*Indicates a student co-author.
Aksoy, Asuman G. and J. M. Almira. “Using the Baire category theorem to explore Lions problem for quasi-Banach spaces.” Advances in Operator Theory, vol. 10, number 34, 2025, pp. 1-34.
Abstract: Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire category theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple (A0, A1). Lions problem, posed in 1960s, is to prove that different parameters (θ,p) produce different interpolation spaces (A0, A1)θ,p. We first establish conditions on A0 and A1 so that interpolation spaces of this couple are strictly intermediate spaces between A0 + A1 and A0 ∩ A1. This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if p ≠ p* the operator ideals 𝓛(a)p,q (X,Y), 𝓛(a)p*,q* (X,Y) generated by approximation numbers are distinct. Moreover, for any fixed p, either all operator ideals 𝓛(a)p,q (X,Y) collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch’s problem for operator ideals based on general s-numbers. However, the BCT can be used to prove a lethargy result for arbitrary s-numbers which guarantees that, under very minimal conditions on X, Y, the space 𝓛(s)p,q (X,Y) is strictly embedded into 𝓛A(X,Y).
Aksoy, Asuman. Review of “Simultaneous minimal extensions with applications,” by Grzegorz, Lewicki and Michael Prophet. MathSciNet Mathematical Reviews, 2025, MR4926040.
Cannon, Sarah and Zarina Dhillon*. “Evaluating Methods used to Quantify Racial Segregation.” Journal of Humanistic Mathematics, vol. 15, issue 2, July 2025, pp. 37-68.
Abstract: Racial segregation has long been a problem in communities across the United States. One approach to help understand such an important issue is to attempt to describe it quantitatively. Many metrics have been developed, all with various strengths and weaknesses, but none fully capture the nuances of this complicated issue. This work provides an overview of four of the mathematical approaches that have been developed to study segregation, explains how they function using small examples, and compares and contrasts their effectiveness in various situations. We then focus on segregation in Los Angeles (LA) County, including a detailed exploration of the most recent score proposed by authors Sousa and Nicosia, which conducts a random walk and outputs the number of steps it takes to reach all racial classes in the system. While we find there is a difference between the average step lengths of LA County vs. an unbiased null model, attempts to standardize outputs erases crucial data and compressing this issue into one score is not representative of its complexity. This suggests that future exploration should attempt to study segregation more comprehensively, rather than distilling an incredibly complicated and important issue into a single statistic. More work is needed to quantitatively represent the complexities of racial segregation in an effective matter.
Cannon, Sarah, Principal Investigator. “CAREER: Random Sampling of Structures on Graphs.” National Science Foundation, Division of Computing and Communication Foundations. Faculty Early Career Development Program (CAREER), 2025-2030, $627,582.
Abstract: When analyzing large systems that have an enormous number of possibilities, studying collections of randomly selected pieces can provide an effective way to understand likely properties or behaviors of the entire system. Random sampling can be applied to polling, estimating quantities in physical systems, detecting gerrymandering, and more. However, it is challenging to do random sampling in a way that is both fast and accurate. This project will study the fundamental mathematics behind methods for random sampling, including introducing new sampling methods, developing new tools to analyze existing sampling methods, and finding problems amenable to the new approaches the investigator develops. One goal is to improve methods used to quantify and detect gerrymandering, making those methods both faster and more reliable. Part of the award will support a summer program where students learn about math, computer science, and data science motivated by problems related to democracy.
This project considers random sampling of structures on graphs, such as spin configurations on the vertices of a graph or partitions of a graph into connected pieces. In one direction, the investigator will consider Pirogov-Sinai theory (PST), an approach from statistical physics that could help advance the state-of-the-art in sampling/counting algorithms for spin systems and more. Specific questions include adapting PST from infinite to finite settings; using PST and the additional probabilistic information it conveys to develop new Markov chain sampling algorithms; and exploring other statistical physics ideas that can lead to algorithmic breakthroughs. In another direction, tree-based methods have emerged as a promising way to sample connected graph partitions, but existing algorithms remain insufficient for fast, provably accurate sampling in general settings. Work on this project will address this gap, building on insights from a recent breakthrough result. This is closely tied to broader questions about the combinatorial and probabilistic structure of random trees and random walks, Markov chain mixing under non-local constraints, and duality in non-planar graphs. As political districting plans can be viewed as connected balanced partitions of population-weighted graphs (and random sampling algorithms are widely used to detect gerrymandering, understand possible plans, and advocate for voting rights, including in court), advances in efficiently generating these structures have broad implications for political science and important societal impact.
