# 2019 Mathematical Sciences Publications and Grants

Aksoy, Asuman Güven. “On a Theorem of Terzioğlu.” *Turkish Journal of Mathematics*, vol. 43, no. 1, 2019, pp. 258-267.

**Abstract**: The theory of compact linear operators acting on a Banach space has a classical core and is familiar to many. Perhaps less known is the characterization theorem of Terzioğlu for compact maps. This theorem has a number of important connections that deserves illumination. In this paper we survey Terzioğlu’s characterization theorem for compact maps and some of its consequences. We also prove a similar characterization theorem for Q-compact maps.

Aksoy, Asuman. Review of “Entropy Numbers and Widths for the Nikol’skii-Besov Classes of Functions of Many Variables in the Space L∞” by A.S. Romanyuk. *American Mathematical Society MathSciNet Mathematical Reviews, **2019, MR3916134.*

Aksoy, Asuman. Review of “Optimal Approximation with Sparsely Connected Deep Neural Networks” by Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, and Philipp Petersen. *American Mathematical Society MathSciNet Mathematical Reviews, **2019, MR3949699.*

Aksoy, Asuman. Review of “An Upper Bound on the Kolmogorov Widths of a Certain Family of Integral Operators” by Duaine Lewis and Bernd Sing. *American Mathematical Society MathSciNet Mathematical Reviews, **2019, MR3926052.*

Aksoy, Asuman Güven, Monairah Al-Ansari, and Qidi Peng. “Representation Theorems of ℝ-Trees and Brownian Motions Indexed by ℝ-Trees.”* Asian-European Journal of Mathematics*, vol. 12, no. 4, 2019, 1950067.

**Abstract**: We provide a new representation of an ℝ-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the ℝ-trees with radial and river metrics, and these sets of metric rays. In stochastic analysis, these graphical representation theorems are of particular interest in identifying Brownian motions indexed by ℝ-trees.

Cannon, Sarah, Joshua J. Daymude, Cem Gökmen, Dana Randall, and Andréa W. Richa. “A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems.”* Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)*, vol. 145, 2019, article 54.

**Abstract**: We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC '16) that can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM '11), the high-temperature expansion of the Ising model, and careful probabilistic arguments.

Cannon, Sarah, David A. Levin, and Alexandre Stauffer. “Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings.”* Combinatorics, Probability, and Computing*, vol. 28, 2019, pp. 365-387.

**Abstract**: We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n nonoverlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{−s} , (a + 1)2^{−s} ] × [b2^{−t },(b + 1)2^{−t} ] for a, b, s, t ∈ ℤ≥0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Ω(n^{1.38}), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

External Grant: American Institute of Mathematics SQuaRE (Structured Quartet Research Ensemble). "Connections between Computational and Physical Phase Transitions.” Antonio Blanca, Sarah Cannon, Tyler Helmuth, Will Perkins, Samantha Petti, and Alexandre Stauffer, 2019.

**Abstract**: There is a deep and fascinating connection between phase transitions in statistical physics models and algorithmic thresholds, but we do not have a full understanding of the precise relationship between them. In part this is because research in statistical physics typically concerns very specific problems, while computational complexity asks about broad classes of problems and graphs. We view this is an opportunity: for physics intuition to help investigate conjectures in computational complexity, for the enrichment of the connection between statistical mechanics, optimization, and theoretical computer science, and for the refinement of our understanding of physical phenomena. We propose to investigate, among other questions: Can slow-mixing Markov chains still be effective algorithms? To which classes of graphs (beyond lattices) can we generalize the Pirogov-Sinai machinery? Does the lace expansion have any algorithmic applications? This AIM SQuaRE grant provides travel funding for the six principal researchers to work together on these questions for a week at AIM Headquarters in San Jose, CA, with the possibility to return in future years.

Böttcher, Albrecht, Simon Eisenbarth, Lenny Fukshansky, Stephan Ramon Garcia, and Hiren Maharaj. “Spherical 2-Designs and Lattices from Abelian Groups.” *Discrete & Computational Geometry*, vol. 61, issue 1, 2019, pp. 123-135.

**Abstract**: We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutectic, which means that the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of automorphism groups of these lattices, strengthening our previous results in this direction.

Chan, Wai Kiu, and Lenny Fukshansky. “Small Representations of Integers by Integral Quadratic Forms.” *Journal of Number Theory*, vol. 201, 2019, pp. 40-52.

