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The American Association of Physics Teachers (AAPT) Committee on Laboratories assembled a task force whose charge was to write an open letter to the physics education community calling for increased investment in accessible lab courses. Contributors to this paper include students, staff, and faculty with and without disabilities who expressed interest in the open letter. In this document, we recognize the need for making physics laboratories more accessible in all spaces (e.g., high school courses, graduate level courses, research labs). We focus on the experiences of students with disabilities in physics lab courses at the undergraduate level because that is the context for which the writing team had the most collective experience. The intended audiences for this document consist of undergraduate physics students, staff, and faculty, especially those who have direct stake in laboratory courses; physics departments; and member societies, including AAPT. We begin by presenting our motivation for the document and the importance of accessibility and diversity in education and the workforce. We start with the broader context of accessibility, narrowing our focus to physics education and the current state of affairs and availability of accessible resources. Accessibility is then discussed in the specific context of physics laboratory courses, focusing on how barriers are created and can be lowered. In exploring ideas and strategies for improving accessibility, we recognize that the development of multiple pathways for laboratory investigation creates opportunities to expand learning opportunities for more students in physics lab programs., Comment: 51 pages; Endorsed by the AAPT Board of Directors on January 5, 2022

We report results from a study designed to identify links between undergraduate students' views about experimental physics and their engagement in multiweek projects in lab courses. Using surveys and interviews, we explored whether students perceived particular classroom activities to be features of experimental physics practice. We focused on 18 activities, including maintaining lab notebooks, fabricating parts, and asking others for help. Interviewees identified activities related to project execution as intrinsic to experimental physics practice based on high prevalence of those activities in interviewees' own projects. Fabrication-oriented activities were identified as conditional features of experimentation based on differences between projects, which interviewees attributed to variations in project resources. Interpersonal activities were also viewed as conditional features of experimentation, dependent upon one's status as novice or expert. Our findings suggest that students' views about experimental physics are shaped by firsthand experiences of their own projects and secondhand experiences of those of others., Comment: 6 pages, 1 table, accepted for publication in 2020 Physics Education Research Conference Proceedings

In many upper-division lab courses, instructors implement multiweek student-led projects. During such projects, students may design and carry out experiments, collect and analyze data, document and report their findings, and collaborate closely with peers and mentors. To better understand cognitive, social, and affective aspects of projects, we conducted an exploratory investigation of student ownership of projects. Ownership is a complex construct that refers to, e.g., students' willingness and ability to make strategic decisions about their project. Using data collected through surveys and interviews with students and instructors at five institutions, we developed a preliminary model for student ownership of projects. Our model describes student interactions with the project during three phases: choice of topic, execution of experiment, and synthesis of results. Herein, we explicate our model and demonstrate that it maps well onto students' and instructors' conceptions of ownership and ideas presented in prior literature., Comment: 6 pages, 1 table, 1 figure. Submitted to 2019 PER Conference Proceedings

In April 2019, Aycock et al. published "Sexual harassment reported by undergraduate female physicists" in Phys. Rev. PER. Their main finding is that 3/4 of undergraduate women in physics in the U.S. report experiencing sexual harassment. Gender minorities experience high rates of harassment also. Many physics departments will want to discuss these findings with the ultimate goal of making our field harassment-free. However, there are major challenges inherent in having these discussions, since many participants will have experienced harassment themselves. We suggest questions for discussion organizers to reflect on, and ideas for how departments can approach these discussions., Comment: Fixed typo; updated acknowledgements

Background: Qualitative interviewing is a common tool that has been utilized by Science, Technology, Engineering, and Mathematics (STEM) education researchers to explore and describe the experiences of students, educators, or other educational stakeholders. Some interviewing techniques use co-creation of an artifact, such as a personal timeline, as a unique way to elicit a detailed narrative from a respondent. The purpose of this commentary is to describe an interview artifact called a life grid. First used and validated in medical sociology to conduct life course research, we adapted the life grid for use in research on undergraduate STEM education. We applied the life grid interview technique to two contexts: 1) students in an advance degree program reflecting on their entire undergraduate career as a biology major, and 2) students in an undergraduate physics program reflecting on a multi-week lab project. Results: We found that the life grid supported four important attributes of an interview: facilitation of the respondents' agency, establishment of rapport between interviewers and respondents, enhanced depth of the respondents' narratives, and the construction of more accurate accounts of events. We situate our experiences with respect to those attributes and compare them with the experiences detailed in literature. Conclusions: We conclude with recommendations for future use of the life grid technique in undergraduate STEM education research. Overall, we find the life grid to be a valuable tool to use when conducting interviews about phenomena with a chronological component., Comment: 39 pages, 2 tables, 2 supplementary tables; submitted to Int. J. STEM Educ

