2021 Mathematical Sciences Publications and Grants

* Indicates student co-author

Aksoy, Asuman G. Review of “Entropy Numbers of Bilinear Operators And Related Classes of Type ℓp,q,” by Dicesar Lass Fernandez, Eduardo Brandani da Silva. Mathematical Reviews/MathSciNet, 2021, MR 4192833.

Aksoy, Asuman G. Review of “Entropy of Sets of Smooth Functions On Compact Homogeneous Manifolds,” by Alexander Kushpel, Kenan Taş, and Jeremy Levesley. Mathematical Reviews/ MathSciNet, 2021, MR4245502.

Aksoy, Asuman G. Review of “The Product Formula For Regularized Fredholm Determinants,” by Thomas Britz, Alan Carey, Fritz Gesztesy, Roger Nichols, Fedor Sukochev, and Dmitriy Zanin. Mathematical Reviews/MathSciNet, 2021, MR4213516.

Li, Shengaki, Bahnisikha Dutta, Sarah Cannon, Joshua J. Daymude, Ram Avinery, Enis Aydin, Andrea W. Richa, Daniel I. Goldman, and Dana Randall. “Programming Active Cohesive Granular Matter with Mechanically Induced Phase Changes.” Science Advances, vol. 7, no. 17, 2021, eabe8494.

Abstract: At the macroscale, controlling robotic swarms typically uses substantial memory, processing power, and coordination unavailable at the microscale, e.g., for colloidal robots, which could be useful for fighting disease, fabricating intelligent textiles, and designing nanocomputers. To develop principles that can leverage physical interactions and thus be used across scales, we take a two-pronged approach: a theoretical abstraction of self-organizing particle systems and an experimental robot system of active cohesive granular matter that intentionally lacks digital electronic computation and communication, using minimal (or no) sensing and control. As predicted by theory, as interparticle attraction increases, the collective transitions from dispersed to a compact phase. When aggregated, the collective can transport non-robot “impurities,” thus performing an emergent task driven by the physics underlying the transition. These results reveal a fruitful interplay between algorithm design and active matter robophysics that can result in principles for programming collectives without the need for complex algorithms or capabilities.

External Grant: Cannon, Sarah. “CRII: AF: RUI: Markov Chains and Random Sampling on Graphs.” National Science Foundation, Division of Computing and Communication Foundations, Research Initiation Initiative (CRII). $174,583, 2021-2023.

Abstract: Given a large collection of mathematical objects, how can you find a “typical” element efficiently? The problem of randomly sampling from a large, complex set arises across many areas including polling, estimating statistics of physical systems, and randomized algorithms; studying randomly selected elements can provide insights about likely properties and behaviors. For example, random samples of political districting plans have been used to build baselines for comparison and detect racial and partisan gerrymandering. However, the problem of efficiently finding random elements in complex settings is often a difficult one, and it can be hard to make rigorous guarantees about the processes involved. More mathematical analysis is needed so that there can be confidence in the conclusions produced. One way to generate random samples is to use a Markov chain: iteratively make random local changes and mathematically bound the mixing time, the number of iterations until the configuration obtained is sufficiently random. Mathematical insight is needed to rigorously understand these Markov chains and their sampling behavior. Another approach is to generate random samples via self-reducibility using approximate counting algorithms. While many approximate counting algorithms come from Markov chains themselves, other approaches include decay of correlations, interpolation, and most recently, the cluster expansion. This project focuses on the following important problems for random sampling on graphs, involving exciting theoretical questions with an eye toward relevant applications: (1) Approximate counting and sampling for spin systems using the cluster expansion; (2) Rigorously analyzing Markov chains used for sampling political districting plans and other problems with similar structure. This project will strengthen the interdisciplinary connections between the theory of random sampling and other disciplines like statistical physics and political science. The investigator will also work to make this research area more accessible to undergraduate students, through continued mentoring, writing and distributing an undergraduate-level introduction to the theory of Markov chains, and creating and teaching a new elective class related to these topics.

Fukshansky, Lenny, Pavel Guerzhoy, and Stefan Kuehnlein. “On Sparse Geometry of Numbers.” Research in the Mathematical Sciences, vol. 8, no. 1, 2021, article 2.

