A certain infinite-dimensional Lie group, obtained originally by regularizing and then exponentiating a singular local current algebra constructed from canonical fields, has been shown over several decades to describe a wide variety of quantum systems, via its continuous unitary representations. The group is a semidirect product of Schwartz space functions under pointwise addition with a diffeomorphism group under composition. Previously unsuspected possibilities were predicted this way, including anyons and nonabelian anyons in two-dimensional space. Last year David Sharp and I offered a much simpler, direct construction. We proposed fundamental reasons why this group serves as a universal kinematical group for quantum systems with mass in an arbitrary physical space. This paper beyond summarizing that construction, discusses more extensively what it means to obtain quantum mechanics without actually quantizing classical observables in phase space, and answers some fundamental questions. Directions are suggested for obtaining still greater generality of kinematical and dynamical possibilities by modifying the initial assumptions.

UDC 517.98 Suppose that we have a canonical Gibbs measure $\mu$ defined on a marked configuration space $\Omega$ that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on $\Omega.$ Then $\mu$ is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to $\mu$-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version $\beta$ of this cocycle. Further, we characterize all the measures on $\Omega$ that possess $\beta$ as their cocycle. If $\mu$ is obtained (e.g.) from a particular two-body potential $\hat{V}$ (satisfying some mild regularity assumptions), then $\beta$ takes a certain explicit form, and the class of canonical Gibbs measures characterized by $\beta$ contains exactly the measures associated with the potential $\hat{V}.$ Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general $G$-spaces for local infinite-dimensional groups.

In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities; notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.

Our distinguished colleague, mathematician Professor Emma Previato, is remembered by the editors for her valuable contributions.

Prediction of "anyons," often attributed exclusively to Wilczek, came first from Leinaas and Myrheim in 1977, and independently from Goldin, Menikoff, and Sharp in 1980-81. In 2020, experimentalists successfully created anyonic excitations. This paper discusses why the possibility of quantum particles in two-dimensional space with intermediate exchange statistics eluded physicists for so long after bosons and fermions were understood. The history suggests ideas for the preparation of future researchers. I conclude by addressing failures to attribute scientific achievements accurately, both inadvertent and intentional. Such practices disproportionately hurt women and minorities in physics.

Drawing on some well-studied ideas, I characterize three sources of human mathematical knowledge as foundational pillars: empirical observation, rational thought, and value. Together, these support the essential concept of truth in mathematics. All three are necessary to a sound, holistic philosophy of mathematics education. Empirical observation underlies learners’ sensory, exploratory processes of discovery, verification, and understanding, prerequisite to mathematical abstraction. Rational thought, informal and formal, underlies definition, axiomatization, and deduction enabled by conventional symbol systems. Value underlies more than learning about applications and practical uses. It is fundamental to experiences of beauty, social purpose, and meaning in mathematics – satisfying fundamental needs of the human psyche. Such a holistic perspective envisions a universally accessible world of mathematics.

“Anyons” are quantum particles or excitations in two space dimensions with exchange statistics intermediate between bosons and fermions. They are associated with surface phenomena in the presence of magnetic flux. Theoretical applications are numerous, and in 2020, experimentalists succeeded in creating anyonic excitations. Their prediction four decades earlier, which required fundamental changes in our understanding of quantum statistics, is often attributed exclusively and incorrectly to Wilczek. This chapter outlines very briefly the actual history, from predecessor ideas to the first clear, independent predictions of intermediate statistics in earlier papers, and immediately subsequent insights derived from those earlier predictions. Why did such an easy concept elude physicists for so long? What then led to three independent predictions within a short time of each other? Major epistemological and cognitive obstacles are identified, and some important developments antecedent to the prediction of anyons are cited—theoretical ideas that helped overcome those obstacles, and facilitated the predictions. The article concludes by addressing painful implications of scientists’ and journalists’ systemic failure or refusal to accurately attribute scientific achievements—breaches of integrity occurring even when there is no dispute. The “anyon” case is not unique in this respect. The social consequences of accepting such dishonesty include non-recognition and career obstacles that intimidate and disillusion younger scientists, disproportionately hurting women, minorities, researchers in developing countries, and others without access to sources of influence.

