*Indicates a student co-author.
Aksoy, Asuman Güven. Fundamentals of Real and Complex Analysis. Springer Undergraduate Mathematics Series, 2024.
Abstract: The primary aim of this text is to help transition undergraduates to study graduate level mathematics. It unites real and complex analysis after developing the basic techniques and aims at a larger readership than that of similar textbooks that have been published, as fewer mathematical requisites are required. The idea is to present analysis as a whole and emphasize the strong connections between various branches of the field. Ample examples and exercises reinforce concepts, and a helpful bibliography guides those wishing to delve deeper into particular topics. Graduate students who are studying for their qualifying exams in analysis will find use in this text, as well as those looking to advance their mathematical studies or who are moving on to explore another quantitative science.
Aksoy, Asuman Güven, and Daniel Akech Thiong. “Approximation spaces for H-operators.” Involve: A Journal of Mathematics, vol. 17, no. 4, 2024, pp. 709-722.
Abstract: We define and establish relations among approximation spaces of certain operators called H-operators, which generalize the notion of self-adjoint to Banach spaces.
Aksoy, Asuman. Review of “Maximal regularity and two-sided estimates of the approximation numbers of the nonlinear Sturm-Liouville equation solutions with rapidly oscillating coefficients in L2(R),” by Madi Muratbekov, Mussakan Muratbekov, and Serik Altynbek. MathSciNet Mathematical Reviews, 2024, MR4631198.
Abstract: In this paper, the authors consider the nonlinear Sturm-Liouville equation in L2(R), and prove the maximum regularity of its solutions in the case of a rapidly oscillating potential.
Aksoy, Asuman. Review of “Optimal recovery and volume estimates,” by Alexander Kushpel. MathSciNet Mathematical Reviews, 2024, MR4610801.
Abstract: In this paper, the author studies volumes of sections of convex origin-symmetric bodies in Rn induced by orthonormal systems on probability spaces.
Aksoy, Asuman. Review of “Gel'fand widths of Sobolev classes of functions in the average setting,” by Yuqi Liu, Huan Li, and Xuehua Li. MathSciNet Mathematical Reviews, 2024, MR 4541465.
Abstract: In this paper, the authors consider Sobolev spaces W2r of univariate functions and Sobolev space MW2r of multivariate functions with mixed derivatives in Lq equipped with Gaussian measure in the average and probabilistic settings. They obtain sharp asymptotic orders of the probabilistic Gel'fand N-widths of the corresponding Sobolev classes in the Lebesgue spaces.
Aksoy, Asuman. Review of “Smooth approximation of mappings with rank of the derivative at most 1,” by Paweł Goldstein and Piotr Hajłasz. MathSciNet Mathematical Reviews, 2024, MR 4537428.
Abstract: A conjecture of J. Gałęski is stated as follows: Conjecture. Let 1≤m<n be integers and let Ω⊂Rn be open. If f∈C1(Ω,Rn) satisfies rank Df≤m everywhere in Ω, then f can be uniformly approximated by smooth mappings g∈C∞(Ω,Rn) such that rank Dg≤m everywhere on Ω. There is also a local version of this conjecture. There are counterexamples to this conjecture. For example, P. Goldstein and P. Hajłasz showed that a C1 mapping in R5 with derivative of rank at most 3 cannot be uniformly approximated by C2 mappings with derivative of rank at most 3. In the paper under review, the authors prove that the conjecture is true when m=1. They prove that if m=1 and Ω is simply connected, then the locally Lipschitz mapping f:Ω→Rn satisfying rank Df≤1 a.e. can be almost uniformly approximated by a C∞ mapping g with rank Dg≤1. The construction of the approximant employs some elegant results on metric spaces, including the methods of factorization through metric trees.
Cannon, Sarah. “Irreducibility of recombination Markov chains in the triangular lattice.” Discrete Applied Mathematics, vol. 347, 2024, pp. 75-130.