**Abstract**: Given an isotropic quadratic form over a number field which assumes a value *t*, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value* t*. Our bounds on height are explicit in terms of the heights of the form, the space, the algebraic set and the value *t*.

Damir, Mohamed Taoufiq, and Lenny Fukshansky. “Canonical Basis Twists of Ideal Lattices from Real Quadratic Number Fields.”* Houston Journal of Mathematics*, vol. 45, no. 4, 2019, pp. 999-1019.

**Abstract**: Ideal lattices in the plane coming from real quadratic number fields have been investigated by several authors in the recent years. In particular, it has been proved that every such ideal has a basis that can be twisted by the action of the diagonal group into a Minkowski reduced basis for a well-rounded lattice. We explicitly study such twists on the canonical bases of ideals, which are especially important in arithmetic theory of quadratic number fields and binary quadratic forms. Specifically, we prove that every fixed real quadratic field has only finitely many ideals whose canonical basis can be twisted into a well-rounded or a stable lattice in the plane. We demonstrate some explicit examples of such twists. We also briefly discuss the relation between stable and well-rounded twists of arbitrary ideal bases.

Fukshansky, Lenny, Deanna Needell, Josiah Park, and Yuxin Xin*. “Lattices from Tight Frames and Vertex Transitive Graphs.” *The Electronic Journal of Combinatorics*, vol. 26, issue 3, 2019, P3.49.

**Abstract**: We show that real tight frames that generate lattices must be rational. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. We use this observation to describe a construction of strongly eutactic lattices from vertex transitive graphs. We show that such lattices exist in arbitrarily large dimensions and discuss some examples arising from complete graphs, product graphs, as well as some other notable examples of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss various properties of this construction and also mention some potential applications of lattices generated by incoherent systems of vectors.

Fukshansky, Lenny, Deanna Needell, and Benny Sudakov. “An Algebraic Perspective on Integer Sparse Recovery.”* Applied Mathematics and Computation*, vol. 340, 2019, pp. 31-42.

**Abstract**: Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is *sparse*. Applications for this technology are ubiquitous, ranging from wireless communications to medical imaging, and there is now a solid foundation of mathematical theory and algorithms to robustly and efficiently reconstruct such signals. However, in many of these applications, the signals of interest do not only have a sparse representation, but have other structure such as lattice-valued coefficients. While there has been a small amount of work in this setting, it is still not very well understood how such extra information can be utilized during sampling and reconstruction. Here, we explore the problem of integer sparse reconstruction, lending insight into when this knowledge can be useful, and what types of sampling designs lead to robust reconstruction guarantees. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also prove sparse versions of Minkowski’s Convex Body and Linear Forms theorems that exhibit some limitations of this framework.

External Grant: CIMPA grant for WAMS Summer Research School on Lattices, Diophantine Approximation and Heights, to be held at Urgench State University, Uzbekistan, August 17 - 22, 2020.

**Abstract**: This is a grant for 4000 Euro from the Italian foundation CIMPA (Centre International de Mathematiques Pures et Appliques) in support of a West Asia Mathematical School to help with funding Summer Research School for graduate and advanced students on Lattices, Diophantine Approximation and Heights, to be held at Urgench State University, Uzbekistan (August 17 - 22, 2020). Diophantine approximation originated as the study of how well a number could be approximated by rational numbers. From this many new problems arose. A few examples are: simultaneous approximation of several rational numbers, approximation of algebraic integers, developing a metric theory of Diophantine approximation, and approximation on powers of the multiplicative group. Classical Diophantine approximation and its many generalizations require familiarity with concepts from algebraic number theory, lattices, geometry of numbers, and heights. This school's aim is to provide an introduction to Diophantine approximation, and the necessary background to study these problems. We plan to have a follow up CIMPA school on various aspect of Diophantine approximation in 2022 in Uzbekistan. The scientific committee for this school consists of Shabnam Akhtari (University of Oregon), Lenny Fukshansky (Claremont McKenna College), Kate Petersen (Florida State University), Valerio Talamanca (Università Roma Tre).

Huber, Mark. ‘Adaptive Monte Carlo Integration.’ *Wiley StatsRef: Statistics Reference Online*, edited by N. Balakrishnan, Theodore Colton, Brian Everitt, Walter W. Piegorsch, Fabrizio Ruggeri, and Jef L. Teugels. John Wiley & Sons, Ltd., 2019, stat07851.