Models of physical systems are used to explain and predict experimental results and observations. The Modeling Framework for Experimental Physics describes the process by which physicists revise their models to account for the newly acquired observations, or change their apparatus to better represent their models when they encounter discrepancies between actual and expected behavior of a system. While modeling is a nationally recognized learning outcome for undergraduate physics lab courses, no assessments of students' model-based reasoning exist for upper-division labs. As part of a larger effort to create two assessments of students' modeling abilities, we used the Modeling Framework to develop and code think-aloud problem-solving activities centered on investigating an inverting amplifier circuit. This study is the second phase of a multiphase assessment instrument development process. Here, we focus on characterizing the range of modeling pathways students employ while interpreting the output signal of a circuit functioning far outside its recommended operation range. We end by discussing four outcomes of this work: (1) Students engaged in all modeling subtasks, and they spent the most time making measurements, making comparisons, and enacting revisions; (2) Each subtask occurred in close temporal proximity to all over subtasks; (3) Sometimes, students propose causes that do not follow logically from observed discrepancies; (4) Similarly, students often rely on their experiential knowledge and enact revisions that do not follow logically from articulated proposed causes., Comment: 18 pages, 5 figures

Multiweek projects in physics labs can engage students in authentic experimentation practices, and it is important to understand student experiences during projects along multiple dimensions. To this end, we conducted an exploratory quantitative investigation to look for connections between students' pre-project views about experimental physics and their post-project sense of project ownership. We administered the Colorado Learning Attitudes About Science Survey for Experimental Physics (E-CLASS) and the Project Ownership Survey (POS) to 96 students enrolled in 6 lab courses at 5 universities. E-CLASS and POS scores were positively correlated, suggesting that students' views about experimentation may be linked to their ownership of projects. This finding motivates future studies that could explore whether these constructs are causally related., Comment: 4 pages, 1 figure, submitted to 2018 PERC Proceedings

Models of physical systems are used to explain and predict experimental results and observations. When students encounter discrepancies between the actual and expected behavior of a system, they revise their models to include the newly acquired observations, or change their apparatus to better represent their models. The Modeling Framework for Experimental Physics describes the process of matching measurements and observations to models by making revisions to resolve discrepancies. As part of a larger effort to create assessments of students' modeling abilities in the context of upper-division electronics courses, we used the Modeling Framework to develop and code think-aloud problem-solving activities centered on troubleshooting an inverting amplifier circuit. We observed that some participants iteratively and continuously made measurements and revisions if they could not immediately propose a cause for an observed discrepancy. This pathway has not been previously discussed in the Modeling Framework. In this paper, we discuss two episodes where students undergo this process to converge on a proposed cause post hoc. We conclude by discussing implications for a modeling assessment based on the observed modeling behavior., Comment: 4 pages, 1 figure, submitted to 2018 PERC Proceedings

The ability to construct, use, and revise models is a crucial experimental physics skill. Many existing frameworks describe modeling in science education at introductory levels. However, most have limited applicability to the context of upper-division physics lab courses or experimental physics. Here, we discuss the Modeling Framework for Experimental Physics, a theoretical framework tailored to labs and experimentation. A key feature of the Framework is recursive interaction between models and apparatus. Models are revised to account for new evidence produced by apparatus, and apparatus are revised to better align with the simplifying assumptions of models. Another key feature is the distinction between the physical phenomenon being investigated and the measurement equipment used to conduct the investigation. Models of physical systems facilitate explanation or prediction of phenomena, whereas models of measurement systems facilitate interpretation of data. We describe the Framework, provide a chronological history of its development, and summarize its applications to research and curricular design. Ultimately, we argue that the Modeling Framework is a theoretically sound and well-tested tool that is applicable to multiple physics domains and research purposes. In particular, it is useful for characterizing students' approaches to experimentation, designing or evaluating curricula for lab courses, and developing instruments to assess students' experimental modeling skills., Comment: 14 pages, 2 figures. Submitted to EJP focus issue on Modelling in Physics Education

