Agency: NSF | Branch: Standard Grant | Program: | Phase: | Award Amount: 248.72K | Year: 2015
Finding online resources for the learning and teaching of college mathematics is not difficult -- resources abound. However, finding resources that are readily available, field-tested in a variety of settings, and known to be high quality is more of a challenge. This exploration project will (1) provide materials such as textbook content, videos, interactive applets, and instructor guides; (2) identify optimal methods for building comprehensive, quality-controlled, and curated libraries of freely reusable instructional materials for college-level mathematics courses; and (3) investigate the impact of such a library on faculty adoption of evidence-based teaching practices. The materials will be created and curated collaboratively, and the project will provide training in the development and use of such content. The project will explore how this new approach enables instructors to locate and utilize a variety of curricular materials in the classroom.
As a starting point for a curated course, the project will focus on linear algebra because these courses are important for real-world applications as well as for the mathematical theory. The project team will develop a platform for hosting and disseminating open mathematics content, identify optimal ways to foster a collaborative approach to content creation, and run workshops to train faculty in the use of open content. By measuring how the open content is being used and reused by instructors, the investigators will discover what features of online content are especially effective at encouraging broad adoption by mathematicians.
Agency: NSF | Branch: Standard Grant | Program: | Phase: | Award Amount: 119.98K | Year: 2011
L-functions are fundamental objects in analytic number theory which encode arithmetic information. The simplest examples are the Riemann zeta-function, Dirichlet L-functions, and the L-functions associated with modular forms. Understanding the statistical behavior of the values and zeros of these L-functions is the primary theme of Dr. Conrey.s research. He and his collaborators propose a variety of projects each involved with some aspect of L-functions. One involves a new formula with the divisor function d(n) which generalizes the classical Voronoi formula and has an application in the Nyman-Beurling approach to the Riemann Hypothesis; Conrey will develop further applications to moment formulae and to the question of understanding low degree L-functions. Conrey will use the Asymptotic Large Sieve (invented with Iwaniec and Soundararajan) to prove a conjecture about the mean square of all Dirichlet L-functions multiplied by an arbitrary Dirichlet polynomial. A third project involves Mazur.s conjecture that the symmetric power L-functions associated with an elliptic curve seldom vanish at their central point.
Professor Conreys research is in the area of number theory. Modern Number Theory has surprisingly diverse applications, from enabling secure internet transactions, to the construction of optimal networks, and even to the question of cataloguing the various types of bodies in the study of low dimensional topology. One of the most successful tools invented by number theorists is the zeta-function. Its original purpose was to help with the study of prime numbers. Now it, and its analogues, are ubiquitous in number theory. However, there are still some very basic properties of zeta-functions which we do not understand, and which if we did would lead to much progress. The main question is Why do all of the zeros of zeta-functions occur on just one line? Professor Conreys research is centered on the study of the zeros of zeta-functions. As part of this project, Professor Conrey will also continue his work with Math Teachers Circles, which are a collection of 39 problem solving groups all across the country that involve professional mathematicians and Middle School math teachers working together to build communities of problem solvers.
Agency: NSF | Branch: Standard Grant | Program: | Phase: IUSE | Award Amount: 399.26K | Year: 2016
The UTMOST 2.0 project will design, develop and study innovative approaches to the teaching and learning of undergraduate mathematics using open software and electronic textbooks. This project will study students use of electronic textbooks when these are made freely available in a variety of formats and on a variety of devices. These textbooks will support interactive computation through a seamless integration of Sage, the leading open-source computing system for mathematics. The main vehicles for this integration will be SageMathCloud, a rich online environment for computation and collaboration between students and their course instructor, and Sage cells, self-contained interactive calculation modules that can be embedded in any web page while requiring no software installation or prior knowledge of Sage, because the commands are pre-loaded. Additionally, UTMOST 2.0 will support the further adoption of MathBook XML, an authoring platform that enables the creation of highly functional, open-source textbooks--in particular, textbooks which can exist both in print and in an online format that contains Sage cells. The project includes technical improvements to Sage cells which will make it easier for instructors to create their own Sage cells for their particular course needs. A catalog of Sage cells will be curated, which will further lower the barrier to widespread adoption.
