# 2016 Mathematical Sciences Publications and Grants

Aksoy, A. G. "Al-Khwarizmı and the Hermeneutic Circle: Reflections on a Trip to Samarkand." *Journal of Humanistic Mathematics*, Volume 6 Issue 2, 2016, 114-127.

**Abstract:** In this paper we discuss al-Khw ̄arzim ̄ı’s life and aspects of his work and suggest a possible hermeneutic avenue into his contributions to mathematics.

Aksoy, A. G. and G. Lewicki. “Bernstein lethargy theorem for Frechet spaces.”* Journal of Approximation Theory*, Vol. 209, 2016, 58-7.

**Abstract:** In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fre ́chet spaces. Let X be an infinite-dimensional Fre ́chet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N. Let en be a decreasing sequence of positive numbers tending to 0. Under one additional but necessary condition on sup{dist(x, Vn)}, we prove that there exist x ∈ X and no ∈ N such that en ≤dist(x,Vn)≤3en 3 for any n ≥ no. By using the above theorem, as a corollary we obtain classical Shapiro’s (1964) and Tyuriemskih’s (1967) theorems for Banach spaces. Also we prove versions of both Shapiro’s (1964) and Tyuriemskih’s (1967) theorems for Fre ́chet spaces. Considering rapidly decreasing sequences, other ver- sions of the BLT theorem in Fre ́chet spaces will be discussed. We also give a theorem improving Konya- gin’s (2014) result for Banach spaces. Finally, we present some applications of the above mentioned result concerning particular classes of Fre ́chet spaces, such as Orlicz spaces generated by s-convex functions and locally bounded Fre ́chet spaces.

Aksoy, A., G. and Z. Ibragimov. “Convexity of the Urysohn Universal Space.”* Journal of Non-linear and Convex Analysis*. (JNCA), Vol. 17, Number 6, 2016, 1239-1247.

**Abstract:** In a paper published posthumously, P.S. Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space, nowadays referred to as the Urysohn universal space. Here we study various convexity properties of the Urysohn universal space and show that it has a finite ball intersection property. We also note that Urysohn universal space is not hyperconvex.

Aksoy, A. G. and G. Lewicki. “Characterization conditions and the numerical index.” *Contemporary Mathematics*, edited by Bernard Russo, Asuman Güven Aksoy, Ravshan Ashurov, and Shavkat Ayupov. Vol. 672, 2016, 17-31.

**Abstract:** In this paper we survey some recent results concerning the numerical index n(·) for large classes of Banach spaces, including vector valued lp-spaces and lp-sums of Banach spaces where 1 ≤ p < ∞. In particular by defining two conditions on a norm of a Banach space X, namely a Local Characterization Condition (LCC) and a Global Characterization Condition (GCC), we are able to show that if a norm on X satisfies the (LCC), then n(X ) = lim n(Xm ). For the case in which N is replaced by a directed, infinite m set S, we will prove an analogous result for X satisfying the (GCC). Our approach is motivated by the fact that n(Lp(μ, X)) = n(lp(X)) = lim n(lmp (X)).

Aksoy, A.G and G. Lewicki. “Minimal projections with respect to numerical radius.” *Ordered Structures and Applications: Positivity VII Trends in Mathematics*, 2016, 1–11.

**Abstract:** In this paper we survey some results on minimality of projections with respect to numerical radius. We note that in the cases Lp, p = 1,2,∞, there is no difference between the minimality of projections measured either with respect to operator norm or with respect to numerical radius. However, we give an example of a projection from l3p onto a two-dimensional subspace which is minimal with respect to norm, but not with respect to numerical ra- dius for p ̸= 1, 2, ∞. Furthermore, utilizing a theorem of Rudin and motivated by Fourier projections, we give a criterion for minimal projections, measured in numerical radius. Additionally, some results concerning strong unicity of minimal projections with respect to numerical radius are given.