Abstract: Let 𝐿 be a lattice of full rank in n-dimensional real space. A vector in 𝐿 is called 𝑖-sparse if it has no more than 𝑖 nonzero coordinates. We define the 𝑖th successive sparsity level of 𝐿, 𝑠𝑖(𝐿), to be the minimal 𝑠 so that 𝐿 has 𝑠 linearly independent 𝑖-sparse vectors, then 𝑠𝑖(𝐿)≤𝑛 for each 1 ≤ 𝑖 ≤ 𝑛. We investigate sufficient conditions for 𝑠𝑖(𝐿) to be smaller than 𝑛 and obtain explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in 𝐿. These results can be viewed as a partial sparse analogue of Minkowski’s successive minima theorem. We then use this result to study virtually rectangular lattices, establishing conditions for the lattice to be virtually rectangular and determining the index of a rectangular sublattice. We further investigate the 2-dimensional situation, showing that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real 𝑗-invariant. We also identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points.

Fukshansky, Lenny and David Kogan. “On the Geometry of Nearly Orthogonal Lattices.” Linear Algebra and its Applications, vol. 629, 2021, pp. 112-137.

Abstract: Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of “near orthogonality” in 2-dimensions goes back to the work of Gauss. In this paper, we focus on well-rounded nearly orthogonal lattices in ~ℝ𝑛 and investigate their geometric and optimization properties. Specifically, we prove that the sphere packing density function on the space of well-rounded lattices in dimension does not have any local maxima on the nearly orthogonal set and has only one local minimum there: at the integer lattice ~ℤ𝑛. Further, we show that the nearly orthogonal set cannot contain any perfect lattices for ~𝑛 ≥ 3 , although it contains multiple eutactic (and even strongly eutactic) lattices in every dimension. This implies that eutactic lattices, while always critical points of the packing density function, are not necessarily local maxima or minima even among the well-rounded lattices. We also prove that a (weakly) nearly orthogonal lattice in ~ℝ𝑛 contains no more than ~4𝑛 - 2 minimal vectors (with any smaller even number possible) and establish some bounds on coherence of these lattices.

Huber, Mark. Probability Adventures. Independent, 2021.

Abstract: This is an undergraduate text for a Calculus-based Probability course. Most traditional texts follow the path of defining probability through set theory. The disadvantage of that approach is that it does not allow for statements such that “the probability that 3 is less than 4 is 1.” It would be nice to be able to say that the probability of a true statement is 1 and the probability of a false statement is 0. Therefore, in writing this book, the first goal to create a text that builds on *logic* rather than *set theory*. A complete axiomatic system is constructed based directly on first order logic. This gives a more intuitive look at probability. The book was written with an eye towards later applications in statistics, stochastic Operations Research, and mathematical finance. The second goal was to create a book that was more striking visually than past texts. Towards this end, the style of the book is that of a Role Playing Game (RPG) such as Dungeons & Dragons. Definitions, lemmas and theorems are color highlighted, making it easy to both skim a chapter and dive in for details. The stories and problems written also have a fantasy flair, and problems at the end of the chapter are presented as probability encounters.

Shapiro, Ilana and Mark Huber. “Markov Chains for Computer Music Generation.” Journal of Humanistic Mathematics, vol. 11, issue 2, 2921, pp. 167-195.

Abstract: Random generation of music goes back at least to the 1700s with the introduction of Musical Dice Games. More recently, Markov chain models have been used as a way of extracting information from a piece of music and generating new music. We explain this approach and give Python code for using it to first draw out a model of the music and then create new music with that model.

Kao, Chiu-Yen and Seyyed Abbas Mohammadi. “Extremal Rearrangement Problems Involving Poisson’s Equation with Robin Boundary Conditions.” Journal of Scientific Computing, vol. 86, 2021, article 40.

Abstract: In this paper, we study both minimization and maximization problems corresponding to a Poisson’s equation with Robin boundary conditions. These rearrangement shape optimization problems arise in many applications including the design of mechanical vibration and fluid mechanics that explore the possibility to control the total displacement and the kinetic energy, respectively. Analytically, we study the properties of the extremizers on general domains including topology and geometry of the optimizers. Asymptotic behaviors of the optimal values are investigated as well. Although the explicit solutions are rare for this kind of optimization problems, we obtain such solutions on 𝑁-balls. Numerically, we propose efficient algorithms based on finite element methods, rearrangement techniques and our analytical results to determine the extremizers in just a few iterations on general domains.

Kao, Chiu-Yen and Seyyed Abbas Mohammadi. “Tuning the Total Displacement of Membranes.” Communications in Nonlinear Science and Numerical Simulation, vol. 96, 2021, 105706.

