitrarily bad maximin share (MMS) approximation for three agents, when each good is allocated to only one agent. We focus on the smallest nontrivial setting of three agents and 2-sharing. First, we show that normalization is necessary: without normalized valuations, no deterministic online algorithm can guarantee any positive constant-factor MMS approximation. Under normalized additive valuations, we present a deterministic online algorithm that guarantees every agent at least a (1-C)-fraction of her MMS, where C is the sharing-cost. Finally, we establish that exact MMS remains impossible for the online 2-sharing setting when C > 1/2, even for normalized identical valuations with three agents.

My research focuses on extending the PACE (Pacing According to Current Estimated Utility) algorithm to fair, real-time resource allocation. While traditional fair division mechanisms rely on static, complete data, real-world systems like recommendation feeds must serve items sequentially without knowing what comes next. By generalizing PACE to multi-item recommendation slates under K-sharing auction rules, our approach splits user attention fairly among the top-paced items. Across both real-world MovieLens data and concentrated preference profiles, K-sharing significantly smooths budget expenditure, prevents item starvation, and speeds up equilibrium convergence reducing maximum relative allocation errors by over 80% as slate size grows.

The online Steiner forest problem asks an algorithm to connect demand pairs arriving sequentially in a weighted undirected graph, without knowledge of future demands and without deleting previously purchased edges. Whereas the offline problem has constant-factor approximations, the optimal ratio in the fully online setting is O(log k) for k demands. We study Steiner forest in the online algorithms with a sample framework, which interpolates between these settings by revealing each demand independently with probability p before the online phase. Previous work by Argue et al. 2022 gives an O(log 1/p) ratio for Steiner tree. We developed an O(1/p)-competitive algorithm for Steiner forest, giving a guarantee independent of the total number of demands.

Large language models (LLMs) are increasingly used by multilingual populations, yet safety evaluation and alignment remain heavily concentrated in English. Prior work has demonstrated that the effectiveness of LLM safeguards can vary across languages and models, raising concerns that safety behavior observed in English may not generalize uniformly to multilingual settings. In addition to translation, adversarial transformations that alter the surface form of harmful requests may further expose inconsistencies in model safeguards. This project investigates poem-based obfuscation as one such transformation and develops the empirical foundation for a broader sequential decision-theoretic framework for multilingual LLM safety auditing. This work focuses on Phase A of the proposed framework: constructing an automated pipeline for translating harmful requests into ten languages, transforming those requests into poems using multiple LLMs, validating the generated poems, submitting valid poems to multiple target LLMs, evaluating the resulting responses, and estimating jailbreak probabilities. The final experiment contained 9,000 generator-target attack combinations. Of these, 8,227 produced valid probability trials, and 1,951 were classified as successful jailbreaks, corresponding to an overall estimated jailbreak probability of 23.71%. The results demonstrate that poem-obfuscated harmful intent can sometimes remain recoverable after substantial changes to linguistic form and that LLM safety is not determined by language or model choice in isolation. Instead, the observed jailbreak probabilities suggest that safety behavior emerges from the interaction between the LLM used to generate the obfuscated poem, the LLM targeted by the attack, and the language in which the interaction occurs. Different combinations of these three experimental factors produced different safety outcomes, emphasizing the importance of evaluating multilingual LLM safety as an interaction among multiple components rather than as a property of a single model or language. The empirical probabilities generated during Phase A provide the measurements required for future work on cross-language dependency modeling and Bellman-based sequential language selection. Rather than exhaustively evaluating every language, this future framework is intended to determine adaptively which language is most valuable to test next and when additional testing is no longer justified.

We study the spectral theory of bounded-degree Borel pmp graphs, deriving descriptive combinatorial results from the spectral properties of the associated adjacency and Laplacian operators. We first adapt a finite spectral graph theoretic result by Nikiforov to give a new lower bound on the approximate measurable chromatic number of a bounded-degree Borel pmp graph in terms of the maximal spectral values of the adjacency and Laplacian operators. We then derive a broader result on the spectra of bounded self-adjoint operators. Finally, we strengthen a characterization of approximate measurable bipartiteness in terms of the spectrum of an adjacency operator, generalizing it to the case of non-regular graphs.