Russo, Bernard, Asuman Güven Aksoy, Ravshan Ashurov, and Shavkat Ayupov, editors. *Topics in Functional Analysis and Algebra*, Contemporary Mathematics, Volume 672, 2016.

Boettcher, A., L. Fukshansky, S. R. Garcia, and H. Maharaj. “Lattices from Abelian groups” a section in "Lattices and Applications in Number Theory.” *Oberwolfach Reports*, vol. 13 no. 1, 2016, 87-154.

**Abstract:** We report on our recent progress investigating geometric properties of lattices obtained via an algebraic construction from finite Abelian groups. These lattices generalize the well-known function field lattices of Rosenbloom and Tsfasman and have many interesting properties. In particular, we prove that many of them have bases of minimal vectors, are strongly eutactic, and have large automorphism groups.

Boettcher, A., L. Fukshansky, S. R. Garcia, and H. Maharaj. “Lattices from Hermitian function fields.” *Journal of Algebra*, vol. 447, 2016, 560-579.

**Abstract:** We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses.

Boettcher, A., L. Fukshansky, S. R. Garcia, H. Maharaj, and D. Needell. “Lattices from tight equiangular frames.”* Linear Algebra and its Applications*, vol. 510, 2016, 395-420.

**Abstract:** We consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k+1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher’s recent observation that this lattice is perfect.

Chan, W. K., L. Fukshansky, and G. Henshaw. “Totally isotropic subspaces of small height in quadratic spaces.” *Advances in Geometry,* vol. 16 no. 2, 2016, 153-164.

**Abstract:** Let K be a global field or Q, F a nonzero quadratic form on K^n , n ≥ 2, and V a subspace of K^n. We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of the quadratic space (V,F) such that each such family spans V as a K-vector space. This result generalizes and extends a well-known theorem of J. Vaaler and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are explicit.

**External grant: **Analytic techniques and algebraic constructions in geometric lattice theory, NSA grant H98230-1510051 (2015-2017).

Huber, Mark. "Nearly optimal Bernoulli factories for linear functions." Combinatorics, Probability and Computing 25.04, 2016, 577-591.

**Abstract:** Suppose that *X*_{1}, *X*_{2}, . . . are independent identically distributed Bernoulli random variables with mean *p*. A Bernoulli factory for a function *f* takes as input *X*_{1}, *X*_{2}, . . . and outputs a random variable that is Bernoulli with mean *f*(*p*). A fast algorithm is a function that only depends on the values of *X*_{1}, . . ., *X*_{T}, where *T* is a stopping time with small mean. When *f*(*p*) is a real analytic function the problem reduces to being able to draw from linear functions *Cp* for a constant *C* > 1. Also it is necessary that *Cp* ⩽ 1 − ε for known ε > 0. Previous methods for this problem required extensive modification of the algorithm for every value of *C* and ε. These methods did not have explicit bounds on as a function of *C* and ε. This paper presents the first Bernoulli factory for *f*(*p*) = *Cp* with bounds on as a function of the input parameters. In fact, sup_{p∈[0,(1−ε)/C]} ≤ 9.5ε^{−1}*C*. In addition, this method is very simple to implement. Furthermore, a lower bound on the average running time of any *Cp* Bernoulli factory is shown. For ε ⩽ 1/2, sup_{p∈[0,(1−ε)/C]} ≥0.004*C*ε^{−1}, so the new method is optimal up to a constant in the running time.

Lai, E., D. Moyer, B. Yuan, E. Fox, B. Hunter, A. L. Bertozzi, and P. J. Brantingham, “Topic Time Series Analysis of Microblogs,”* IMA Journal of Applied Mathematics*, 81(3), 2016, 409-431.