Abstract: In this paper we study a design problem to tune the robustness of a membrane by changing its vulnerability. Consider an energy functional corresponding to solutions of Poisson’s equation with Robin boundary conditions. The aim is to find functions in a rearrangement class such that their energies would be a given specific value. We prove that this design problem has a solution and also we propose a way to find it. Furthermore, we derive some topological and geometrical properties of the configuration of the vulnerability. In addition, some explicit solutions are found analytically when the domain is an 𝑁-ball. For general domain we develop a numerical algorithm based on rearrangements to find the solution. The algorithm evolves both minimization and maximization processes over two different rearrangement classes. Our algorithm works efficiently for various domains and the numerical results obtained coincide with our analytical findings.

Kao, Chiu-Yen, Seyyed Abbas Mohammadi, and Braxton Osting. “Linear Convergence of a Rearrangement Method for the One-dimensional Poisson Equation.” Journal of Scientific Computing, vol. 86, no. 1, 2021, article 6.

Abstract: In this paper, we study a rearrangement method for solving a maximization problem associated with Poisson’s equation with Dirichlet boundary conditions. The maximization problem is to find the forcing within a certain admissible set as to maximize the total displacement. The rearrangement method alternatively (i) solves the Poisson equation for a given forcing and (ii) defines a new forcing corresponding to a particular super-level-set of the solution. Rearrangement methods are frequently used for this problem and a wide variety of similar optimization problems due to their convergence guarantees and observed efficiency; however, the convergence rate for rearrangement methods has not generally been established. In this paper, for the one-dimensional problem, we establish linear convergence. We also discuss the higher dimensional problem and provide computational evidence for linear convergence of the rearrangement method in two dimensions.

Oudet, Éduoard, Chiu-Yen Kao and Braxton Osting. “Computation of Free Boundary Minimal Surfaces Via Extremal Steklov Eigenvalue Problems.” ESAIM: Control, Optimisation and Calculus of Variations, vol. 27, 2021, article 34.

Abstract: Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally. In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus 𝛾 and 𝑏 boundary components, we maximize σ𝑗(Σ, 𝑔) 𝐿(∂Σ, 𝑔) over a class of smooth metrics, 𝑔, where σ𝑗(Σ, 𝑔) is the 𝑗th nonzero Steklov eigenvalue and 𝐿(∂Σ, 𝑔) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus 𝛾 = 0 and 𝑏 = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with 𝑏 = 2 boundary components and for the third and fifth eigenvalues with 𝑏 = 3 boundary components.

Shilman, Mikhail Marrtcheneko, Gloria Bartolo, Saleem Alameh, Johnny W Peterson, William S Lawrence, Jenniefer E Peel, Satheesh K Sivasubramani, David W C Beasley, Christopher K Cote, Samandra T Demons, Stephanie A Halashoris, Lynda L Miller, Christopher P Klimko, Jennifer L Shoe, David P Fetterer, Ryan McComb, Chi-Lee C Ho, Kenneth A Bradley, Stella Hartmann, Luisa W Cheng, Marina Chugunova, Chui-Yen Kao, Jennifer K Tran, Aram Debedroissan, Leeor Zibermintz, Emiene Amali-Adekwu, Anastasia Levitin, Joel West. “InVivo Activity of Repurposed Amodiaquine as a Host-Targeting Therapy for the Treatment of Anthrax.” ACS Infectious Diseases, vol. 7, issue 8, 2021, pp. 2176-2191.

Abstract: Anthrax is caused by Bacillus anthracis and can result in nearly 100% mortality due in part to anthrax toxin. Antimalarial amodiaquine (AQ) acts as a host-oriented inhibitor of anthrax toxin endocytosis. Here, we determined the pharmacokinetics and safety of AQ in mice, rabbits, and humans as well as the efficacy in the fly, mouse, and rabbit models of anthrax infection. In the therapeutic-intervention studies, AQ nearly doubled the survival of mice infected subcutaneously with a B. anthracis dose lethal to 60% of the animals (LD60). In rabbits challenged with 200 LD50 of aerosolized B. anthracis, AQ as a monotherapy delayed death, doubled the survival rate of infected animals that received a suboptimal amount of antibacterial levofloxacin, and reduced bacteremia and toxemia in tissues. Surprisingly, the anthrax efficacy of AQ relies on an additional host macrophage-directed antibacterial mechanism, which was validated in the toxin-independent Drosophila model of Bacillus infection. Lastly, a systematic literature review of the safety and pharmacokinetics of AQ in humans from over 2,000 published articles revealed that AQ is likely safe when taken as prescribed, and its pharmacokinetics predicts anthrax efficacy in humans. Our results support the future examination of AQ as adjunctive therapy for the prophylactic anthrax treatment.

