# 2020 Mathematical Sciences Publications and Grants

*Indicates student co-author

Asuman G. Aksoy. “The Bieberbach Conjecture.” The World of Mathematics, I and II, 2020, pp. 48-51. (In Turkish)

Abstract: One of the celebrated conjectures in classical analysis which has stood as a challenge to mathematicians for nearly 70 years is called the Bieberbach conjecture (BC). This conjecture appeared in a footnote to a paper [1] of a German mathematician Ludwig Bieberbach in 1916 and was solved by Louis de Branges of Purdue University in 1984 [2]. The Bieberbach conjecture is appealing partly because it is simple to pose, and it states that under reasonable restrictions the coefficients of a power series is not too large. In this paper, we prove several elementary results of univalent functions and normalized Schlicht class S, only using undergraduate analysis. It is important to realize that, in the decades that BC stood unsolved, mathematicians discovered many properties of univalent functions. In fact, most of the theory of univalent functions arouse from partial results about S and its subclasses. Thus, the significance of the Bieberbach conjecture does not lie only with its solution; but rather it belongs to the theory that was developed to solve it.

Aksoy, Asuman. G. Review of “Optimal Spline Spaces for L^2 n-width Problems with Boundary Conditions,” by Michael S. Floater and Espen Sande. Mathematical Reviews/ MathSciNet, 2020, MR3975879.

Aksoy, Asuman. G. Review of “Probabilistic and Average Linear Widths of Weighted Sobolev Spaces on the Ball Equipped with a Gaussian Measure,” by Heping Wang. Mathematical Reviews/ MathSciNet, 2020, MR3902758.

Aksoy, Asuman Güven, Mehmet Kılıç, and Şahin Koçak. “Isometric Embeddings of Finite Metric Trees into (ℝ n, d1) and (ℝ n, d∞).” Advances in Analysis and Geometry: The Mathematical Legacy of Victor Lomonosov, edited by Richard M. Aron, Eva A. Gallardo Gutiérrez, Miguel Martin, Dmitry Ryabogin, Ilya M. Spitkovsky, and Artem Zvavitch. De Gruyher, 2020, pp. 15-24.

Abstract: We investigate isometric embeddings of finite metric trees into (ℝn, d1) and (ℝn, d). We prove that a finite metric tree can be isometrically embedded into (ℝn, d1) if and only if the number of its leaves is at most 2n. We show that a finite star tree with at most 2n leaves can be isometrically embedded into (ℝn, d) and a finite metric tree with more than 2n leaves cannot be isometrically embedded into (ℝn, d). We conjecture that an arbitrary finite metric tree with at most 2n leaves can be isometrically embedded into (ℝn, d).

Cannon, Sarah, and Will Perkins. "Counting Independent Sets in Unbalanced Bipartite Graphs." Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA20), 2020, pp. 1456-1466.

Abstract: Understanding the complexity of approximately counting the number of weighted or unweighted independent sets in a bipartite graph (#BIS) is a central open problem in the field of approximate counting. Here we consider a subclass of this problem and give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides (LR) of the bipartition. This includes, among others, the biregular case when λ = 1 (approximating the number of independent sets of G) and the degree on the left side is bigger than the degree on the right side by a logarithmic factor. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. Further consequences of this method for unbalanced bipartite graphs include an efficient sampling algorithm for the hard-core model and zero-freeness results for the partition function with complex fugacities. By utilizing connections between the cluster expansion and joint cumulants of certain random variables, we go beyond previous algorithmic applications of the cluster expansion to prove that the hard-core model exhibits exponential decay of correlations for all graphs and fugacities satisfying our conditions. This illustrates the applicability of statistical mechanics tools to algorithmic problems and refines our understanding of the connections between different methods of approximate counting.

Aletheia-Zomlefer*, Soren Laing, Lenny Fukshansky and Stephan Ramon Garcia. “The Bateman-Horn Conjecture: Heuristic, History, and Applications.” Expositiones Mathematicae, vol. 38, issue 4, 2020, pp. 430-479.

Abstract: The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins.

Böttcher, Albrecht, and Lenny Fukshansky. “Representing Integers by Multilinear Polynomials.” Research in Number Theory, vol. 6 no. 4, 2020, article 38.