Huber, Mark. “An Optimal (*є, δ*)-Randomized Approximation Scheme for the Mean of Random Variables with Bounded Relative Variance.”* Random Structures & Algorithms*, vol. 55, issue 2, 2019, pp. 356-370.

**Abstract**: Randomized approximation algorithms for many #P-complete problems (such
as the partition function of a Gibbs distribution, the volume of a convex body,
the permanent of a {0,1}-matrix, and many others) reduce to creating random
variables X1,X2,... with finite mean 𝜇
and standard deviation 𝜎 such that 𝜇
is the solution for the problem input, and the relative standard
deviation |𝜎∕𝜇|≤c for known c. Under
these circumstances, it is known that the number of samples from the {X_{i}}
needed to form an (𝜖,𝛿)- approximation 𝜇^
that satisfies P(|̂𝜇^−𝜇|>𝜖𝜇)≤𝛿
is at least(2−*o*(1))𝜖^{−2}c^{2}ln(1∕[√2𝜋𝛿]). We
present here an easy to implement (𝜖,𝛿)-approximation
𝜇^
that uses (2+*o*(1))c^{2}𝜖^{−2}ln(4∕𝛿)
samples. This achieves the same optimal running time as other estimators, but
without the need for extra conditions such as bounds on third or fourth moments.

Huber, Mark. *Probability: Lectures and Labs*. Mark Huber, 2019.

**Abstract**: This is an undergraduate textbook in probability aimed at math majors or for those intending to take courses such as mathematical statistics or stochastic processes. The book contains the content of a typical one-semester course for students with familiarity with calculus and linear algebra. The book also contains eight laboratory experiments using R. These can be used in class in place of lectures, or as supplemental activities for students.Topics include: basic probability definitions, conditional probability, Bayes' rule, the common distributions, densities, expectation, conditional expectation, joint densities, Bernoulli and Poisson point processes, covariance and correlation, counting measure, Lebesgue measure, Markov and Chebyshev tail inequalities, the Strong Law of Large Numbers, and the ever popular Central Limit Theorem.

Huber, Mark and Bo Jones. “Faster Estimates of the Mean of Bounded Random Variables.” *Mathematics and Computers in Simulation*, vol. 161, 2019, pp. 93-101.

**Abstract**: This work presents a new adaptive algorithm for robustly estimating the mean of a bounded random variable with unknown variance. Previous algorithms came within a constant factor of the best possible average number of samples for this problem, but the constant was large enough to discourage use. Here we present an algorithm that uses at most 40% as many samples on average as the previous approach, and runs as quickly as Central Limit Theorem based heuristic estimates (to first order) on a large class or problems. The method is illustrated using a network reliability problem and importance sampling.

Huber, Mark and Nevena Marić. “Admissible Bernoulli Correlations.” *Journal of Statistical Distributions and Applications*, vol. 6, 2019, article no. 2.

**Abstract**: A multivariate symmetric Bernoulli distribution has marginals that are uniform over the pair {0, 1}. Consider the problem of sampling from this distribution given a prescribed correlation between each pair of variables. Not all correlation structures can be attained. Here we completely characterize the admissible correlation vectors as those given by convex combinations of simpler distributions. This allows us to bijectively relate the correlations to the well-known CUT_{n }polytope, as well as determine if the correlation is possible through a linear programming formulation.

Feghahati, Amir, and Mike Izbicki. "Automatic Discovery of Language Dialects via Explainable Machine Learning." *Southern California Symposium on Natural Language Processing (SoCalNLP)*, 2019.

Izbicki, Mike, Vagelis Papalexakis, Vassilis Tsotras. "Exploiting the Earth's Spherical Geometry to Geolocate Images." *European Conference on Machine Learning and Principles and Practice of Knowledge Discovery (ECML/PKDD)*,* Wurzburg, Germany,* 2019.

**Abstract**: Existing methods for geolocating images use standard classification or image retrieval techniques. These methods have poor theoretical properties because they do not take advantage of the earth's spherical geometry. In some cases, they require training data sets that grow exponentially with the number of feature dimensions. This paper introduces the first image geolocation method that exploits the earth's spherical geometry. Our method is based on the Mixture of von-Mises Fisher (MvMF) distribution, which is a spherical analogue of the popular Gaussian mixture model. We prove that this method requires only a dataset of size linear in the number of feature dimensions, and empirical results show that our method outperforms previous methods with orders of magnitude less training data and computation.