Online dynamic textbooks provide a unique opportunity to directly measure all aspects of how they are used. UTMOST 2.0 is designed to leverage technology to enhance the quality and breadth of the mathematical education research into how students and teachers use electronic textbooks and modern online tools for communication about technical subjects in real classrooms. UTMOST 2.0 will work with instructors at multiple institutions, all of whom will use MathBook XML textbooks, some in the online version and some in the printed version. The online version will be hosted in the cloud, on a server that will be configured to record all user activity at an extremely fine-grained level. The automatic online data collection will be combined with classroom data collection methods, providing a means to validate the information collected that can later be correlated with student gains in course content knowledge. UTMOST 2.0 will also assist authors of undergraduate mathematics textbooks that have already converted to the Mathbook XML format in the next steps of incorporating interactive features such as Sage cells and WeBWorK exercises. This will greatly increase the number of books with those advanced features and simultaneously expand the areas in which the UTMOST project can perform its research investigations.
Agency: NSF | Branch: Standard Grant | Program: | Phase: ALGEBRA,NUMBER THEORY,AND COM | Award Amount: 101.98K | Year: 2016
This research project centers on the connections between number theory and random matrix theory. In particular, the study of prime numbers and divisors is now influenced by the work of physicists in modeling high energy systems by understanding the statistics of random matrix theory. This project explores the relationship between these apparently independent areas of research. The work studies the analytic theory of L-functions, which are fundamental objects in number theory that encode arithmetic information. Examples include the Riemann zeta-function, which encodes information about prime numbers, Dirichlet L-functions, which encode information about the equidistribution of primes in arithmetic progressions, and the L-functions associated with modular forms, which encode the equidistribution of more complex sequences, including rational points on elliptic curves. The investigator aims to develop a theoretical framework to explain by number theoretic means the statistical behavior of the values and zeros of such L-functions.
The investigator and collaborators plan work in a variety of projects, each involved with some aspect of L-functions. One project begins with a novel approach to understanding moments of the Riemann zeta-function through a study of convolutions of correlations of shifted divisor functions. This research incorporates a multidimensional discrete analogue of the Hardy-Littlewood Circle method. Another project is to improve bounds on the proportion of zeros of the Riemann zeta function on the critical line. A third project involves work related to an approach to the Riemann Hypothesis as a mollification problem; it is to understand an exact formula for the second moment of the zeta-function multiplied by a specific long Dirichlet polynomial. A fourth project is to prove that at least 60% of the zeros of Dirichlet L-functions are on the critical line.
Agency: NSF | Branch: Standard Grant | Program: | Phase: GEOMETRIC ANALYSIS | Award Amount: 130.58K | Year: 2012
Award: DMS 1206284, Principal Investigator: Yanir Rubinstein
This project focuses on problems mainly in differential geometry and geometric analysis that can be formulated as real and complex Monge-Ampere type equations. These include (1) the existence and regularity of Kahler-Einstein metrics with conic singularities and their applications; (2) the existence and regularity of solutions to and well-posedness of the homogeneous Monge-Ampere equation; (3) new equations of Monge-Ampere type that arise in convex geometry and their relations to PDEs and the Legendre transform. One feature in key parts of this project is to combine tools of microlocal analysis to study these equations, in addition to the more traditional methods of PDEs, convex analysis, pluripotential theory, and several complex variables. Another theme is to investigate novel relations between convex analysis and geometry and complex analysis and geometry.
In general terms, the analytic techniques developed in this proposal should be useful to researchers working in geometry, physics and elsewhere. On the one hand, deepening our understanding of canonical geometries on Kahler manifolds seems to be of interest to physicists trying to model the geometry of the universe. On the other hand, these canonical geometries have relations to a wide variety of established fields in mathematics. Moreover, Monge-Ampere type equations arise in a wide variety of problems in pure and applied mathematics and have a wide range of real-world applications, such as meteorology and optimal design of networks. Developing methods and techniques to construct and approximate solutions to such equations and to study their regularity could have applications in other instances where these equations appear. Finally, the Legendre transform is a classical tool in mathematics, mechanics and economics, and seeking generalizations of this theory to other settings, as in this project, could find a broad range of applications.