Cass, Robert and Yujie Xu. “Geometrization of the Satake transform for mod p Hecke algebras.” Forum of Mathematics, Sigma, February 2025.
Abstract: We geometrize the mod p Satake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric mod p Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod p Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a mod p Local Langlands Correspondence.
Cass, Robert, Thibaud van den Hove, and Jakob Scholbach. “The geometric Satake equivalence for integral motives.” Compositio Mathematica, vol. 161, issue 11, November 2025, pp. 2755-2851.
Abstract: We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney–Tate stratifications of Beilinson–Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne’s modification of the dual group and a modified form of Vinberg’s monoid over the integers.
Fukshansky, Lenny and Sehun Jeong*. “Integral zeros of quadratic polynomials avoiding sublattices,” The Ramanujan Journal, vol. 66, number 26, January 2025.
Abstract: Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an explicit bound. This is a contribution related to the celebrated theorem of Cassels on small-height zeros of quadratic forms, which builds on some previous work in this area. We also demonstrate an application of these results to the problem of effective distribution of angles between vectors in the integer lattice.
Fukshansky, Lenny. “On lattice illumination of smooth convex bodies.” Archiv der Mathematik, vol. 125, number 2, May 2025, pp. 133-143.
Abstract: The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body K in ℝn can be illuminated by a set of no more than 2n points. If K has smooth boundary, it is known that n + 1 points are necessary and sufficient. We consider an effective variant of the illumination problem for bodies with smooth boundary, where the illuminating set is restricted to points of a lattice and prove the existence of such a set close to K with an explicit bound on the maximal distance. We produce improved bounds on this distance for certain classes of lattices, exhibiting additional symmetry or near-orthogonality properties. Our approach is based on the geometry of numbers.
Huber, Mark. “Martingales, Markov chains, and Brownian motion.” Data Science Adventures (Independent), December 2025.
Abstract: This book is intended for an upper-division or Master's level course in stochastic processes. Beginning with a rigorous development of mathematical probability, expectation, conditional expectation, and the main theorems linking limits and expectation, the text goes on to present the most commonly used stochastic processes. These include martingales, Markov chains (both discrete time, continuous time, and Harris chains), Brownian motion, and the Ito integral. Intended for students that have already completed a one semester undergraduate course in Probability using Calculus based methods.
Huber, Mark, and Gizem Karaali. “Rehumanizing Mathematics.” Journal of Humanistic Mathematics (introduction), vol. 15, issue 1, pp. 1-3.
Abstract: We feel strongly about the human nature of mathematics; there is a reason “humanistic” is in the title of this Journal after all! So it is quite appropriate that several articles in this issue remind us that mathematics has always flourished as a human endeavor and it is perhaps the reduction to sterile algorithms that has stripped it of interest for many students. So we are actually trying to rehumanize mathematics, to bring back that spark that leads to students looking upon their math journey in a more positive light.
Huber, Mark, and Gizem Karaali, “What Can Mathematics Do For Us?” Journal of Humanistic Mathematics (introduction), vol. 15, issue 2, pp. 1-3.
Abstract: Mathematics is often considered pure or abstract, but many mathematical advances have come from real-world needs or practical questions. This issue has a wide variety of articles addressing how math and realistic problems intersect.
Fukshansky, Lenny and Sehun Jeong*. “Integral zeros of quadratic polynomials avoiding sublattices,” The Ramanujan Journal, vol. 66, number 26, January 2025.
Abstract: Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an explicit bound. This is a contribution related to the celebrated theorem of Cassels on small-height zeros of quadratic forms, which builds on some previous work in this area. We also demonstrate an application of these results to the problem of effective distribution of angles between vectors in the integer lattice.
Kao, Chiu-Yen, Junshan Lin, and Braxton Osting. “A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals.” Journal of Computational Physics, vol. 521, part 1, January 2025.
Abstract: Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.
Kao, Chiu-Yen, Principal Investigator. “Collaborative Research: Computational Methods for Geometric Extremal Eigenvalue Problems.” National Science Foundation, 2025-2028, $147,731.