Izbicki, Mike, Vagelis Papalexakis, and Vassilis Tsotras. “Geolocating Tweets Written in Any Language Sent from Any Location.” Conference on Information and Knowledge Management (CIKM): Proceedings of the 28^{th} ACM International Conference on Information and Knowledge Management, 2019, pp. 89-98.

**Abstract**: Most social media messages are written in languages other than English, but commonly used text mining tools were designed only for English. This paper introduces the Unicode Convolutional Neural Network (UnicodeCNN) for analyzing text written in any language. The UnicodeCNN does not require the language to be known in advance, allows the language to change arbitrarily mid-sentence, and is robust to the misspellings and grammatical mistakes commonly found in social media. We demonstrate the UnicodeCNN's effectiveness on the challenging task of content-based tweet geolocation using a dataset with 900 million tweets written in more than 100 languages. Whereas previous work restricted itself to predicting a tweet's country or city of origin (and only worked on tweets written in certain languages from highly populated cities), we predict the exact GPS locations of tweets (and our method works on tweets written in any language sent from anywhere in the world). We predict GPS coordinates using the mixture of von Mises-Fisher (MvMF) distribution. The MvMF exploits the Earth's spherical geometry to improve predictions, a task that previous work considered too com- putationally difficult. On English tweets, our model's predictions average more than 300km closer to the true location than previous work, and in other languages our model's predictions are up to 1500km more accurate. Remarkably, the UnicodeCNN can learn geographic knowledge in one language and automatically transfer that knowledge to other languages.

Izbicki, Mike, Evangelos Papalexakis and Vassilis Tsotras. "The MvMF Loss for Predicting Locations on the Earth's Surface." MAChine Learning for EArth ObservatioN (MACLEAN), 2019.

Mike Izbicki, Christian R. Shelton. "Distributed Learning of Neural Networks with One Round of Communication." *2 ^{nd} International Workshop on Distributed Machine Learning at the Edge (DMLE)*, 2019.

**Abstract**: The optimal weighted average (OWA) is an algorithm for distributed learning of linear models. It achieves statistically optimal theoretical guarantees with only a single round of communication [3]. This paper introduces the non-linear OWA (NOWA) algorithm, which extends the linear OWA into the non-linear setting of neural networks. Due to the diﬃculty of proving theoretical results in this more complex setting, NOWA loses the theoretical guarantees of the OWA algorithm. Nevertheless, we show that NOWA works well empirically. We follow an evaluation procedure introduced by McMahan et. al. [16] for federated learning and show signiﬁcantly improved results on a simple MNIST baseline task.

Izbicki, Mike, and Christian Shelton. "Distributed Learning of Non-Convex Linear Models with One Round of Communication." *European Conference on Machine Learning and Principles and Practice of Knowledge Discovery (ECML/PKDD), Wurzburg, Germany*, 2019.

**Abstract**: We present the optimal weighted average (OWA) distributed learning algorithm for linear models. OWA achieves statistically optimal learning rates, uses only one round of communication, works on nonconvex problems, and supports a fast cross validation procedure. The OWA algorithm first trains local models on each of the compute nodes; then a master machine merges the models using a second round of optimization. This second optimization uses only a small fraction of the data, and so has negligible computational cost. Compared with similar distributed estimators that merge locally trained models, OWA either has stronger statistical guarantees, is applicable to more models, or has a more computationally efficient merging procedure.

Alhejaili, Weaam* and Chiu-Yen Kao. “Maximal Convex Combinations of Sequential Steklov Eigenvalues.” *Journal of Scientific Computing*, vol. 79, issue 3, 2019, pp. 2006-2026.

**Abstract**: In this paper, we study a shape optimization problem in two dimensions where the objective function is the convex combination of two sequential Steklov eigenvalues of a domain with a fixed area constraint. We show the existence of the optimal domain and the nondecreasing, Lipschitz continuity, and convexity of the optimal objective function with respect to the convex combination constant. On one-parameter family of rectangular domains, asymptotic behaviors of lower eigenvalues are found. For general shapes, numerical approaches are used to find optimal shapes. The range of the first two Steklov eigenvalues are discussed for several one-parameter families of shapes including Cassini oval shapes and Hippopede shapes.