Abstract: In the United States, regions (such as states or counties) are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can have a profound effect on who is elected, and drawing the districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect when gerrymandering is occurring, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used to do this random sampling: randomly choose two districts, consider their union, and split this union up in a new way. This approach works well in practice and has been widely used, including in litigation, but the theory behind it remains underdeveloped. For example, it is not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a planar graph G, is the space of all partitions of G into k connected subgraphs (k districts) connected by recombination moves? While the answer is yes when districts can be as small as one vertex, this is not realistic in real-world settings where districts must have approximately balanced populations. Here we fix district sizes to be k1 ± 1 vertices, k2 ± 1 vertices,… for fixed k1, k2,…, a more realistic setting. We prove for arbitrarily large triangular regions in the triangular lattice, when there are three simply connected districts, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples. The triangular lattice is the most natural setting in which to first consider such a question, as graphs representing states/regions are frequently triangulated. The proof uses a sweep-line argument, and there is hope it will generalize to more districts, triangulations satisfying mild additional conditions, and other redistricting Markov chains.
Blanca, Antonio, Sarah Cannon, and Will Perkins. “Fast and perfect sampling of subgraphs and polymer systems.” ACM Transactions on Algorithms, vol. 20, issue 1, 2024, pp. 1-30.
Abstract: We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.
Cannon, Sarah, Wesley Pegden, and Jamie Tucker-Foltz. “Sampling Balanced Forests of Grids in Polynomial Time.” STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of Computing, 2024, pp. 1676-1687.
Abstract: We prove that a polynomial fraction of the set of k-component forests in the m × n grid graph have equal numbers of vertices in each component, for any constant k. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each k-partition according to the product, across its k pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative (1 ± ε) constant with constant probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. This implies polynomial-time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into k population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.
External Grant: Cannon, Sarah, and Evan Rosenman. “A Statistical Study of Alternative Representational Systems for the LA City Council.” John Randolph Haynes and Dora Haynes Foundation, Governance and Democracy Planning Grant. $110,000.
Abstract: In partnership with advocacy groups, we will study redistricting reform proposals for the Los Angeles City Council using a rigorous, data-driven, hybrid approach that will enable deeper insights and higher confidence than standard methods.
Fukshansky, Lenny, Pavel Guerzhoy, and Tanis Nielsen. “Deep hole lattices and isogenies of elliptic curves.” Research in Number Theory, vol. 10, number 33, 2024.
Abstract: Given a lattice L in the plane, we define the affiliated deep hole lattice H(L) to be spanned by a shortest vector of L and a deep hole of L contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on L under which H(L) is well-rounded and prove that H(L) is defined over the same field as L. For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of SL2(Z) on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice.
Fukshansky, Lenny, and Sehun Jeong. “Diophantine avoidance and small-height primitive elements in ideals of number fields.” Combinatorics and Number Theory, vol. 13, no. 4, 2024, pp. 333-350.
Abstract: Let K be a number field of degree d. Then every ideal I in the ring of integers OK contains infinitely many primitive elements, i.e., elements of degree d. A bound on the smallest height of such an element in I follows from some recent developments in the direction of a 1998 conjecture of W. Ruppert. We prove an explicit bound on the smallest height of such a primitive element in the case of quadratic fields. Further, we consider primitive elements in an ideal outside of a finite union of other ideals and prove a bound on the height of a smallest such element. Our main tool is a result on points of small norm in a lattice outside of an algebraic hypersurface and a finite union of sublattices of finite index, which we prove by blending two previous Diophantine avoidance results. We also obtain a bound for small-norm lattice points in the positive orthant in Rd with avoidance conditions and use it to obtain a small-height totally positive primitive element in an ideal of a totally real number field outside of a finite union of other ideals. Additionally, we use our avoidance method to prove a bound on the Mahler measure of a generating nonsparse polynomial for a given number field. Finally, we produce a bound on the height of a smallest primitive generator for a principal ideal in a quadratic number field.
Forst, Maxwell, and Lenny Fukshansky. “On lattice extensions.” Monatshefte für Mathematik, vol. 203, no. 3, 2024, pp. 613-634.
Abstract: A lattice Λ is said to be an extension of a sublattice L of smaller rank if L is equal to the intersection of Λ with the subspace spanned by L. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.
Forst, Maxwell and Lenny Fukshansky. “On a new absolute version of Siegel's lemma.” Research in Mathematical Sciences, vol. 11, no. 1, 2024.