Agency: NSF | Branch: Continuing grant | Program: | Phase: INFRASTRUCTURE PROGRAM | Award Amount: 195.48K | Year: 2016
Enabling more Americans to earn undergraduate and graduate degrees in science, technology, engineering and mathematics (STEM) is important to improving American innovation capabilities. Participating in research as undergraduates improves retention in STEM majors and encourages students to pursue graduate degrees. Faculty at colleges and universities that focus on undergraduate education are critical to this mission, yet in many cases such faculty receive little support to do research with the students they teach or continue their own research, and doctoral programs often fail to train their graduates to mentor undergraduate research. To address this national need, the American Institute of Mathematics (AIM) and the Institute for Computational and Experimental Research in Mathematics (ICERM) will conduct a series of four annual Research Experiences for Undergraduate Faculty (REUF) workshops during the summers of 2016, 2017, 2018, and 2019. Each REUF workshop and ongoing activities will provide a new group of faculty participants at undergraduate institutions that have limited research support with a research experience investigating open questions in the mathematical sciences and equip them to engage in research with undergraduate students at their home institutions.
At each REUF workshop four senior mathematicians (leaders) who have experience doing research with undergraduate students will present problems in a variety of areas, after which the participants and leaders will divide into groups and work on the problems; the workshop will conclude with group presentations of each project. Most of the time will be spent doing research, but there will also be whole group discussions about topics such as best practices in undergraduate research led by the (co)-principal investigators, who also meet daily with the leaders. There are opportunities to continue work on the projects started in the workshop, and some participants will become involved in long-term research collaborations with other faculty at the workshop. The recruitment efforts for REUF will target faculty at undergraduate colleges and universities that serve a large proportion of students who are underrepresented minorities, person with disabilities, or first generation college students, as well as faculty who are themselves underrepresented or have disabilities. The REUF project will lead to more and better undergraduate research experiences for the students of the faculty participants, and greater engagement in mathematical research by the faculty participants, thus expanding and diversifying the mathematical workforce.
Agency: NSF | Branch: Standard Grant | Program: | Phase: | Award Amount: 91.43K | Year: 2011
Principal Investigator: Sikimeti Mau
A-infinity algebra structures have recently emerged in symplectic topology and will be investigated and extended by these projects. The principal investigator intends to study these algebraic structures in concrete examples based on Hilbert schemes of points on complex curves or complex surfaces. Part of the goal will be to extend the structures to higher categorical structures, using symplectic constructions closely related to the string diagrams of physicists. The extended algebraic structures are motivated by constructions in algebraic geometry for the same Hilbert schemes, which should have symplectic analogues by mirror symmetry. Two examples in particular that have been well studied on the symplectic side, and can function as guides, are the Heegaard Floer theory developed by low-dimensional topologists, and Seidel-Smiths symplectic Khovanov homology, a symplectically constructed invariant of knots and links. The short-term objective is to find concrete illustrations, and potential applications, of a new theory that is largely abstract, but has the potential to explain algebraic phenomena in these fields. The broader goal is to describe as much of the algebraic structure of Lagrangian Floer theory as possible in a single algebraic language coming from quilts, a recent technique in symplectic topology due to Wehrheim and Woodward.
A symplectic structure is the geometric face of Hamiltonian mechanics, in which the position and momentum coordinates of a system of moving particles are tracked and used to write out equations of motion that correspond to Newtons laws. Spaces that carry such structures are always even-dimensional, and their underlying geometry is about two-dimensional area and higher-dimensional volume rather than about length and angle, which are at the root of much of familiar geometry. New methods are coming into symplectic geometry from other subjects such as low--dimensional topology, and it appears that an algebraic formalism can be devised to carry a number of these new constructions and to reveal useful properties of them. ˇ
Agency: NSF | Branch: Continuing grant | Program: | Phase: MATHEMATICAL SCIENCES RES INST | Award Amount: 10.04M | Year: 2012
The American Institute of Mathematics (AIM) Research Conference Center will continue in its mission of promoting focused collaborative research on important topics from all areas of the mathematical sciences. These workshops are distinguished by their emphasis on identifying key problems in a given field and in establishing teams of researchers to work collaboratively toward resolving these problems. The workshop topics encompass outstanding fundamental problems as well as important applied and interdisciplinary problems facing 21st century mathematical scientists. In addition to conducting 20 workshops annually, AIM will run 30 Structured Quartet Research Ensembles per year; these are groups of four to six mathematical scientists who have an ambitious research goal and meet for a week at a time at AIM, multiple times over a multi-year period. AIM has committed to increasing diversity in the mathematical sciences workplace and has put a structure in place to ensure that such diversity occurs in all of AIMs programs. In addition, AIM holds special workshops that directly address workforce issues and strives to ensure that the discipline has a healthy pipeline to the future.