The ability to develop, use, and refine models of experimental systems is a nationally recognized learning outcome for undergraduate physics lab courses. However, no assessments of students' model-based reasoning exist for upper-division labs. This study is the first step toward development of modeling assessments for optics and electronics labs. In order to identify test objectives that are likely relevant across many institutional contexts, we interviewed 35 lab instructors about the ways they incorporate modeling in their course learning goals and activities. The study design was informed by the Modeling Framework for Experimental Physics. This framework conceptualizes modeling as consisting of multiple subtasks: making measurements, constructing system models, comparing data to predictions, proposing causes for discrepancies, and enacting revisions to models or apparatus. We found that each modeling subtask was identified by multiple instructors as an important learning outcome for their course. Based on these results, we argue that test objectives should include probing students' competence with most modeling subtasks, and test items should be designed to elicit students' justifications for choosing particular modeling pathways. In addition to discussing these and other implications for assessment, we also identify future areas of research related to the role of modeling in optics and electronics labs., Comment: 24 pages, 2 figures, 5 tables; submitted to Phys. Rev. PER

We investigate students' sense of ownership of multiweek final projects in an upper-division optics lab course. Using a multiple case study approach, we describe three student projects in detail. Within-case analyses focused on identifying key issues in each project, and constructing chronological descriptions of those events. Cross-case analysis focused on identifying emergent themes with respect to five dimensions of project ownership: student agency, instructor mentorship, peer collaboration, interest and value, and affective responses. Our within- and cross-case analyses yielded three major findings. First, coupling division of labor with collective brainstorming can help balance student agency, instructor mentorship, and peer collaboration. Second, students' interest in the project and perceptions of its value can increase over time; initial student interest in the project topic is not a necessary condition for student ownership of the project. Third, student ownership is characterized by a wide range of emotions that fluctuate as students alternate between extended periods of struggle and moments of success while working on their projects. These findings not only extend the literature on student ownership into a new educational domain---namely, upper-division physics labs---they also have concrete implications for the design of experimental physics projects in courses for which student ownership is a desired learning outcome. We describe the course and projects in sufficient detail that others can adapt our results to their particular contexts., Comment: 22 pages, 3 tables, submitted to Phys. Rev. PER

Proficiency with calculating, reporting, and understanding measurement uncertainty is a nationally recognized learning outcome for undergraduate physics lab courses. The Physics Measurement Questionnaire (PMQ) is a research-based assessment tool that measures such understanding. The PMQ was designed to characterize student reasoning into point or set paradigms, where the set paradigm is more aligned with expert reasoning. We analyzed over 500 student open-ended responses collected at the beginning and the end of a traditional introductory lab course at the University of Colorado Boulder. We discuss changes in students' understanding over a semester by analyzing pre-post shifts in student responses regarding data collection, data analysis, and data comparison.

Physics lab courses are an essential part of the physics undergraduate curriculum. Learning goals for these classes often include the ability to interpret measurements and uncertainties. The Physics Measurement Questionnaire (PMQ) is an established open-response survey that probes students' understanding of measurement uncertainty along three dimensions: data collection, data analysis, and data comparison. It classifies students' reasoning into point-like and set-like paradigms, with the set-like paradigm more aligned with expert reasoning. In the context of a course transformation effort at the University of Colorado Boulder, we examine over 500 student responses to the PMQ both before and after instruction in the pre-transformed course. We describe changes in students' overall reasoning, measured by aggregating four probes of the PMQ. In particular, we observe large shifts towards set-like reasoning by the end of the course.

Although developing proficiency with modeling is a nationally endorsed learning outcome for upper-division undergraduate physics lab courses, no corresponding research-based assessments exist. Our longterm goal is to develop assessments of students' modeling ability that are relevant across multiple upper-division lab contexts. To this end, we interviewed 19 instructors from 16 institutions about optics lab activities that incorporate photodiodes. Interviews focused on how those activities were designed to engage students in some aspects of modeling. We find that, according to many interviewees, iteration is an important aspect of modeling. In addition, interviewees described four distinct types of iteration: revising apparatuses, revising models, revising data-taking procedures, and repeating data collection using existing apparatuses and procedures. We provide examples of each type of iteration, and discuss implications for the development of future modeling assessments., Comment: 4 pages, 1 figure; under review