Yousefinezhad, Mohsen, Chiu-Yen Kao, and Seyyed Abbas Mohammadi. “Optimal Chemotherapy for Brain Tumor Growth in a Reaction-Diffusion Model.” SIAM Journal on Applied Mathematics, vol. 81, issue 3, 2021, pp. 1077-1097.

Abstract: In this paper we address the question of determining optimal chemotherapy strategies to prevent the growth of brain tumor population. To do so, we consider a reaction-diffusion model which describes the diffusion and proliferation of tumor cells and a minimization problem corresponding to it. We shall establish that the optimization problem admits a solution and obtain a necessary condition for the minimizer. In a specific case, the optimizer is calculated explicitly, and we prove that it is unique. Then, a gradient-based efficient numerical algorithm is developed in order to determine the optimizer. Our results suggest a bang-bang chemotherapy strategy in a cycle which starts at the maximum dose and terminates with a rest period. Numerical simulations based upon our algorithm on a real brain image show that this is in line with the maximum tolerated dose (MTD), a standard chemotherapy protocol.

Markaki, Yolanda, Johnny Gan Chong, Yuying Wang, Elsie C. Jacobson, Christy Luong, Shawn Y.X. Tan, Joanna W. Jachowicz, Mackenzie Strehle, Davide Maestrini, Abhik K. Banerjee, Bhaven A. Mistry, Iris Dror, Francois Dossin, Johannes Schöneberg, Edith Heard, Mitchell Guttman, Tom Chou, Kathrin Plath. “𝑋𝑖𝑠𝑡 Nucleates Local Protein Gradients to Propagate Silencing Across the X Chromosome.” Cell, vol. 184, issue 25, 2021, pp. 6174-6192.

Abstract: The lncRNA 𝑋𝑖𝑠𝑡 forms ∼50 diffraction-limited foci to transcriptionally silence one X chromosome. How this small number of RNA foci and interacting proteins regulate a much larger number of X-linked genes is unknown. We show that 𝑋𝑖𝑠𝑡 foci are locally confined, contain ∼2 RNA molecules, and nucleate supramolecular complexes (SMACs) that include many copies of the critical silencing protein SPEN. Aggregation and exchange of SMAC proteins generate local protein gradients that regulate broad, proximal chromatin regions. Partitioning of numerous SPEN molecules into SMACs is mediated by their intrinsically disordered regions and essential for transcriptional repression. Polycomb deposition via SMACs induces chromatin compaction and the increase in SMACs density around genes, which propagates silencing across the X chromosome. Our findings introduce a mechanism for functional nuclear compartmentalization whereby crowding of transcriptional and architectural regulators enables the silencing of many target genes by few RNA molecules.

Adams, Colin, Erica Flapan, Allison Henrich, Louis H. Kauffman, Lewis D. Ludwig, and Sam Nelson, eds. Encyclopedia of Knot Theory. Taylor and Francis Group, 2021.

Aggarwal, Laira*, Sam Nelson, and Patricia Rivera*. “Quantum Enhancements via Tribracket Brackets.” Mediterranean Journal of Mathematics, vol. 18, issue 1, 2021.

Abstract: We enhance the tribracket counting invariant with tribracket brackets, skein invariants of tribracket-colored oriented knots and links analogously to biquandle brackets. This infinite family of invariants includes the classical quantum invariants and tribracket cocycle invariants as special cases, as well as new invariants. We provide explicit examples as well as questions for future work.

Ceniceros, Jose, Mohamed Elhamdadi, Sam Nelson. “Legendrian Rack Invariants of Legendrian Knots.” Communications of the Korean Mathematical Society, vol. 36, no. 3, 2021, pp. 623-639.

Abstract: We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian structures on racks with 3 and 4 elements. We use Legendrian racks to distinguish certain Legendrian knots which are equivalent as smooth knots.

Ceniceros, Jose and Nelson, Sam. “Cocycle Enhancements of Psyquandle Counting Invariants.” International Journal of Mathematics, vol. 32, no. 5, 2021.

Abstract: We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant and are not determined by the Jablan polynomial.