Alhejaili, Weaam* and Chiu-Yen Kao. “Numerical Studies of the Steklov Eigenvalue Problem via Conformal Mappings.”*Applied Mathematics and Computation*, vol. 347, 2019, pp. 785-802.

**Abstract**: In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use a gradient ascent approach to find the optimal domain which maximizes *k*th Steklov eigenvalue with a fixed area for a given *k*. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different *k*.

Anderson, Heather A., Melissa D. Bailey, and Chiu-Yen Kao. “Ciliary Muscle Thickness in Adults with Down Syndrome. *Investigative Ophthalmology and Visual Science*, vol. 60, issue 9, 2019, 4306.

**Purpose**: The purpose of this study was to investigate the relationship between age and refractive error (RE) with ciliary muscle thickness (CMT) in adults with Down syndrome. **Methods****: **Thirty three adults with Down syndrome were recruited for imaging of the ciliary muscle with an anterior segment optical coherence tomographer. Images were collected on the right eye of each subject 45 minutes post-dilation with 1% tropicamide and 2.5% phenylephrine. Images were analyzed to calculate thickness at 1, 2, and 3 mm posterior to the scleral spur (CMT1, CMT2, CMT3). Maximum thickness (CMTMAX) and the apical thickness at 1mm posterior to the scleral spur (CMT1 – CMT2) were also calculated. Spherical equivalent RE was based upon a best clinical refraction that utilized both non-dilated and dilated refractive findings for its determination. Multivariate regression analysis was performed to evaluate the relationship between CMT and RE while controlling for subject age at three locations (CMT1, CMT2, CMT3), as well as for apical thickness at CMT1. **Results****:** Good quality images were analyzed from 27 subjects (mean age = 29 ± 9 years) with mean RE of -1.16 ± 5.10 D (range = -15.75 to +5.13 D). Mean CMT decreased with more posterior position (CMT1: 805 ± 82µm; CMT2: 539 ± 131 µm; CMT3: 309 ± 99 µm). Mean CMTMAX was 870 ± 56µm and mean apical fiber thickness was 266 ± 88 µm at CMT1. Overall there was a significant linear correlation indicating thinning CMT with increasing age for CMT1 and CMT2 (p≤0.05) and a significant interaction between RE and age for CMT2 (p = 0.05), but no significant correlation between CMT and RE was found for either location. CMT3 was not associated with age, but had a significant negative correlation with RE (p = 0.01) indicating thicker muscle with increasing myopic RE. Apical thickness was not associated with age, but had a significant positive correlation with RE (p = 0.01) indicating thicker muscle with increasing hyperopic RE. **Conclusions****: **CMT measures in adults with Down syndrome were similar to those previously reported in typical adults without Down syndrome. Also similar to previous reports, CMT was negatively correlated with RE in the posterior region (thicker muscles with increasing myopic RE) and positively correlated with refractive error for the anteriorly located apical fibers (thicker muscles with increasing hyperopic RE).

Kao, Chiu-Yen and Braxton Osting. “Extremal Spectral Gaps for Periodic Schrödinger Operators.”* ESAIM*, vol. 25, 2019, 40.

**Abstract**: The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed *m*, we consider the problem of maximizing the gap-to-midgap ratio for the *m*th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the *m*th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly *m* intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.

External Grant: National Center for Theoretical Sciences 2019 Research Pairs: Theoretical and Numerical Methods for Shape Optimizations, 2019.

**Abstract**: 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 shape optimization, to work on open problems and give a series of lectures to share their knowledge with mathematicians and students in Taiwan during their NCTS visit 6/15/2019-6/30/2019. Shape optimization problems involve many areas of mathematics including differential geometry and geometric analysis, partial differential equations, 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. Several subsets of this group of people have successfully collaborated on shape optimization problems in the past. CYK and BO have worked with each other for several years, publishing articles on minimizing sequential Laplace-Dirichlet eigenvalues, maximizing Laplace-Beltrami eigenvalues on closed Riemannian surfaces, computational methods for extremal Steklov problems, and extremal spectral gaps for periodic Schrödinger operators. BO and EO worked together on developing efficient methods for computing Dirichlet partitions on graphs. SAM and EO have exchanged some research ideas with CYK and BO during the 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, July 5 - July 9, 2018 at Taipei, Taiwan, and the Steklov eigenproblems workshop, April 30 to May 4, 2018 at AIM, San Jose, California. This research pairs program provides a great opportunity for this international group of people from four different time zones to work together in person, interact with members of the mathematics community in Taiwan, and develop both theoretical tools and numerical approaches for the shape optimization problems including extremal rearrangement problems involving Poisson's equation with Robin boundary conditions, extremal eigenvalues of Steklov problems, and isometric embedding of manifolds.