The American Institute of Mathematics was founded with the goal of fostering mathematical research through collaboration. AIMs guiding premise is that collaboration is essential to the development of modern mathematics, where the depth and breadth of fields have become so great that understanding the connections between various mathematical areas is increasingly difficult, yet many of the most important and interesting developments occur at the confluence of different research areas. AIM has created a collaborative model for workshops that facilitates collaboration, bridging the gap between the various subject areas and between the different mathematical communities. This model is complemented by the Structured Quartet Research Ensembles (SQuaREs) program, which hosts smaller groups of four to six participants to work on ambitious projects over a three-year period. The American Institute of Mathematics Research Conference Center will host 20 AIM-style workshops (each with up to thirty participants) and 30 SQuaREs per year.
Agency: NSF | Branch: Standard Grant | Program: | Phase: DISCOVERY RESEARCH K-12 | Award Amount: 449.98K | Year: 2011
The Math Teachers Circles project (MTC) is connecting mathematicians and mathematics teachers in middle schools by offering summer workshops and continued communication throughout the year. The workshops focus on mathematical problem solving and include activities that offer multiple entry points. The goal of the workshops is to increase teachers knowledge of mathematics for teaching and to help teachers use their knowledge to improve student learning of mathematics. In addition to conducting workshops, researchers are investigating what mathematics teachers learn by participating in the workshops and how teachers use what they have learned in their mathematics teaching. The American Institute of Mathematics (AIM) is facilitating Math Circles in 26 states with research sites in Albuquerque, Denver and San Francisco Bay area. Their research questions include: (1) How is the MTC model being implemented at local sites? (2) What are the effects of participation in a MTC on Teachers Mathematical Knowledge for Teaching? (3) What is the impact of MTC involvement on teachers approaches toward mathematics and classroom practice? Twelve case studies, based on classroom observations, are offering insights into how teachers use their mathematical knowledge in planning, implementing, assessing, and reflecting on their instruction.
Math Teachers Circle leaders and participants are connected by a digital network organized by AIM. Workshops are offered for mathematicians who would like to be leaders and organizers of local Math Teachers Circles, and help is provided to local Circles. The purpose of the local workshops is to develop teachers content knowledge, problem-solving skills, and mathematical habits of mind. The Math Circles supplement other professional development efforts that focus on pedagogy. The MTC model includes five criteria: content focus, active learning, coherence, approximately 50 hours of professional development, and collective participation. Participants are expected to continue to work within the networked community to develop their mathematical knowledge. The research effort is measuring teachers mathematical knowledge and conducting case studies to investigate the impact of the MTC on mathematics teaching. They are videotaping lessons and using the Mathematical Quality of Instruction observation protocol. The project evaluator is from Colorado State University.
Agency: NSF | Branch: Standard Grant | Program: | Phase: | Award Amount: 224.44K | Year: 2012
The American Institute of Mathematics (AIM) will conduct a series of three annual Research Experiences for Undergraduate Faculty (REUF) workshops during the summers of 2013, 2014, and 2015, and additional activities for participants afterwards to support continuation of research engagement sparked by the workshop. At each REUF workshop four senior mathematicians (leaders) who have experience doing research with undergraduate students will present problems in a variety of areas, after which the participants and leaders will divide into groups and work on the problems; the workshop will conclude with group presentations of each project. Most of the time will be spent doing mathematics, but there will also be whole group discussions about topics such as best practices in undergraduate research led by the organizers. Since numerical experiments and computational investigations are crucial to much of the research done with undergraduates, the workshop will also include instruction in using the free open-source mathematics software Sage. AIM directors will facilitate the management of the workshop, and participants will receive full funding.
The REUF workshop and ongoing activities will provide faculty participants whose undergraduate institutions have limited research activity with a research experience investigating open questions in the mathematical sciences and equip them to engage in research with undergraduate students at their home institutions. In addition, some participants will become involved in long-term research collaborations with other faculty at the workshop. The recruitment efforts for REUF will particularly focus on faculty at minority-serving institutions as well as underrepresented minority faculty at undergraduate colleges. The REUF project will lead to more and better undergraduate research experiences for students of the faculty participants, and greater engagement in mathematical research by the faculty participants, thus expanding and diversifying the mathematical workforce.