We provide a comprehensive landscape of possibilities and impossibilities for NOM fair division across both goods and chores. For additive valuations, we design NOM mechanisms that guarantee EF1 for any number of agents, and MMS for two agents. When items are unit-demand valuations, we show that NOM mechanisms can simultaneously achieve EF1, EFX, and MMS. To capture complementary items in real-world environment, we analyze k-minded valuations, proving a stark dichotomy: while EF1 and EFX remain achievable by NOM mechanisms (under mild agent-count conditions for chores), guaranteeing MMS is fundamentally impossible for both goods and chores. Our results establish sharp boundaries on what cognitive relaxations of strategyproofness can achieve in fair division.

Assortment optimization studies the problem of selecting a subset of products to offer to customers in order to maximize expected revenue under uncertain customer preferences. This problem is commonly modeled using probabilistic choice models, including Mallows-based ranking models that capture substitution behavior through distributions over customer rankings. However, full rankings may be unrealistic in settings where customers primarily distinguish between their most preferred alternatives, motivating the study of the Top-k Mallows Model, introduced by Chierichetti et al. in 2018. We investigate assortment optimization under the Top-k Mallows Model, where only a customer's top-k alternatives are ranked and all remaining alternatives are treated as indistinguishable. We first establish structural properties of choice probabilities arising from symmetries in the underlying Kendall-Tau distance. We show that unranked alternatives exhibit uniform choice behavior and that the contribution of unranked products to the optimal assortment depends on their cardinality. Using these properties, we develop an exact algorithm for assortment optimization with runtime \(\Theta(2^k n)\), improving upon exhaustive enumeration over all possible assortments. Finally, we develop a framework for approximation algorithms by conditioning on the position of the no-purchase option and derive probability bounds that support further approximation results.

Our study focused towards finding explicit polynomial equations for the fake projective planes (FPPs) labeled (C18, p = 3, {2},D3), (C18, p = 3, {2}, (dD)3), and (C18, p = 3, {2}, (d2D)3) in the Cartwright-Steger FPP classification, under their bicanonical embeddings into CP9. We do so by starting with equations for a known FPP, and then working through a common finite-sheeted Galois covering space to find equations of the unknown FPPs.

Ribosomes are complex macromolecules that translate messenger RNA into proteins. Their large subunit contains an exit tunnel through which the growing peptide chain passes before leaving the ribosome. While most studies have focused on the proteins lining the tunnel, comparatively little is known about the highly conserved ribosomal RNA surrounding it. As a result, the three-dimensional organization of this ribosomal RNA is not fully understood. This project examined how RNA structural motifs are arranged near the exit tunnel and whether parts of this organization are conserved across species. Ribosome structures and motif annotations were obtained from the RCSB Protein Data Bank and DSSR, respectively, and the resulting data were analyzed and visualized using RiboVision 2.0, PyMOL, and Python. The main analysis focused on nucleotides within 20 Å of the exit-tunnel centerline in the Escherichia coli ribosome structure 5AFI. A Python program was also developed to superimpose ribosome structures from different species to better compare the exit tunnels. This work provides a computational and visual framework for studying RNA motif organization and future cross-species comparisons.

This project studies whether top-(k) rankings can be trusted when many machine learning models achieve nearly identical predictive performance. Rather than analyzing a single trained model, we examine the surrounding Rashomon set of near-optimal models. We develop optimization methods to track how each item's score, rank, and top-(k) membership can change across plausible models. These tools help identify items that are consistently highly ranked, items whose positions are fundamentally uncertain, and comparisons that can reverse without meaningfully affecting model accuracy. The project combines machine learning, mathematical optimization, and algorithm design to create interpretable diagnostics for ranking reliability in applications such as recommender systems, hiring tools, search engines, and LLM response ranking.

Topological data analysis (TDA) provides tools for studying time-varying data, but many existing methods struggle to distinguish data that are isometric at each time but differ qualitatively. Kim and Memoli's spatiotemporal persistent homology addresses this shortcoming but produces multiparameter modules that are computationally challenging. To mitigate this, we introduce dynamic curvature-set persistent homology, extending Gomez and Memoli's curvature-set persistence to dynamic metric spaces. The resulting modules are interval-decomposable and admit efficient algorithms for computing interleaving distances. Our construction is stable with respect to the dynamic Gromov-Hausdorff distance introduced by the same authors, and we implement both the erosion distance $d_E$ (due to Puuska) and a new Hausdorff-inspired distance $d_2$ with cubic and quadratic runtime, respectively. This enables a robust computational pipeline for distinguishing dynamic data, as demonstrated in experiments with the boids model, where we successfully detect parameter changes.