Chien, Julien* and Sam Nelson. “Virtual Links with Finite Medial Bikei.” *Journal of Symbolic Computation*, vol. 92, 2019, pp. 211-221.

**Abstract**: We consider the question of which virtual knots have finite fundamental medial bikei. We describe and implement an algorithm for completing a presentation matrix of a medial bikei to an operation table, determining both the cardinality and isomorphism class of the fundamental medial bikei, each of which are link invariants. As an example, we compute the fundamental medial bikei for all of the prime virtual knots with up to four classical crossings as listed in the knot atlas.

Cho, Karina*, and Sam Nelson. “Quandle Cocycle Quivers.” *Topology and Its Applications*, vol. 268, 2019, 106908.

**Abstract**: We incorporate quandle cocycle information into the quandle coloring quivers we defined in [2] to define weighted directed graph-valued invariants of oriented links we call quandle cocycle quivers. This construction turns the quandle cocycle invariant into a small category, yielding a categorification of the quandle cocycle invariant. From these graphs we define several new link invariants including a 2-variable polynomial which specializes to the usual quandle cocycle invariant. Examples and computations are provided.

Cho, Karina*, and Sam Nelson. “Quandle Coloring Quivers.”* Journal of Knot Theory and Its Ramifications*, vol. 29, no. 1, 2019, 1950001.

**Abstract**: We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.

Gügümcü, Neslihan, and Sam Nelson. “Biquandle Coloring Invariants of Knotoids.”* Journal of Knot Theory and Its Ramifications*, vol. 28, no. 4, 2019, 1950029.

**Abstract**: In this paper, we consider biquandle colorings for knotoids in ℝ^{2} or *S*^{2}, and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a knotoid diagram is endowed with. We generalize Niebrzydowski’s biquandle longitude invariant for virtual long knots to obtain new invariants for knotoids. We show that biquandle invariants can detect mirror images of knotoids and show that our enhancements are proper in the sense that knotoids which are not distinguished by the counting invariant are distinguished by our enhancements.

Nelson, Sam. “A Survey of Quantum Enhancements.” *Knots, Low-Dimensional Topology and Applications: Knots in Hellas, International Olympic Academy, Greece, July 2016*, edited by Colin C. Aams, Cameron McA. Gordon, Vaughan F.R. Jones, Louis H. Kauffman, Sofia Lambropoulou, Kenneth C. Millett, Jozaef H. Przytycki,, Renzo Ricca, and Radmila Sazdanovic. Springer, 2019, pp. 163-178.

**Abstract**: In this short survey article, we collect the current state of the art in the nascent field of quantum enhancements, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.

Nelson, Sam, Kanako Oshiro, and Natsumi Oyamaguchi. “Local Biquandles and Niebrzydowski’s Tribracket Theory.” *Topology and Its Applications*, vol. 258, 2019, pp. 474-512.

**Abstract**: We introduce a new algebraic structure called* local biquandles *and show how colorings of oriented classical link diagrams and broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski’s tribracket (co)homology. This implies that Niebrzydowski’s (co)homology theory can be interpreted similarly as biquandle (co)homology theory. Moreover through the isomorphism between two cohomology groups, we show that Niebrzydowski’s cocycle invariants and local biquandle cocycle invariants are the same.

Nelson, Sam, Kanako Oshiro, Ayaka Shimizu, and Yoshiro Yaguchi. “Biquandle Virtual Brackets.”* Journal of Knot Theory and Its Ramifications*, vol. 28, no. 11, 2019, 1940003.

**Abstract**: We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call *biquandle virtual brackets*. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring* R* using virtual crossings as smoothings, these invariants take the form of multisets of elements of* R *and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, *J. Knot Theory Ramifications *26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

Nelson, Sam, and Shane Pico*. “Virtual Tribrackets.” *Journal of Knot Theory and Its Ramifications*, vol. 28, no. 4, 2019, 1950026.