**Abstract:** Social media data tend to cluster around events and themes. Local newsworthy events, sports team victories or defeats, abnormal weather patterns, and globally trending topics all influence the content of online discussion. The automated discovery of these underlying themes from corpora of text is of interest to numerous academic fields as well as to law enforcement organizations and commercial users. One useful class of tools to deal with such problems are topic models, which attempt to recover latent groups of word associations from the text.However, it is clear that these topics may also exhibit patterns in both time and space. The recovery of such patterns complements the analysis of the text itself, and in many cases provides additional context. In the present work we describe two methods for mining interesting spatio-temporal dynamics and relations among topics, one that compares the topic distributions as histograms in space and time, and another that models topics over time as temporal or spatio-temporal Hawkes process with exponential trigger functions. Both methods may be used to discover topics with abnormal distributions in space and time. The second method also allows for self-exciting topics, and can recover inter-topic relationships (excitation or inhibition) in both time and space. We apply these methods to a geo-tagged Twitter dataset, and provide analysis and discussion of the results.

D’Orsogna, M., S. Alvaro, B. Hunter, M. Oiva, I. Meirelles, C. Bardiot, A. Jofre, and L. Manovich. “Collaboration and Translational Data Science for the Study of Culture.” *NSF White Paper*, edited by T.R. Tangherlini, 2016.

**Abstract:** The unprecedented growth of cultural data created, collected and shared digitally over the past decade presents enormous opportunities for a better understanding of cultural trends, popular interests and human needs. Extracting information from heterogeneous and, at times, poorly defined data sources and then applying this information in a cogent way presents significant challenges, but holds the promise of advancing the state of the art across many of the core disciplines of Culture Analytics: the Humanities, the Social Sciences, the Library Sciences, Applied Mathematics, Physics and Computer Science. Meaningful advances in the study of culture can derive from a deliberately translational approach to the datadriven study of culture at scale, engaging in a vigorous transdisciplinary approach to problems that range from archiving historical cultural artifacts, to helping individuals navigate the cultural complexities of the dating scene, from tailoring culturally sensitive medical interventions for at risk patients to creating personalized advertisement, while recognizing the demands of users as they navigate these complex collections.

**External Grant:** NSF IPAM Long Program Grant - Culture Analytics ($10,000)

The explosion in the widespread use of the Internet and social media and the ubiquity of low cost computing have increased the possibilities for understanding cultural behaviors and expressions, while at the same time have facilitated opportunities for making cultural artifacts both accessible and comprehensible. The rapidly proliferating digital footprints that people leave as they crisscross these virtual spaces offer a treasure trove of cultural information, where culture is considered to be expressive of the norms, beliefs and values of a group. This program encourages the exploration of the unsolved mathematical opportunities that are emerging in this cultural information space. Many successful approaches to the analysis of cultural content and activities have been developed, yet there is still a great deal of work to be done. In this program, we aim to promote a vigorous collaboration across disciplines and devise new approaches and novel mathematics to address these problems of culture analytics, by bringing together leading scholars in the social sciences and humanities with those in applied mathematics, engineering, and computer science.

**External grant: ** NSF ICERM Travel Grant ($2,000)

This workshop is a one-week program aimed at 20-25 researchers interested in the opportunity to shape the future of research on the mathematics of crime. Small teams will come together to work on real problems with real crime and policing data provided by the Providence Police Department. Five teams will be assembled, each with a technical advisor who will share their expertise and serve as an anchor point and leader for hands-on research that will take place over the course of the week. This will be a truly hands-on experience in which groups will spend time brainstorming mathematical methods and models to approach the problem at hand, analyzing data provided, and creating code to implement ideas as necessary. There will also be research presentations from the technical advisors throughout the week, as well as closing presentations by each team to present their ideas and progress at the end of the workshop.