Abstract: In a variety of real-world applications, eigenvalues of linear partial differential operators describe physical phenomena of interest, e.g., light propagation, mechanical vibrations, and liquid sloshing. It is of practical and fundamental interest to study the dependence of an eigenvalue on a control variable, such as the material coefficient or the domain shape, and to engineer/design/optimize control variables to enhance relevant spectral properties. This project will develop and analyze new computational methods for solving extremal eigenvalue problems, especially involving challenging geometric constraints. The research activities will advance discovery and understanding in computational mathematics and mathematical physics, as well as more general areas of science and engineering through applications. Educational activities are integrated with research activities in four specific ways: (i) training of students (including K-12, undergraduate, and graduate students across different schools) and junior researchers at different levels, (ii) encouraging participation of researchers in the area (iii) dissemination and sharing of research results publicly, and (iv) organization of international workshops on proposed research topics. Due to the collaborative nature of this proposal, students will engage in activities across R1 and primarily undergraduate institutions.
The aim of this project is to tackle two canonical extremal eigenvalue problems from the mathematical and engineering communities: (1) study the Steklov eigenvalue problem on a compact Riemannian surface with boundary and seek to maximize of an eigenvalue over the class of smooth metrics; (2) address a key challenge in the design of topological photonic crystals (TPCs): find materials that have large shared spectral bandgaps where the adjacent dispersion surfaces have prescribed topological invariants (e.g., the Chern number, a topological invariant obtained from Berry curvature). A technical challenge in these problems is to handle geometric constraints - either stemming from topological constraints on Riemannian surfaces or topological invariants of dispersion surfaces. The proposed research activities will develop analytical and computational tools to tackle this challenge.
Kao, Chiu-Yen, Principal Investigator. “Optimal Shape Designs Involving Partial Differential Equations” Simons Foundation, Collaboration Grants for Mathematicians, 9/01/2025-8/30/2030.
Abstract: Optimal shape designs involving partial differential equations provide a vast number of interesting and challenging mathematical problems. We have studied many problems including two-phase conductors, topological photnic crystals design, optimization problems for rods and plates, extremal Steklov eigenvalues, maximal total population, and optimal treatment designs. In each problem, not only theoretical properties of optimizers are studied but also numerical approaches are developed to find optimizers efficiently and effectively. Projects that we plan to tackle in the near future include (1) optimization of Steklov eigenvalue problems and construction of free boundary minimal surfaces with high genus, (2) computational optimization for Dirac point in topological photonic crystals, (3) convergence rate study of rearrangement methods in high dimensions, (4) radiotherapy plan optimization, and (5) analyticity of eigenvalues with respect to domain perturbations. The grant will be used to support travel and related expenditures for PI and her collaborators including her students to work closely.
Kao, Chiu-Yen, Seyyed Abbas Mohammadi, Braxton Osting, and Edouard Oudet. “Theoretical and Numerical Methods for Shape Optimization Involving Steklov Eigenvalues.” International Centre for Mathematical Sciences (ICMS) Research-in-Groups (RIGs) program, July 14-25, 2025, Edinburgh.
Abstract: We aim to study theoretical and numerical methods for shape optimization, including their convergence properties and optimality conditions. We focus on some specific shape optimization problems, where the objective function involves Steklov eigenvalues (eigenvalues of the Dirichlet-to-Neumann operator). The proposed work builds on previous work by the participants and this RIG provides an excellent opportunity for this international group of people from four different time zones to work together in person on our proposed research goals in shape optimization.
Alhejaili, Weaam, Chiu-Yen Kao, Braxton Osting, and Chee-Han Tan. “AIM SQuaRE on Theoretical, Asymptotic, and Numerical Analysis of Extremal Steklov Eigenvalue Problems.” American Institute of Mathematics, March 10-14, 2025.
Abstract: This proposal aims to study problems related to the dependence of Steklov eigenvalues on the domain shape from analytic, asymptotic, and computational perspectives.
Kim, Jieon, and Sam Nelson. “Entropic Niebrzydowski Tribrackets.” Journal of Knot Theory and Its Ramifications, vol. 34, number 9, 2025.
Abstract: In this paper, we introduce the notion of entropic Niebrzydowski tribrackets or just entropic tribrackets, analogous to entropic (also known as abelian or medial) quandles and biquandles. We show that if X is a finite entropic tribracket then for any tribracket T, the homset Hom(T,X) (and in particular, for any oriented link L, the homset Hom(T(L),X)) also has the structure of an entropic tribracket. This operation yields a product on the category of entropic tribrackets; we compute the operation table for entropic tribrackets of small cardinality and prove a few results. We conjecture that this structure can be used to distinguish links which have the same counting invariant with respect to a chosen entropic coloring tribracket X.