Abstract: We establish a new version of Siegel’s lemma over a number field k, providing a bound on the maximum of heights of basis vectors of a subspace of kNkN, N≥2N≥2. In addition to the small-height property, the basis vectors we obtain satisfy certain sparsity condition. Further, we produce a nontrivial bound on the heights of all the possible subspaces generated by subcollections of these basis vectors. Our bounds are absolute in the sense that they do not depend on the field of definition. The main novelty of our method is that it uses only linear algebra and does not rely on the geometry of numbers or the Dirichlet box principle employed in the previous works on this subject.
Fukshansky, Lenny. “Soviet Antisemitism at California’s Claremont Colleges.” The Wall Street Journal, October 2, 2024.
Abstract: Today’s ‘anti-Zionism’ is hard to differentiate from the communist slogans I heard in my youth.
Fukshansky, Lenny. “Antisemitism and Anti-Zionism Through the Lens of the Soviet Experience.” The Claremont Independent, November 6, 2024.
Abstract: Today’s ‘anti-Zionism’ is hard to differentiate from the communist slogans I heard in my youth.
Huber, Mark. Stochastic Operations Research. Independent, 2024.
Abstract: This book covers a one semester course in stochastic operations research for students who have had an undergraduate Calculus based course in probability. It covers queuing networks, simulation, decision and game theory, forecasting, and inventory management. The book does not assume prior programming knowledge, and introduces the reader to R in order to do calculations and simulations.
Huber, Mark. Verovatnoća: Predavanja i vežbe (elektronska knjiga), translated by Nevena Marić. CET Computer Equipment and Trade, 2024.
Abstract: A Serbian translation of Prof. Huber’s 2023 book Probability: Lectures and Labs.
Huber, Mark, and Gizem Karaali. "Our Histories, Our Values, Our Mathematics," Journal of Humanistic Mathematics, vol. 14, issue 1, January 2024, pp. 1-3.
Abstract: One way to form a connection with mathematics is to learn about its history. Another is to dig into one's own personal history with the subject. This winter issue offers our readers a neat selection of articles exploring how personal histories intertwine with the history of the discipline to tell a richer story about ourselves and our mathematics.
Huber, Mark, and Gizem Karaali. “Picturing Mathematics (Education) in New Ways.” Journal of Humanistic Mathematics, vol. 14, issue 2, July 2024, pp. 1-3.
Abstract: Mathematics comes to most through the efforts of those who educate us in school during our learning years. This issue contains a variety of articles exploring not only how children learn mathematics, but how educators might gain a deeper understanding of concepts before jumping into the teaching world.
Huber, Mark. Review of How to expect the unexpected, by Kit Yates. Journal of Humanistic Mathematics, vol. 14, issue 2, July 2024, pp. 585-590.
Abstract: Humans think about the future all the time. Prediction is a part of how we prepare for the coming of both good and bad events in our lives. Kit Yates' book, How to Expect the Unexpected, concentrates primarily on the question of why prediction is difficult, and what mental shortcuts people take in prediction that can lead to incorrect results. Unfortunately, a lack of concern for details and several omissions undermine the quality of the book.
Bertozzi, Andrea, Ron Fedkiw, Frederic Gibou, Chiu-Yen Kao, Chi-Wang Shu, Richard Tsai, Wotao Yin, and Hong-Kai Zhao, eds. “Focused Issue in Honor of Prof. Stanley Osher on His 80th Birthday.” Communications on Applied Mathematics and Computation, vol. 6, issue 2, June 2024.
Abstract: In this special issue, we curated scholarly articles that reflect the broad and dynamic scope of research areas championed by Professor Stanley Osher. These include advancements in numerical methods for nonlinear partial differential equations, optimization, interface problems, image processing, machine learning, and artificial intelligence. We sincerely wish that this issue brings the excitement and enthusiasm we all experienced as we discussed new ideas with Professor Osher.
Chen, Weitao, and Chiu-Yen Kao. "Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates." Communications on Applied Mathematics and Computation, vol. 6, issue 1, 2024, pp. 236-256.
Abstract: In this paper, we review computational approaches to optimization problems of inhomogeneous rods and plates. We consider both the optimization of eigenvalues and the localization of eigenfunctions. These problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration displacement. We demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.