Earlier research has characterized recurrent patterns of cognition, affect, and behavior during in-the-moment mathematical activity. Each pattern, termed an “engagement structure,” is named by a specific motivating desire that evokes it: e.g., Get The Job Done, I’m Really Into This, Value My Culture, etc. This study explores prospective teachers’ motivating desires as they engage in small-group problem solving sessions. Participants were enrolled in courses required for teaching certification at two eastern U.S. state universities. Based on survey, individual interview, and focus group data, we identify the most frequently occurring desires, their perceived importance and accompanying emotional feelings. We present and discuss some findings briefly, including the motivating desire to Carry My Weight with a team of peers.

Symmetry groups describe invariances or partial invariances in physical systems under transformations. Locality refers to the association between physical effects and spatial or spacetime regions, with “action at a distance” forbidden. Local symmetry joins these ideas mathematically in the theory of certain infinite-dimensional groups and their representations. This chapter is an extended abstract of lectures by the author, surveying how unitary representations of diffeomorphism groups and the corresponding current algebras provide a unifying framework for understanding or predicting a wide variety of different quantum and statistical systems.

Based on research and practical experience, this chapter argues that adequate foundations for studying and enhancing mathematical learning and development must incorporate the affective domain—not as an architectural add-on, but as a structurally essential building block. It focuses on some psychological aspects of affect in relation to mathematical thinking and modeling processes. The chapter highlights briefly three affective/cognitive constructs that are not commonly associated with mathematics—integrity, identity, and intimacy. Regarding affect as a representational system renders plausible the hypotheses that it functions crucially during mathematical activity, and that ultimately it is an essential component of mathematical ability. The chapter discusses possible relations between modeling and affect. A fundamental psychological component of modeling with mathematics is the assignment of personal meanings to mathematical symbols and configurations. These "semiotic acts" enable mathematical representations of non-mathematical situations to be constructed.

A general approach is presented to describe nonlinear classical Maxwell electrodynamics with conformal symmetry in flat spacetime. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functions of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.

Diffeomorphism groups and their unitary representations unify the description of a wide variety of diverse nonrelativistic (Galilean) quantum systems. In recent work, we have suggested an approach to relativistic quantum field theory that begins with hierarchies of diffeomorphism group representations and the corresponding current densities. We introduce (noncovariant) creation and annihilation fields that intertwine the representations, and restore the (relativistic) spacetime symmetry by redefining local quantum fields and the Hamiltonian operator in terms of the intertwining fields and current densities. Here we outline the main ideas, and point to next steps and possible future directions of this line of inquiry.

Mathematical engagement is a complex, multidimensional, and dynamic construct. It involves giving attention to one or more objects of engagement – e.g. a mathematical concept, a problem to solve, a situation to be modeled, and/or a person or group in the immediate environment. For a student, engagement often entails social interactions with a teacher, a parent, a tutor, or peers. Sometimes it is characterized as involving interacting cognitive, affective, and behavioral aspects (although these do not constitute distinct types of engagement). This chapter explores briefly another of its dimensions – the conative, which encompasses individuals’ experienced needs, goals, desires, and meaningful purposes, and how these are (or are not) fulfilled. Relationships among mathematical engagement, fundamental human needs, conative feelings, motivating desires, and engagement structures are discussed. Then I outline a possible model describing students’ in-the-moment mathematical engagement during challenging classroom activity such as mathematical modeling. The crucial question for educators becomes how immediate mathematical experiences can meet fundamental, universal needs. This points toward ways of removing barriers to motivation and productive engagement associated specifically with mathematics.

This paper describes certain characteristics of discrete mathematics that can enable teachers to evoke student interest and engagement, and develop students' powerful affect in relation to math-emotions, attitudes, beliefs, and values. Special affordances of discrete mathematics include interesting topics arising in familiar settings, special cases that are easy to set up and explore, a variety of natural representations embodying mathematical structures, and few prerequisites needed for in-depth inquiry. We also list several possible domain-specific sources of commonly-occurring math anxiety (long-term negative affect) which can be ameliorated through effective teaching making use of these features of discrete mathematics. An example from game theory illustrates our suggested approach. Pitfalls are also identified, including the too-early introduction of formal definitions, theorems, and problem-solving algorithms, or (alternatively) the relegation of discrete mathematics to a "slow track" in the curriculum.

We remember a valued colleague and dear friend, S. Twareque Ali, who passed away unexpectedly in January 2016.

Earlier research with younger students finds considerable diversity in the in-the-moment motivating desires that occur during mathematical problem solving. These motivating desires evoke differing patterns of engagement. The present study uses survey, individual interview, and focus group techniques to explore prospective K-6 teachers' motivating desires as they work in groups to solve mathematical problems in university methods courses. We identify and discuss some potentially important motivating desires not previously described.