**External grant: ** Alfred P. Sloan Foundation - MSRI speaker series and summer school (co-PI) ($124,982) Grant Number G-2016-7296 Helene Barcelo (PI)

The Mathematical Sciences Research Institute (MSRI) is hosting a two-week summer graduate school in 2018 with the goal to engage the next generation of researchers in a topic that will continue to gain interest: Analysis of large-scale data. Although the ability to gather vast amount of data has grown, there is a serious shortage of reliable, accurate, and efficient techniques for analyzing this data. Such mathematical techniques are imperative in order to synthesize the abundance of information in large-scale data and truly harness its power to promote significant scientific advances. The two-week summer workshop will be taught by Professors Blake Hunter of Claremont McKenna College’s mathematical sciences and Deanna Needell of UCLA’s mathematics department.

Chen, Weitao, Ching-Shan Chou, and Chiu-Yen Kao. "Minimizing Eigenvalues for Inhomogeneous Rods and Plates." *Journal of Scientific Computing*, 2016, 1-31.

**Abstract:** Optimizing eigenvalues of biharmonic equations appears in the frequency control based on density distribution of composite rods and thin plates with clamped or simply supported boundary conditions. In this paper, we use a rearrangement algorithm to find the optimal density distribution which minimizes a specific eigenvalue. We answer the open question regarding optimal density configurations to minimize k-th eigenvalue for clamped rods and analytically show that the optimal configurations are distinct for clamped rods and simply supported rods. Many numerical simulations in both one and two dimensions demonstrate the robustness and efficiency of the proposed approach.

Chugunova, Marina, Baasansuren Jadamba, Chiu-Yen Kao, Christine Klymko, Evelyn Thomas, and Bingyu Zhao. "Study of a mixed dispersal population dynamics model." *Topics in Numerical Partial Differential Equations and Scientific Computing*, Springer New York, 2016, 51-77.

**Abstract:** In this paper, we consider a mixed dispersal model with periodic and Dirichlet boundary conditions and its corresponding linear eigenvalue problem. This model describes the time evolution of a population which disperses both locally and nonlocally. We investigate how long time dynamics depend on the parameter values. Furthermore, we study the minimization of the principal eigenvalue under the constraints that the resource function is bounded from above and below, and with a fixed total integral. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for the species to die out more slowly or survive more easily. Our numerical simulations indicate that the optimal favorable region tends to be a simply connected domain. Numerous results are shown to demonstrate various scenarios of optimal favorable regions for periodic and Dirichlet boundary conditions.

Kao, Chiu-Yen, Chih-Wen Shih, and Chang-Hong Wu. "Absolute stability and synchronization in neural field models with transmission delays." *Physica D: Nonlinear Phenomena* 328, 2016, 21-33.

**Abstract:** Neural fields model macroscopic parts of the cortex which involve several populations of neurons. We consider a class of neural field models which are represented by integro-differential equations with propagation time delays which are space-dependent. The considered domains underlying the systems can be bounded or unbounded. A new approach, called sequential contracting, instead of the conventional Lyapunov functional technique, is employed to investigate the global dynamics of such systems. Sufficient conditions for the absolute stability and synchronization of the systems are established. Several numerical examples are presented to demonstrate the theoretical results.

**External grant: ** NSF Grant DMS 1318364: Closest point methods for eigenvalue problems from inhomogeneous structures (PI), 2013-2016

**External grant: ** NSF Grant DMS 1346466: AWM-SIAM Workshop and Kovalevsky Lecture, 2014 (co-PI), 2013-2016

Boettcher, A., L. Fukshansky, S. R. Garcia, H. Maharaj, and D. Needell. “Lattices from tight equiangular frames,” *Linear Algebra and its Applications*, vol. 510, 2016, 395-420.

**Abstract:** We consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k+1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher’s recent observation that this lattice is perfect.

Needell, Deanna, Nathan Srebro, and Rachel Ward. "Stochastic gradient descent, weighted sampling, and the randomized Kaczmarz algorithm." *Mathematical Programming* 155.1-2, 2016, 549-573.