Kao, Chiu-Yen, Seyyed Abbas Mohammadi, and Mohsen Yousefnezhad. "Is maximum tolerated dose (MTD) chemotherapy scheduling optimal for glioblastoma multiforme?." Communications in Nonlinear Science and Numerical Simulation, vol. 139, December 2024, 108292.
Abstract: In this study, we investigate a control problem involving a reaction-diffusion partial differential equation (PDE). Specifically, the focus is on optimizing the chemotherapy scheduling for brain tumor treatment to minimize the remaining tumor cells post-chemotherapy. Our findings establish that a bang-bang increasing function is the unique solution, affirming the MTD scheduling as the optimal chemotherapy profile. Several numerical experiments on a real brain image with parameters from clinics are conducted for tumors located in the frontal lobe, temporal lobe, or occipital lobe. They confirm our theoretical results and suggest a correlation between the proliferation rate of the tumor and the effectiveness of the optimal treatment.
Abbas Mohammadi, Seyyed (Lead Principal Investigator), Chiu-Yen Kao, 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), Edinburgh, Scotland.
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 (various subsets) of the participants and this RIGs at IMCS 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.
Chao-Haft*, Max, and Sam Nelson. “Biracks and Switch Braid Quivers. Journal of Knot Theory and its Ramifications, vol. 33, no. 12, 2024.
Abstract: We consider birack and switch colorings of braids. We define a switch structure on the set of permutation representations of the braid group and consider when such a representation is a switch automorphism. We define quiver-valued invariants of braids using finite switches and biracks and use these to categorify the birack 2-cocycle invariant for braids. We obtain new polynomial invariants of braids via decategorification of these quivers.
Choi, Seonmi, and Sam Nelson. “MC-Biquandles and MC-Biquandle Coloring Quivers.” Journal of Knot Theory and its Ramifications, vol. 33, no. 11, 2024.
Abstract: We introduce the notion of mc-biquandles, algebraic structures which have possibly distinct biquandle operations at single-component and multi-component crossings. These structures provide computable homset invariants for classical and virtual links. We categorify these homsets to obtain mc-biquandle coloring quivers and define several new link invariants via decategorification from these invariant quivers.
Chang*, Melody, and Sam Nelson. “Skew Brace Enhancements and Virtual Links.” Communications of the Korean Mathematical Society, vol. 39, issue 1, 2024, pp. 247-257.
Abstract: We use the structure of skew braces to enhance the biquandle counting invariant for virtual knots and links for finite biquandles defined from skew braces. We introduce two new invariants: a single-variable polynomial using skew brace ideals and a two-variable polynomial using the skew brace group structures. We provide examples to show that the new invariants are not determined by the counting invariant and hence are proper enhancements.
Sarnquist, Clea, Rina Friedberg, Evan T.R. Rosenman, Mary Amuyunzu-Nyamongo, Gavin
Nyairo, and Michael Baiocchi. “Sexual Assault Among Young Adolescents in Informal Settlements in Nairobi, Kenya: Findings from the IMPower and SOS Cluster-Randomized Controlled Trial.” Prevention Science, vol. 25, 2024, pp. 578-589.
Abstract: Sexual assault is a global threat to adolescent health, but empowerment self-defense (ESD) interventions have shown promise for prevention. This study evaluated the joint implementation of a girls’ ESD program and a concurrent boys’ program, implemented via a cluster-randomized controlled trial in informal settlements of Nairobi, Kenya, from January 2016 to October 2018. Schools were randomized to the 12-h intervention or 2-h standard of care. Students were randomly sampled to complete surveys at baseline and again at 24 months post-intervention. A total of 3263 girls, ages 10–14, who completed both baseline and follow-up surveys were analyzed; weights were adjusted for dropout. At follow-up, 5.9% (n = 194/3263) of girls reported having been raped in the prior 12 months. Odds of reporting rape were not significantly different in the intervention versus SOC group (OR: 1.21; 95% CI (0.40, 5.21), p = 0.63). Secondary outcomes, social self-efficacy (OR: 1.08; 95% CI (0.95, 1.22), p = 0.22), emotional self-efficacy (OR 1.07; 95% CI (0.89, 1.29), p = 0.49), and academic self-efficacy (OR: 0.90; 95% CI (0.82, 1.00), p = 0.06) were not significantly different. Exploratory analyses of boys’ victimization and perpetration are reported. This study improved on previous ESD studies in this setting with longitudinal follow-up of individuals and independent data collection. This study did not show an effect of the intervention on self-reported rape; findings should be interpreted cautiously due to limitations. Sexual assault rates are high in this young population, underscoring a dire need to implement and rigorously test sexual assault prevention interventions in this setting. The trial was registered with Clinical Trials.gov # NCT02771132. Version 3.1 registered on May 2017, first participant enrolled January 2017.