One way to foster a supportive culture in physics departments is for instructors to provide students with personal attention regarding their academic difficulties. To this end, we have developed the Guided Reflection Form (GRF), an online tool that facilitates student reflections and personalized instructor responses. In the present work, we report on the experiences and practices of two instructors who used the GRF in an introductory physics lab course. Our analysis draws on two sources of data: (i) post-semester interviews with both instructors and (ii) the instructors' written responses to 134 student reflections. Interviews focused on the instructors' perceptions about the goals and framing of the GRF activity, and characteristics of good or bad feedback. Their GRF responses were analyzed for the presence of up to six types of statement: encouraging statements, normalizing statements, empathizing statements, strategy suggestions, resource suggestions, and feedback to the student on the structure of their reflection. We find that both instructors used all six response types, in alignment with their perceptions of what counts as good feedback. This exploratory qualitative investigation demonstrates that the GRF can serve as a mechanism for instructors to pay personal attention to their students. In addition, it opens the door to future work about the impact of the GRF on student-teacher interactions., Comment: 12 pages, 2 tables

Developing students' ability to troubleshoot is an important learning outcome for many undergraduate physics lab courses, especially electronics courses. In other work, metacognition has been identified as an important feature of troubleshooting. However, that work has focused primarily on individual students' metacognitive processes or troubleshooting abilities. In contrast, electronics courses often require students to work in pairs, and hence students' in-class experiences likely have significant social dimensions that are not well understood. In this work, we use an existing framework for socially mediated metacognition to analyze audiovisual data from think-aloud activities in which eight pairs of students from two institutions attempted to diagnose and repair a malfunctioning electric circuit. In doing so, we provide insight into some of the social metacognitive dynamics that arise during collaborative troubleshooting. We find that students engaged in socially mediated metacognition at multiple key transitions during the troubleshooting process. Reciprocated metacognitive dialogue arose when students were collectively strategizing about which measurements to perform, or reaching a shared understanding of the circuit's behavior. In addition to elaborating upon these findings, we discuss implications for instruction, and we identify areas for potential future investigation.

In this exploratory qualitative study, we describe instructors' self-reported practices for teaching and assessing students' ability to troubleshoot in electronics lab courses. We collected audio data from interviews with 20 electronics instructors from 18 institutions that varied by size, selectivity, and other factors. In addition to describing participants' instructional practices, we characterize their perceptions about the role of troubleshooting in electronics, the importance of the ability to troubleshoot more generally, and what it means for students to be competent troubleshooters. One major finding of this work is that, while almost all instructors in our study said that troubleshooting is an important learning outcome for students in electronics lab courses, only half of instructors said they directly assessed students' ability to troubleshoot. Based on our findings, we argue that there is a need for research-based instructional materials that attend to both cognitive and non-cognitive aspects of troubleshooting proficiency. We also identify several areas for future investigation related to troubleshooting instruction in electronics lab courses., Comment: 17 pages, 1 table; Submitted to Phys. Rev. Phys. Educ. Res

Physics and physics education in the United States suffer from severe (and, in some cases, worsening) underrepresentation of Black, Latina/o, and Native people of all genders and women of all races and ethnicities. This underrepresentation is a symptom with multiple causes and myriad potential solutions. In this paper, we describe an approach to addressing the causes of underrepresentation through physics students' collective and continued education about racism, sexism, other dimensions of marginalization, as well as models of allyship and social change. Specifically, we focus on the efforts of undergraduate students, graduate students, and postdocs who are members of a student-run diversity-oriented organization in a physics department at a large, selective, predominantly white university with high research activity. This group's education was accomplished through quarterly Diversity Workshops. Here we report on six Diversity Workshops that were co-designed and facilitated by the authors. We describe the context, motivation, and goals of the workshops, the theories underlying their design and implementation, and their content. In addition, we discuss workshop attendance and suggest strategies for maintaining high attendance in the future. Because the details of our workshops were tailored to the specific needs and interests of a particular student organization, our workshop agendas may not be widely applicable beyond our local context. Therefore, we share our model, design principles, and facilitation strategies in this paper., Comment: 6 pages, 2 figures, 1 table; in press

The ability to troubleshoot is an important learning outcome for undergraduate physics laboratory courses. To better understand the role of troubleshooting in electronics laboratory courses, we interviewed 20 electronics instructors from multiple institution types about their beliefs and teaching practices related to troubleshooting. In these interviews, instructors articulated the idea that \emph{nothing works the first time} in multiple contexts pertaining to troubleshooting. We argue that this idea is an expert epistemology and show how it informs instructors' beliefs that (i) students need to know how to troubleshoot, (ii) students should expect to troubleshoot, (iii) all circuit-building lab activities provide opportunities for students to troubleshoot, and (iv) students' ability to construct functional circuits can be a proxy for their ability to troubleshoot malfunctioning circuits. Moreover, we discuss implications for instruction and assessment of troubleshooting in electronics courses., Comment: 4 pages; submitted to 2016 PERC Proceedings