Abstract: Let F(xx) be a homogeneous polynomial in n≥1 variables of degree 1≤dn with integer coefficients so that its degree in every variable is equal to 1. We give some sufficient conditions on F to ensure that for every integer b there exists an integer vector aa such that F(a)=b. The conditions provided also guarantee that the vector aa can be found in a finite number of steps.

Fukshansky, Lenny, and Yingqi Shi*. “Positive Semigroups and Generalized Frobenius Numbers Over Totally Real Number Fields.” Moscow Journal of Combinatorics and Number Theory, vol. 9 no. 1, 2020, pp. 29-41.

Abstract: Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers.

Huber, Mark. “Estimating the Presidential Winner through the Electoral College.” Medium, September 29, 2020.

Huber, Mark. “Halving the Bounds for the Markov, Chebyshev, and Chernoff Inequalities through Smoothing.” American Mathematical Monthly, vol. 126, issue 10, 2019, pp. 915-927. [Published in 2019 but not available until 2020]

Abstract: The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equalities. Here we will show, through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.

Huber, Mark. “A Probabilistic Approach to the Fibonacci Sequence.” The Mathematical Intelligencer, vol. 42, 2020, pp. 29-33.

Abstract: One of the fascinating things about the Fibonacci sequence is that the values quickly approach a particular exponential function. In this work, the constants of this limiting function are explained through the limiting distribution of a Markov chain that encapsulates the entire Fibonacci sequence.

Huber, Mark. Probability: Lectures and Labs (Learning College Mathematics). Independent, 2020.

Abstract: This is the 2020 edition of the undergraduate textbook in Probability first published in 2019.

Stoikos*, Stefanos, and Mike Izbicki. “Multilingual Emoticon Prediction of Tweets about COVID-19.” Proceedings of the Third Workshop on Computational Modeling of People's Opinions, Personality, and Emotions in Social Media, 2020, pp. 109-118.

Abstract: Emojis are a widely used tool for encoding emotional content in informal messages such as tweets, and predicting which emoji corresponds to a piece of text can be used as a proxy for measuring the emotional content in the text. This paper presents the first model for predicting emojis in highly multilingual text. Our BERTmoticon model is a fine-tuned version of the BERT model, and it can predict emojis for text written in 102 different languages. We trained our BERTmoticon model on 54.2 million geolocated tweets sent in the first 6 months of 2020,and we apply the model to a case study analyzing the emotional reaction of Twitter users to news about the coronavirus. Example findings include a spike in sadness when the World Health Organization (WHO) declared that coronavirus was a global pandemic, and a spike in anger and disgust when the number of COVID-19 related deaths in the United States surpassed one hundred thousand. We provide an easy-to-use and open source python library for predicting emojis with BERTmoticon so that the model can easily be applied to other data mining tasks.

Stringham*, Nathan, and Mike Izbicki. “Evaluating Word Embeddings on Low-Resource Languages.” Proceedings of the First Workshop on Evaluation and Comparison of NLP Systems, 2020, pp. 176-186.

Abstract: The analogy task introduced by Mikolov et al. (2013) has become the standard metric for tuning the hyperparameters of word embedding models. In this paper, however, we argue that the analogy task is unsuitable for low-resource languages for two reasons: (1) it requires that word embeddings be trained on large amounts of text, and (2) analogies may not be well-defined in some low-resource settings. We solve these problems by introducing the OddOneOut and Topk tasks, which are specifically designed for model selection in the low-resource setting. We use these metrics to successfully tune hyperparameters for a low-resource emoji embedding task and word embeddings on 16 extinct languages. The largest of these languages (Ancient Hebrew) has a 41 million token dataset, and the smallest (Old Gujarati) has only a 1813 token dataset.

Kang*, Di, Patrick Choi*, and Chiu-Yen Kao. "Minimization of the First Nonzero Eigenvalue Problem for Two-Phase Conductors with Neumann Boundary Conditions." SIAM Journal on Applied Mathematics, vol. 80, issue 4, 2020, pp. 1607-1628.