**Abstract:** We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning (L/μ)2(L/μ)2 (where LL is a bound on the smoothness and μμ on the strong convexity) to a linear dependence on L/μL/μ. Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the *randomized Kaczmarz algorithm*, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate.

Shi, H.M., M. Case, X. Gu, S. Tu, and D. Needell. “Methods for Quantized Compressed Sensing.”* Proc. Information Theory and Applications* (ITA), La Jolla CA, Jan. 2016.

**Abstract:** In this paper, we compare and catalog the performance of various greedy quantized compressed sensing algorithms that reconstruct sparse signals from quantized compressed measurements. We also introduce two new greedy approaches for reconstruction: Quantized Compressed Sampling Matching Pursuit (QCoSaMP) and Adaptive Outlier Pursuit for Quantized Iterative Hard Thresholding (AOP-QIHT). We compare the performance of greedy quantized compressed sensing algorithms for a given bit-depth, sparsity, and noise level.

**External grant: ** The Sloan Foundation, Grant 2016-7296, “ 2018 Summer Graduate Workshop: Representations of High Dimensional Data”, co-PI, $150,000, 2016-18

**External grant: ** MSRI Summer Graduate School grant (with Blake Hunter, CMC), Representations of High Dimensional Data, 2018, $18,000, co-PI and co-organizer.

**Abstract:** In today's world, data is exploding at a faster rate than computer architectures can handle. For that reason, mathematical techniques to analyze large-scale objects must be developed. One mathematical method that has gained a lot of recent attention is the use of sparsity. Sparsity captures the idea that high dimensional signals often contain a very small amount of intrinsic information. Using this notion, one may design efficient low-dimensional representations of large-scale data as well as robust reconstruction methods for those representations. Moreover, in many applications one does not desire to reconstruct the full signal but rather perform some inference task such as classification, clustering, parameter estimation, and feature selection. In this course, we study the mathematical representations, reconstruction approaches, and data mining techniques for such large-scale data. We will explore various mathematical notions used in high dimensional signal processing including an overview of compressive signal processing, clustering and classification, and topic modeling. Students will learn the mathematical theory, solve small problems, and perform lab activities working with these techniques on real world data.

**External Grant:** NSF CAREER #1348721,“Practical Compressive Signal Processing, $413,527, co-PI.

**Abstract:** A "signal" is any data set that one would like to acquire, for example, an image, a large block of data, or an audio clip. One can imagine asking how quickly one would need to sample an audio clip so that from those samples alone, the audio clip could be accurately recovered. Would you need to sample every nanosecond, every millisecond, or every second? Compressive Signal Processing (CSP) shows that the important information in many signals can be obtained and recovered from far fewer samples than traditionally thought. The applications of CSP are widespread and include imaging (medical, hyperspectral, microscopy, biological), analog-to-information conversion, radar, large scale information synthesis, geophysical data analysis, computational biology, and many more. Although these applications are astounding, there has been a disconnect between the theoretical work in CSP and the use of CSP in practical settings. The goals of this project will bridge this gap by providing methods and analysis for CSP that apply to real-world signals and settings. Such work will lead to decreased scan time in MRI, reduced cost and energy consumption in computing infrastructures, improved detection of diseased crops from hyperspectral images, increased accuracy in radar, and improved compression and analysis in many other large-data applications. In addition, this project will involve students at all levels and introduce them to rigorous scientific research. The PI actively recruits members from under-represented populations, and will continue to promote diversity through her own research and outreach programs.