Fishman, Nic, and Evan Rosenman. “Estimating Vote Choice in U.S. Elections with Approximate Poisson-Binomial Logistic Regression.” NeurIPS Optimization for Machine Learning Workshop, 2024, paper 28.
Abstract: We develop an approximate method for maximum likelihood estimation in Poisson-Binomial Logistic regression. The resulting approximate log-likelihood is generally non-convex but easy to optimize in practice. We investigate the geometry of the likelihood and propose simple but effective optimization procedures. We use these methods to fit logistic regressions in all statewide U.S. elections between 2016 and 2020, a total of 544 offices and over 1.75 billion votes.
External Grant: Rosenman, Evan, Principal Investigator. “Empirical Bayes Evidence Synthesis Techniques for Hybridizing Observational and Experimental Data.” National Science Foundation, Division of Mathematical Sciences, Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences (LEAPS-MPS), 2024-2026, $249,063.
Abstract: When evaluating the efficacy of a health intervention, researchers may have access to two distinct types of data: observational data from sources such as electronic health records or insurance claims databases, and experimental data from randomized trials. Observational data are useful for their scope and representativeness, but the individuals who receive the intervention may fundamentally differ from those who do not, making causal estimation difficult. In contrast, experimental data use random treatment assignment to facilitate accurate causal effect estimation, but such experiments are often costly and time-consuming to conduct. Hence, regulatory agencies and statisticians have advocated for methodologies that integrate observational data with randomized trials. Despite increased research attention, there is little consensus on how to effectively combine these two data types for many common statistical procedures. This proposal aims to develop evidence-synthesis techniques to enhance causal effect estimation and inference and to design more efficient experiments.
The primary toolkit for this research will be Empirical Bayes procedures, a flexible paradigm for weighting between competing estimators. The goal is to enable practitioners to estimate causal effects more efficiently in diverse settings. The principal investigator has identified three causal estimation tasks where current data-integration methods can be improved: 1) average treatment effect (ATE) estimation; 2) modeling of flexible functional estimands, such as dose-response curves and conditional average treatment effect (CATE) functions; and 3) the incorporation of multiple datasets of each type. Additionally, this proposal includes projects to narrow confidence intervals by incorporating observational data and to design more efficient and adaptive randomized trials that explicitly complement existing observational data. To demonstrate their efficacy and utility in practical settings, methods developed under this proposal will be deployed on two datasets: the Women’s Health Initiative, a study of the health effects of hormone therapy that includes observational and experimental components; and an air quality and Medicare insurance claims database maintained by the National Studies on Air Pollution and Health group.
External Grant: Cannon, Sarah, and Evan Rosenman. “A Statistical Study of Alternative Representational Systems for the LA City Council.” John Randolph Haynes and Dora Haynes Foundation, Governance and Democracy Planning Grant. $110,000.
Abstract: In partnership with advocacy groups, we will study redistricting reform proposals for the Los Angeles City Council using a rigorous, data-driven, hybrid approach that will enable deeper insights and higher confidence than standard methods.
Moon, Han-Bom, and Helen Wong. “Consequences of the compatibility of skein algebra and cluster algebra on surfaces.” New York Journal of Mathematics, vol. 30, 2024, pp. 1648-1682.
Abstract: We investigate two algebras consisting of curves on a surface with interior punctures -- the cluster algebra defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. We establish their compatibility, and use it to prove Roger-Yang’s conjecture that the skein algebra is a deformation quantization of the decorated Teichmuller space. We also obtain several structural results on the cluster algebra of surfaces. The cluster algebra of a positive genus surface is not finitely generated, and it differs from its upper cluster algebra.
External Grant: Wong, Helen. “Research Grant for Visitors.” Max Planck Institute for Mathematics, January-May 2026.
Abstract: This grant provides monetary support for a visit to Max Planck Institute for Mathematics.