Abstract: We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that the region that has lower conductivity is simply connected. On a rectangular domain in two dimensions, we show that the strip configuration of two conducting materials can be a local minimizer. For general domains, we propose a rearrangement algorithm to find the optimal distribution numerically. Many results on various domains are shown to demonstrate the efficiency and robustness of the algorithms. Topological changes of the optimal configurations are discussed on circles, ellipses, annuli, and L-shaped domains.

Stokkermans, Thomas J., Jeremy C. Reitinger, George Tye, Chiu-Yen Kao, Sangeetha Ragupathy, Huachun A. Wang, and Carol B. Toris. “Accommodative Exercises to Lower Intraocular Pressure.” Journal of Ophthalmology, vol. 2020, 2020, article 6613066.

Abstract: Purpose. This study investigated how a conscious change in ocular accommodation affects intraocular pressure (IOP) and ocular biometrics in healthy adult volunteers of different ages. Methods. Thirty-five healthy volunteers without ocular disease or past ocular surgery, and with refractive error between −3.50 and +2.50 diopters, were stratified into 20, 40, and 60 year old (y.o.) age groups. Baseline measurements of central cornea thickness, anterior chamber depth, anterior chamber angle, cornea diameter, pupil size, and ciliary muscle thickness were made by autorefraction and optical coherence tomography (OCT), while IOP was measured by pneumotonometry. Each subject’s right eye focused on a target 40 cm away. Three different tests were performed in random order: (1) 10 minutes of nonaccommodation (gazing at the target through lenses that allowed clear vision without accommodating), (2) 10 minutes of accommodation (addition of a minus 3 diopter lens), and (3) 10 minutes of alternating between accommodation and nonaccommodation (1-minute intervals). IOP was measured immediately after each test. A 20-minute rest period was provided between tests. Data from 31 subjects were included in the study. ANOVA and paired t-tests were used for statistical analyses. Results. Following alternating accommodation, IOP decreased by 0.7 mmHg in the right eye when all age groups were combined ( = 0.029). Accommodation or nonaccommodation alone did not decrease IOP. Compared to the 20 y.o. group, the 60 y.o. group had a thicker ciliary muscle within 75 μm of the scleral spur, a thinner ciliary muscle at 125–300 μm from the scleral spur, narrower anterior chamber angles, shallower anterior chambers, and smaller pupils during accommodation and nonaccommodation (’s < 0.01). Conclusion. Alternating accommodation, but not constant accommodation, significantly decreased IOP. This effect was not lost with aging despite physical changes to the aging eye. A greater accommodative workload and/or longer test period may improve the effect.

External Grant: Kao, Chiu-Yen. “Research Pairs: Theoretical and Numerical Methods for Geometrical Optimization,” Centre International de Rencontres Mathematiques (CIRM), Luminy, France, 2020.

Abstract: Geometrical optimization problems involve many areas of mathematics including differential geometry and geometric analysis, partial differential equations (PDE), numerical analysis, and scientific computation. These optimization problems arise naturally in many different problems from science and technology, such as mechanical vibrations, photonic crystals, and population dynamics, to name just a few, but also in pure mathematics like in the study of free boundary minimal surfaces, isometric embeddings, and extremal manifolds. The aim of this proposal is to bring together four mathematicians: (CYK) Chiu-Yen Kao, Claremont McKenna College, United States, (SAM) Seyyed Abbas Mohammadi, Yasouj University, Iran, (BO) Braxton Osting, University of Utah, United States, and (EO) Edouard Oudet, Universite Grenoble Alpes, France, who share common research interests on geometric optimization, to work on the open problems which are described below. The specifics of this research project requires specialists from different areas with different points of view to make significant progress. A research pairs context would give us the opportunity to focus together on these questions and share our knowledge which seems to us mandatory to obtain new interesting results for the mathematical community. Our project will focus on the following two directions and the described goals: (1) Computing free boundary minimal surfaces in the unit ball using extremal Steklov eigenfunctions: investigate numerically the existence of minimal surfaces with prescribed genius and a fixed number of boundary connected components. (2) A spectral approach for computing Nash embeddings: provide an alternative constructive approach by spectral optimization to Gromov's convex integration theory to build C1 isometric embedding.