We explore the overlap of two nationally-recognized learning outcomes for physics lab courses, namely, the ability to model experimental systems and the ability to troubleshoot a malfunctioning apparatus. Modeling and troubleshooting are both nonlinear, recursive processes that involve using models to inform revisions to an apparatus. To probe the overlap of modeling and troubleshooting, we collected audiovisual data from think-aloud activities in which eight pairs of students from two institutions attempted to diagnose and repair a malfunctioning electrical circuit. We characterize the cognitive tasks and model-based reasoning that students employed during this activity. In doing so, we demonstrate that troubleshooting engages students in the core scientific practice of modeling., Comment: 20 pages, 6 figures, 4 tables; Submitted to Physical Review PER

This paper describes a new approach for learning from homework, called Peer-Assisted Reflection (PAR). PAR involves students using peer feedback to improve their work on open-ended homework problems. Collaborating with peers and revising one's work based on the feedback of others are important aspects of doing and learning physics. While notable exceptions exist, homework and exams are generally individual activities that do not support collaboration and refinement, which misses important opportunities to use assessment for learning. In contrast, PAR provides students with a structure to iteratively engage with challenging, open-ended problems and solicit the input of their peers to improve their work., Comment: 14 pages (preprint format), 2 figures; accepted for publication in The Physics Teacher

In undergraduate research experiences, student development of an identity as a scientist is coupled to their sense of ownership of their research projects. As a a first step towards studying similar connections in physics laboratory courses, we investigate student ownership of projects in a lasers-based upper-division course. Students spent the final seven weeks of the semester working in groups on final projects of their choosing. Using data from the Project Ownership Survey and weekly student reflections, we investigate student ownership as it relates to students' personal agency, self-efficacy, peer interactions, and complex affective responses to challenges and successes. We present evidence of students' project ownership in an upper-division physics lab. Additionally, we find that there is a complex relationship between student affect and their sense of ownership., Comment: 4 pages, Submitted to PERC Summer 2015

Students in physics laboratory courses, particularly at the upper division, are often expected to engage in troubleshooting. Although there are numerous ways in which students may proceed when diagnosing a problem, not all approaches are equivalent in terms of providing meaningful insight. It is reasonable to believe that metacognition, by assisting students in making informed decisions, is an integral component of effective troubleshooting. We report on an investigation of authentic student troubleshooting in the context of junior-level electronics courses at two institutions. Think-aloud interviews were conducted with pairs of students as they attempted to repair a malfunctioning operational-amplifier circuit. Video data from the interviews have been analyzed to examine the relationship between each group's troubleshooting activities and instances of socially mediated metacognition. We present an analysis of a short episode from one interview., Comment: 4 pages, 1 figure; Submitted to 2015 PERC Proceedings

Troubleshooting systems is integral to experimental physics in both research and instructional laboratory settings. The recently adopted AAPT Lab Guidelines identify troubleshooting as an important learning outcome of the undergraduate laboratory curriculum. We investigate students' model-based reasoning on a troubleshooting task using data collected in think-aloud interviews during which pairs of students attempted to diagnose and repair a malfunctioning circuit. Our analysis scheme is informed by the Experimental Modeling Framework, which describes physicists' use of mathematical and conceptual models when reasoning about experimental systems. We show that this framework is a useful lens through which to characterize the troubleshooting process., Comment: 4 pages, 2 figures; Submitted to 2015 PERC Proceedings

The American Physical Society recently released a statement calling on all university physics departments to provide or facilitate access to research experiences for all undergraduate students. In response, we investigated the current status of access to undergraduate research at University of Colorado Boulder (CU), a large research institution where the number of undergraduate physics majors outnumber faculty by roughly ten to one. We created and administered two surveys within CU's Physics Department: one probed undergraduate students' familiarity with, and participation in, research; the other probed faculty members' experiences as research mentors to undergraduates. We describe the development of these instruments, our results, and our corresponding evidence-based recommendations for improving local access to undergraduate research experiences. Reflecting on our work, we make several connections to an institutional change framework and note how other universities and colleges might adapt our process., Comment: 4 pages, 3 figures, 1 table; Submitted to 2015 PERC Proceedings