**Abstract**: We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.

Valenza, Robert. “Spectacular Exponents: A Semi Modular Approach to Fast Exponentiation.” *Journal of Advances in Mathematics,* vol. 16, 2019, pp. 8430-8448.

**Abstract**: This paper introduces a computational scheme for calculating the exponential *b ^{w}* where

*b*and

*w*are positive integers. This two-step method is based on elementary number theory that is used routinely in this and similar contexts, especially the Chinese remainder theorem (CRT), Lagrange’s theorem, and a variation on Garner’s algorithm for inverting the CRT isomorphism. We compare the performance of the new method to the standard fast algorithm and show that for a certain class of exponents it is significantly more efficient as measured by the number of required extended multiplications.

Bonahon, Francis, and Helen Wong. “Representations of the Kauffman Bracket Skein Algebra III: Closed Surfaces and Naturality.” *Quantum Topology*, vol. 10, issue 2, 2019, pp. 325-398.

**Abstract**: This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface *S*. In [BonWon3] we associated a classical shadow to an irreducible representation ρ of the skein algebra, which is a character rρ ∈ RSL_{2}(C)(*S*) represented by a group homomorphism π_{1}(*S*)→SL_{2}(C). The main result of the current article is that, when the surface *S* is closed, every character r ∈ RSL_{2}(C)(*S*) occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism r:π_{1}(*S*)→SL_{2}(C) a representation of the skein algebra *S*^{A}(*S*) that is uniquely determined up to isomorphism.

Cui, Shawn X., Kevin T. Tian, Jennifer F. Vasquez, Zhenghan Wang, and Helen M. Wong. “The Search for Leakage-Free Entangling Fibonacci Braiding Gates.”* Journal of Physics A: Mathematical and Theoretical*, vol. 52, no. 45, 2019, 455301.

**Abstract**: It is an open question if there are leakage-free entangling Fibonacci braiding gates. In this article, we give a construction of a large family of leakage-free braiding gates which are then proved to be non-entangling. We also conducted brute-force numerical searches for braids with a word-length up to seven and found no leakage-free entangling gates. These suggest the negative for the conjecture. On the other hand, we provide a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature.

Flapan, Erica, Adam He, and Helen Wong. “Topological Descriptions of Protein Folding.” *Proceedings of the National Academy of Sciences of the United States of America*, vol. 116, no. 19, 2019, pp. 9360-9369.

**Abstract**: How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010) *PLoS Comput Biol*6:e1000731] for the folding of the **6 _{1}**-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor’s twisted hairpin theory of knot folding for the

**3**

**-knotted proteins and the**

_{1}**4**-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the

_{1}**4**-knotted phytochromes and the

_{1}**5**- and

_{2}**6**-knotted proteins.

_{1}External Grant: National Science Foundation Grant, DMS-1906323, $229,315, 2019-2022. Principal investigator for "RUI: Knots in Three-dimensional Manifolds: Quantum Topology, Hyperbolic Geometry, and Applications". Funded by NSF programs: MPS/DMS-Topology, BIO/MCB-Molecular bio- phyics, MSPA-Interdisciplinary, and Office of Interdisciplinary Activities.

**Abstract**: This project is split into two different areas of research concerning a field of mathematics called topology, which studies the properties of objects that remain the same even when they are twisted or deformed continuously. One direction relates to quantum physics, and the other to molecular biology. In 2016, physicists won the Nobel Prize for applying topology to research in condensed matter physics, and the underlying mathematical framework is called a topological quantum field theory (TQFT). The first part of the project focuses on topological constructions from TQFTs and conjectures about them. The PI aims to further advance the basic understanding of the connections between the mathematical and the theoretical physical sides of the subject. This work may be relevant to practical applications, such as the theoretical foundations and development of a topological quantum computer. The second part of the project is about the topology of proteins, which are long and flexible enough to exhibit knotting or linking. It is believed that such topological characteristics affect a protein's functionality, which is governed by its three-dimensional placement. However, little is known about how the proteins fold into a knotted state, and this project analyzes theories of protein folding from a topological viewpoint. In particular, knotted proteins are implicated in neurodegenerative disorders like Parkinson's and are found in bacteria used for bioremediation; a better understanding of the molecular knotting mechanism may lead to novel ways to target topological characteristics which affect specific biological functions. The award also supports undergraduate students participating in this research.