Istanbouli*, Karma, and Sam Nelson. “Quandle Module Quivers.” Journal of Knot Theory and Its Ramifications, vol. 29, no. 12, 2020, 2050084.

Astract: We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant which specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same quandle module polynomial.

Joung, Yewon, and Sam Nelson. “Biquandle Module Invariants of Oriented Surface-Links.” Proceedings of the American Mathematical Society, vol. 148, no. 7, 2020, pp. 3135-3148

Abstract: We define invariants of oriented surface-links by enhancing the biquandle counting invariant using biquandle modules, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface links. We provide examples illustrating the computation of the invariant and demonstrate that these invariants are not determined by the first and second Alexander elementary ideals and characteristic polynomials.

Kobayashi, Forest* and Sam Nelson. “Kaestner Brackets.” Topology and Its Applications, vol. 282, 2020, 107324.

Abstract: We introduce Kaestner brackets, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle 2-cocycle invariants and the classical biquandle brackets as special cases, coinciding with them for oriented classical knots and links but defining generally stronger invariants for oriented virtual knots and links. We provide examples to illustrate the computation of the new invariant and to show that it is stronger than the classical biquandle bracket invariant for virtual knots.

Needell, Deanna, Sam Nelson, Sam, and Yingqi Shi*. “Tribracket Modules.” International Journal of Mathematics, vol. 31, no. 4, 2020, 2050028.

Abstract: Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modules and use these structures to enhance the tribracket counting invariant. We provide examples to illustrate the computation of the invariant and show that the enhancement is proper.

Nelson, Sam, Natsumi Oyamaguchi, and Radmila Sazdanovic. “Psyquandles, Singular Knots and Pseudoknots.” Tokyo Journal of Mathematics, vol. 42, no. 2, 2019, pp. 405-429. [Published in 2019, but not included in previous Celebration]

Abstract: We generalize the notion of biquandles to \textit{psyquandles} and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and p-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks we call the \textit{Jablan polynomial} and compute the invariant for all pseudoknots with up to five crossings and all 2-bouquet graphs with up to 6 classical crossings.

External Grant: Nelson, Sam. Simons Foundation Collaboration Grant for Mathematicians, 2020, #702597

Abstract: This five-year grant will provide me with $6,000 per year for research travel and$1,000 per year to bring research collaborators and speakers to CMC for five years.

Pike, James A., and Robert J. Valenza. “Invitation to the Conversation.” Under Six Eyes Blog, December 2020.

Pike, James A., and Robert J. Valenza. Under Six Eyes: A Dialog on God in the World. Austin Macauley Publishers, 2020.

Abstract: A mind drawn to and trained in mathematics and science meets a mind drawn to and trained in religion. Both are fascinated by the other's worldview and see the chance for a reciprocal expansion. A long conversation takes place with relentless questions exchanged--not to win an argument, but rather to find the truth. In this exploratory dialogue, scores of great images and ideas are brought to bear from both the scientific and religious worldviews, adding depth and color to a conversation that is as organic as it is profound. New ideas and new convictions arise; and where there cannot be resolution, there is at least clarity.

Flapan, Erica, and Helen Wong, eds. Topology and Geometry of Biopolymers, vol. 746. American Mathematical Society, 2020.

Abstract: This book contains the proceedings of the AMS Special Session on Topology of Biopolymers, held from April 21–22, 2018, at Northeastern University, Boston, MA. The papers cover recent results on the topology and geometry of DNA and protein knotting using techniques from knot theory, spatial graph theory, differential geometry, molecular simulations, and laboratory experimentation. They include current work on the following topics: the density and supercoiling of DNA minicircles; the dependence of DNA geometry on its amino acid sequence; random models of DNA knotting; topological models of DNA replication and recombination; theories of how and why proteins knot; topological and geometric approaches to identifying entanglements in proteins; and topological and geometric techniques to predict protein folding rates. All of the articles are written as surveys intended for a broad interdisciplinary audience with a minimum of prerequisites. In addition to being a useful reference for experts, this book also provides an excellent introduction to the fast-moving field of topology and geometry of biopolymers.