Early CSP models restrict the class of signals to those compressible in a very specific sense (sparse with respect to an orthonormal or incoherent basis). One goal of this project is to relax this restriction to allow for signals actually encountered in practice, such as those sparse in redundant, coherent, and highly overcomplete dictionaries. We will utilize both greedy approaches and optimization-based methods, tailored to specific dictionaries of interest, as well as more general methods for arbitrary bases. In addition, this project will develop adaptive CSP sampling schemes, where measurements of the signal are designed "on the fly," as they are being taken. Traditional measurement schemes ignore this information, while adaptive schemes have the potential to significantly reduce reconstruction error, number of measurements, and computation time. We will identify optimal measurement strategies for constrained and unconstrained settings, and analyze how much one can actually gain from adaptivity from an information theoretic point of view. The project will also involve work in "one-bit CSP", a new and exciting branch of CSP which handles extreme (and often more realistic) quantization. We will draw on sub-linear methods, where large errors appear naturally, and also use optimization based techniques along with adaptive quantization thresholds to reduce the recovery error below the best possible for non-adaptive quantization. In studying these topics, the research will bridge a large gap in the theory of CSP and provide a unified framework for both practitioners and researchers.

**External grant:** Sloan Fellowship award, $50,000.

**Abstract:** Traditional signal acquisition measures the entire signal of interest using costly sampling hardware, only to discard it as the processing hardware compresses the signal for storage or analysis. Recently, a new field of compressive signal processing (CSP) has emerged in an effort to resolve this seemingly wasteful dilemma. The applications of CSP range from imaging, analog-to-information conversion and radar to geophysical data analysis and computational biology The Sloan Research Fellowships seek to stimulate fundamental research by early-career scientists and scholars of outstanding promise. These two-year, $50,000 fellowships are awarded yearly to 126 researchers in recognition of distinguished performance and a unique potential to make substantial contributions to their field. Nominations are reviewed and candidates selected by an independent selection committee of distinguished scientists in each eligible field. Fellows are selected on the basis of their independent research accomplishments, creativity, and potential to become leaders in the scientific community through their contributions to their field.

Kaestner, Aaron, Sam Nelson, and Leo Selker, “Parity biquandle invariants of virtual knots,” *Topology and Its Applications*, Volume 209, 2016, 207-219.

**Abstract:** We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle phi^0 and a map phi^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide examples to show that the parity cocycle invariants can distinguish virtual knots which are not distinguished by the corresponding non-parity invariants.

Nelson, Sam. "WHAT IS... a Quandle?" *Notices of the American Mathematical Society* 63.4, 2016, 378-380.

**Abstract:** In this invited survey paper for the AMS's "What is a..." series, we introduce the basics of quandle theory.

Nelson, Sam, and Patricia Rivera. "Bikei invariants and gauss diagrams for virtual knotted surfaces." *Journal of Knot Theory and Its Ramifications* 25.03, 2016.

**Abstract:** Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in ℝ^4; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

Nelson, Sam, and Sherilyn Tamagawa. "Quotient quandles and the fundamental Latin Alexander quandle." *New York J. Math* 22, 2016, 251-263.

**Abstract:** Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gr"obner basis-valued invariants, which generalize the Alexander polynomial. We show via example that the invariant is not determined by the generalized Alexander polynomial for virtual knots.

Valenza, Robert J. “Reflections on Education and Self-expression,” *CMC Magazine*, Spring 2016.

**Abstract:** The essay argues for a dynamic understanding of educational discipline not as an end in itself or as a pre-professional enterprise, but ultimately as the foundation of self-expression, particularly as related to the culture of CMC and the future careers of its students. The key theme is illustrated most directly via art-historical examples. The key quote is, “[B]e together by being different.”

Valenza, Robert J. “What Does a Particle Know? Information and Entanglement.” *Intuition in Mathematics and Physics: A Whiteheadian Approach* (Toward Ecological Civilization, Volume 10), edited by Ronny Desmet, Process Century Press, 2016, 182-193.

**Abstract:** This paper addresses the following topics: The analysis of the concept of information in terms of physical substrate, abstract structure and semantic efficacy (three concepts defined within the article) and the cyclical interactions among them. I consider how the last of these bypasses mere physical efficacy, and then relate these ideas to particle physics, in particular to the notion of entanglement and the freewill theorem of Conway and Kochen.