This paper describes a tool, the Guided Reflection Form (GRF), which was used to promote reflection in a modeling-based physics course. Each week, students completed a guided reflection and received feedback from their instructors. These activities were intended to help students become better at the process of reflection, developing skills that they could apply in their future learning. We analyzed student reflections: (1) to provide insight into the reflection process itself and (2) to describe common themes in student reflections. Most students were able to use the GRF to reflect on their learning in meaningful ways. Moreover, the themes present in student reflections provide insights into struggles commonly faced by physics students. We discuss the design of the GRF in detail, so that others may use it as a tool to support student reflections., Comment: 13 pages, 3 figures, 2 tables, submitted to AJP

We have designed an introductory laboratory course that engaged first-year undergraduate students in two complementary types of iteration: (1) iterative improvement of experiments through cycles of modeling systems, designing experiments, analyzing data, and refining models and designs; and (2) iterative improvement of self through cycles of reflecting on progress, soliciting feedback, and implementing changes to study habits and habits of mind. The course consisted of three major activities: a thermal expansion activity, which spanned the first half of the semester; final research projects, which spanned the second half of the semester; and guided student reflections, which took place throughout the duration of the course. We describe our curricular designs and report examples of student work that demonstrate students' iterative improvements in multiple contexts., Comment: 23 pages (preprint format), 3 figures; published in American Journal of Physics

We describe a simple experiment for measuring the thermal expansion coefficient of a metal wire and discuss how the experiment can be used as a tool for exploring the interplay of measurement uncertainty and scientific models. In particular, we probe the regimes of applicability of three models of the wire: stiff and massless, elastic and massless, and elastic and massive. Using both analytical and empirical techniques, we present the conditions under which the wire's mass and elasticity can be neglected. By accounting for these effects, we measure nichrome's thermal expansion coefficient to be 17.1(1.3) ppm/K, which is consistent with the accepted value at the 8% level., Comment: 6 pages, 6 figures, 1 table, accepted for publication in American Journal of Physics

The Compass Project is a self-formed group of graduate and undergraduate students in the physical sciences at UC Berkeley. Our goals are to improve undergraduate physics education, provide opportunities for professional development, and increase retention of students-especially those from populations typically underrepresented in the physical sciences. Compass fosters a diverse, collaborative student community by providing a wide range of services, including a summer program and fall/spring seminar courses. We describe Compass's cultural values, discuss how community members are introduced to and help shape those values, and demonstrate how a single set of values informs the structure of both our classroom and organization.We emphasize that all members of the Compass community participate in, and benefit from, our cultural values, regardless of status as student, teacher, or otherwise., Comment: 4 pages, to be submitted to PERC 2012 proceedings

The universal Baxter operator is an element of the Archimedean spherical Hecke algebra H(G,K), K be a maximal compact subgroup of a Lie group G. It has a defining property to act in spherical principle series representations of G via multiplication on the corresponding local Archimedean L-factors. Recently such operators were introduced for G=GL_{\ell+1}(R) as generalizations of the Baxter operators arising in the theory of quantum Toda chains. In this note we provide universal Baxter operators for classical groups SO_{2\ell}, Sp_{2\ell} using the results of Piatetski-Shapiro and Rallis on integral representations of local Archimedean L-factors., Comment: 12 pages

Recently we propose a class of infinite-dimensional integral representations of classical gl(n+1)-Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors given by U(n+1)-equivariant symplectic volumes of appropriate Kahler U(n+1)-manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids GL(n+1,R), R being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over Q_p by taking the formal limit p->1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field Q_1. The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L-factors coincide with L-factors introduced previously by Kurokawa., Comment: 45 pages, Sections 6 and 7 are extended

Looking for a quantum field theory model of Archimedean algebraic geometry a class of infinite-dimensional integral representations of classical special functions was introduced. Precisely the special functions such as Whittaker functions and Gamma-function were identified with correlation functions in topological field theories on a two-dimensional disk. Mirror symmetry of the underlying topological field theory leads to a dual finite-dimensional integral representations reproducing classical integral representations for the corresponding special functions. The mirror symmetry interchanging infinite- and finite-dimensional integral representations provides an incarnation of the local Archimedean Langlands duality on the level of classical special functions. In this note we provide some directions to higher-dimensional generalizations of our previous results. In the first part we consider topological field theory representations of multiple local L-factors introduced by Kurokawa and expressed through multiple Barnes's Gamma-functions. In the second part we are dealing with generalizations based on consideration of topological Yang-Mills theories on non-compact four-dimensional manifolds. Presumably, in analogy with the mirror duality in two-dimensions, S-dual description should be instrumental for deriving integral representations for a particular class of quantum field theory correlation functions and thus providing a new interesting class of special functions supplied with canonical integral representations., Comment: 21 pages, typos are fixed and important reference is added