Gügümcü, Neslihan, and Sam Nelson. “Biquandle Power Brackets of Oriented Links.” Turkish Journal of Mathematics, vol. 49, number 2, 2025, pp. 157-172.
Abstract: In this paper, we introduce biquandle power brackets, an infinite family of invariants of oriented links containing the classical skein invariants and the quandle and biquandle 2-cocycle invariants as special cases. Biquandle power brackets are generalizations of biquandle brackets in which the values of Kauffman states also depend on the biquandle colors they admit. We provide example computations and discuss the relationship between these new invariants and the previous cases.
Cai, Mason*, and Sam Nelson. “Quandle Action Quivers.” Journal of Knot Theory and Its Ramifications, vol. 34, number 7, 2025.
Abstract: Quandle Coloring Quivers are directed graph-valued invariants of classical and virtual knots and links associated to finite quandles. Quandle action quivers are subquivers of the full quandle coloring quiver associated to quandle actions by elements of the coloring quandle. These quivers provide a categorification of the quandle counting invariant associated to each element of the quandle. We obtain new polynomial invariants called quandle action polynomials from these quivers as decategorifications.
Nelson, Sam, and Migiwa Sakurai. “Categorification of Biquandle Arrow Weights via Quivers.” Journal of Knot Theory and Its Ramifications, vol. 34, number 11, 2025.
Abstract: Introduced in S. Nelson and M. Sakurai [Biquandle arrow weight enhancements, Internat. J. Math. 34(8) (2023) 2350046], biquandle arrow weight invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian group satisfying certain properties. In this paper, we categorify the biquandle arrow weight polynomial invariant using biquandle coloring quivers, obtaining new infinite families of polynomial invariants of oriented virtual and classical knots.
Rosenman, Evan T. “Methods for Combining Observational and Experimental Causal Estimates: A Review.” Wiley Interdisciplinary Reviews: Computational Statistics, vol. 17, issue 2, June 2025.
Abstract: Recent years have seen an explosion in methodological work on combining causal effects estimated from observational and experimental datasets. Observational data have the advantage of being inexpensive and increasingly available from sources such as electronic health records, insurance claims databases, and online learning platforms. These data are representative of target populations, but because treatment assignments are not randomized, they suffer from unmeasured confounding bias. By contrast, as a consequence of randomization, experimental data yield unbiased causal effects. Yet experiments are costly, often involve relatively few units, and may incorporate stringent inclusion criteria that make the studied populations somewhat artificial. A challenge for researchers is how to integrate these two types of data to leverage their respective virtues. Over roughly the past 5 years, many novel approaches have been proposed. As in this review, we restrict our focus to techniques for integrating individual-level experimental and observational data, without assuming all confounding variables are studied in the observational data. We first “locate” the problem by detailing important considerations from the causal inference and transportability literature. We next discuss three important research traditions that predate modern methodological work: meta-analysis, Empirical Bayes shrinkage, and historical borrowing. In organizing the growing literature on data-combination methods, we use a categorization involving five distinct approaches: auxiliary methods, control-arm augmentation, debiasing, test-then-merge, and weighting. Within each category, we summarize recently proposed methodologies, highlighting the strengths and weaknesses of each. We conclude with a discussion of how practitioners might choose between competing approaches when conducting applied work.
Witter, R. Teal, Yurong Liu, and Christopher Musco. “Regression-adjusted Monte Carlo Estimators for Shapley Values and Probabilistic Values.” Proceedings of the Conference on Neural Information Processing Systems, December 2025.
Abstract: With origins in game-theory, probabilistic values like Shapley values, Banzhaf values, and semi-values have emerged as a central tool in explainable AI. They are used for feature attribution, data attribution, data valuation, and more. Since all of these values require exponential time to compute exactly, research has focused on efficient approximation methods using two techniques: Monte Carlo sampling and linear regression formulations. In this work, we present a new way of combining both of these techniques. Our approach is more flexible than prior algorithms, allowing for linear regression to be replaced with any function family whose probabilistic values can be computed efficiently. This allows us to harness the accuracy of tree-based models like XGBoost, while still producing unbiased estimates. From experiments across eight datasets, we find that our methods give state-of-the-art performance for estimating probabilistic values. For Shapley values, the error of our methods is up to 6x lower than Permutation SHAP (the most popular Monte Carlo method), 2.75x lower than Kernel SHAP (the most popular linear regression method), and 1.75x lower than Leverage SHAP (the prior state-of-the-art Shapley value estimator). For more general probabilistic values, we can obtain error up to 60x lower than prior work.