Kim, Jieon, Sam Nelson, and Minju Seo. “Quandle Coloring Quivers of Surface-Links.” Journal of Knot Theory Its Ramifications, vol. 30, no. 1, 2021.

Abstract: Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle 𝑋 and set 𝑆 ⊂ Hom(𝑋,𝑋) of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the in-degree quiver polynomial is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.

Nelson, Sam. “Forbidden Moves, Welded Knots and Virtual Unknotting.” Encyclopedia of Knot Theory, edited by Colin Adams, Erica Flapan, Allison Henrich, Louis H. Kauffman, Lewis D. Ludwig, and Sam Nelson. Taylor and Francis Group, 2021.

Nelson, Sam. “Racks, Biquandles and Biracks.” Encyclopedia of Knot Theory, edited by Colin Adams, Erica Flapan, Allison Henrich, Louis H. Kauffman, Lewis D. Ludwig, and Sam Nelson. Taylor and Francis Group, 2021.

Valenza, Robert. “Consciousness and Self-awareness.” Academia Letters, 2021, article 1704.

Abstract: The mathematician-turned-philosopher Alfred North Whitehead (1861-1947) proposed in his metaphysics that creation of all actual entities was governed by a process he called concrescence, whereby a new thing comes into existence as a kind of hub for subjective connections with previous actual entities (Whitehead, 2010). This is a complicated business, to say the least, but suffice it to mention that at the instant of its becoming, an actual entity is frozen into the past. Thus process takes precedence over substance, and the latter may then be seen as a kind of pattern woven by Whitehead’s evanescent precipitations of subjectivity. My point in mentioning this admittedly simplistic summary of process metaphysics is that it is founded on subjectivity and relationality. About the former one need only notice that the so-called hard problem of consciousness, that something experiential appears out of parts to which no interiority whatsoever is ascribed, is given a sharp tug at its roots by the notion that experience “goes all the way down,” as process philosophers are wont to say. Nonetheless, even Whitehead would not have ascribed consciousness to a quark or an electron. How exactly conscious beings arise from countless such components remains the biggest mystery in philosophy—although not quite so big as it might be from the perspective of reductionism, sitting atop mechanism and atomism.

Bakshi, Rhea Palak, Jozef Przytycki, and Helen Wong. “Skein Modules.” Encyclopedia of Knot Theory, edited by Colin Adams, Erica Flapan, Allison Heinrich, Louis H. Kauffman, Louis H. Ludwig, Sam Nelson. CRC Press, 2021, pp. 617-624.

Bakshi, Rhea Palak, Jozef Przytycki, and Helen Wong. “Kauffman Bracket Skein Modules and Algebras.” Encyclopedia of Knot Theory, edited by Colin Adams, Erica Flapan, Allison Heinrich, Louis H. Kauffman, Louis H. Ludwig, Sam Nelson. CRC Press, 2021, pp. 666-667.

Moon, Han-Bom, and Helen Wong. “The Roger-Yang Kauffman Skein Algebra And the Decorated Teichmüller Space.” Quantum Topology, vol. 12, issue 2, 2021, pp. 265-308.

Abstract: Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper, we consider surfaces with punctures which are not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang’s Poisson algebra homomorphism is injective, and the skein algebra has no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.

Wong, Helen. “Protein Knots, Links, and Non-Planar Graphs.” Encyclopedia of Knot Theory, edited by Colin Adams, Erica Flapan, Allison Heinrich, Louis H. Kauffman, Louis H. Ludwig, Sam Nelson. CRC Press, 2021, pp. 911-918.

Wong, Helen, Rhea Palak Bakshi, and Jozef Przytycki. “Kauffman Bracket Skein Modules and Algebras.” Encyclopedia of Knot Theory, edited by Colin Adams, Erica Flapan, Allison Heinrich, Louis H. Kauffman, Lewis Ludwig, and Sam Nelson. Taylor and Francis Group, 2021.

External Grant: Wong, Helen. Joan and Joseph Birman Fellowship, American Mathematical Society, $50,000, 2021-22.

Abstract: Awarded to exceptionally talented women for extra research support during their mid-career years

External Grant: Wong, Helen. “Topology in Low Dimensions: Quantum Topology and Applications to Molecular Biology.” Simons Fellow in Mathematics, Simons Foundation, $110,245, 2021–22.

Abstract: Awarded for scientific accomplishments. Extends a one-term sabbatical leave to a full year.