Arabi, Kasra, R. Teal Witter, Chinmay Hegde, and Niv Cohen. “SEAL: Semantic Aware Image Watermarking.” Proceedings of the International Conference on Computer Vision, August 2025.
Abstract: Generative models have rapidly evolved to generate realistic outputs. However, their synthetic outputs increasingly challenge the clear distinction between natural and AI-generated content, necessitating robust watermarking techniques. Watermarks are typically expected to preserve the integrity of the target image, withstand removal attempts, and prevent unauthorized replication onto unrelated images. To address this need, recent methods embed persistent watermarks into images produced by diffusion models using the initial noise. Yet, to do so, they either distort the distribution of generated images or rely on searching through a long dictionary of used keys for detection.
In this paper, we propose a novel watermarking method that embeds semantic information about the generated image directly into the watermark, enabling a distortion-free watermark that can be verified without requiring a database of key patterns. Instead, the key pattern can be inferred from the semantic embedding of the image using locality-sensitive hashing. Furthermore, conditioning the watermark detection on the original image content improves robustness against forgery attacks. To demonstrate that, we consider two largely overlooked attack strategies: (i) an attacker extracting the initial noise and generating a novel image with the same pattern; (ii) an attacker inserting an unrelated (potentially harmful) object into a watermarked image, possibly while preserving the watermark. We empirically validate our method's increased robustness to these attacks. Taken together, our results suggest that content-aware watermarks can mitigate risks arising from image-generative models.
Musco, Christopher, and R. Teal Witter. “Provably Accurate Shapley Value Estimation via Leverage Score Sampling.” Proceedings of the International Conference on Learning Representations, March 2025.
Abstract: Originally introduced in game theory, Shapley values have emerged as a central tool in explainable machine learning, where they are used to attribute model predictions to specific input features. However, computing Shapley values exactly is expensive: for a general model with n features, O(2n) model evaluations are necessary. To address this issue, approximation algorithms are widely used. One of the most popular is the Kernel SHAP algorithm, which is model agnostic and remarkably effective in practice. However, to the best of our knowledge, Kernel SHAP has no strong non-asymptotic complexity guarantees. We address this issue by introducing Leverage SHAP, a light-weight modification of Kernel SHAP that provides provably accurate Shapley value estimates with just O(n log n) model evaluations. Our approach takes advantage of a connection between Shapley value estimation and agnostic active learning by employing leverage score sampling, a powerful regression tool. Beyond theoretical guarantees, we show that Leverage SHAP consistently outperforms even the highly optimized implementation of Kernel SHAP available in the ubiquitous SHAP library [Lundberg & Lee, 2017].
Arabi Kasra, Benjamin Feuer, R. Teal Witter, Chinmay Hegde, and Niv Cohen. “Hidden in the Noise: Two-Stage Robust Watermarking for Images.” Proceedings of the International Conference on Learning Representations, 2025.
Abstract: As the quality of image generators continues to improve, deepfakes become a topic of considerable societal debate. Image watermarking allows responsible model owners to detect and label their AI-generated content, which can mitigate the harm. Yet, current state-of-the-art methods in image watermarking remain vulnerable to forgery and removal attacks. This vulnerability occurs in part because watermarks distort the distribution of generated images, unintentionally revealing information about the watermarking techniques.
In this work, we first demonstrate a distortion-free watermarking method for images, based on a diffusion model's initial noise. However, detecting the watermark requires comparing the initial noise reconstructed for an image to all previously used initial noises. To mitigate these issues, we propose a two-stage watermarking framework for efficient detection. During generation, we augment the initial noise with generated Fourier patterns to embed information about the group of initial noises we used. For detection, we (i) retrieve the relevant group of noises, and (ii) search within the given group for an initial noise that might match our image. This watermarking approach achieves state-of-the-art robustness to forgery and removal against a large battery of attacks.