AMERICAN INSTITUTE OF MATHEMATICS
600 E BROKAW ROAD, SAN JOSE, CA 95112 www.aimath.org

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Organizations Filed Purposes: THE INSTITUTE WAS FOUNDED WITH THE PRIMARY GOAL OF IDENTIFYING AND SOLVING IMPORTANT MATHEMATICAL PROBLEMS.

AIM RESEARCH CONFERENCE CENTER (ARCC): A MODEL FOR COLLABORATIVE RESEARCHARCC HOSTS FOCUSED WORKSHOPS IN ALL AREAS OF THE MATHEMATICAL SCIENCES. ARCC FOCUSED WORKSHOPS ARE DISTINGUISHED BY THEIR EMPHASIS ON A SPECIFIC MATHEMATICAL GOAL, SUCH AS MAKING PROGRESS ON A SIGNIFICANT UNSOLVED PROBLEM, UNDERSTANDING THE PROOF OF AN IMPORTANT NEW RESULT, OR EXAMINING THE CONVERGENCE OF TWO DISTINCT AREAS OF MATHEMATICS.AIM FOCUSED WORKSHOPS PROVIDE AN IDEAL FORUM FOR A TEAM OF RESEARCHERS WORKING TOGETHER TO MAP OUT STRATEGIES, SET PRIORITIES, WORK TOWARD A SOLUTION, AND SET IN PLACE A FRAMEWORK FOR PROGRESS ON IMPORTANT MATHEMATICAL PROBLEMS. THE LEADERS IN EACH FIELD ARE INVOLVED IN THE PLANNING OF THE WORKSHOPS, AND JUNIOR SCIENTISTS AND GRADUATE STUDENTS ARE ACTIVE PARTICIPANTS. SPECIAL ATTENTION IS DEVOTED TO FACILITATE COLLABORATIONS WHICH INCLUDE WOMEN, MINORITIES, AND RESEARCHERS AT PRIMARILY UNDERGRADUATE INSTITUTIONS.THE PURPOSE OF AIM'S RESEARCH PROGRAM CALLED SQUARES (STRUCTURED QUARTET RESEARCH ENSEMBLES) IS TO ALLOW A DEDICATED GROUP OF FOUR TO SIX MATHEMATICIANS TO SPEND A WEEK AT AIM IN SAN JOSE, CALIFORNIA WITH THE POSSIBILITY OF RETURNING IN FOLLOWING YEARS. A SQUARE COULD ARISE AS A FOLLOW-UP TO AN AIM WORKSHOP OR IT COULD BE A FREE-STANDING ACTIVITY. AIM WILL PROVIDE BOTH THE RESEARCH FACILITIES AND THE FINANCIAL SUPPORT FOR EACH SQUARE GROUP. AIM WILL SOLICIT SQUARE PROPOSALS IN ALL AREAS OF PURE AND APPLIED MATHEMATICS. PREFERENCE IS GIVEN TO GROUPS WHICH CONTAIN A MIX OF JUNIOR AND SENIOR RESEARCHERS AND TO GROUPS WHICH HAVE PARTICIPANTS FROM NORTH AMERICA. SQUARES USUALLY MEET DURING WEEKS WHEN THERE ARE NO ARCC WORKSHOPS SCHEDULED.TWENTY WORKSHOPS WERE HELD IN 2019 INVOLVING OVER 400 PARTICIPANTS.FIFTEEN SQUARES WORKSHOPS WERE HELD IN 2019 INVOLVING OVER 200 PARTICIPANTS.

DIRECTED GRANTS:THE FOLLOWING DIRECTED GRANTS WERE ACTIVE DURING 2019:1. RESEARCH EXPERIENCES FOR UNDERGRADUATE FACULTY (REUF)THE GOALS OF REUF ARE (A) TO REKINDLE OR ENHANCE A LOVE OF DOING ORIGINAL MATHEMATICS AMONG UNDERGRADUATE FACULTY PARTICIPANTS, (B) TO PREPARE PARTICIPANTS TO ENGAGE IN RESEARCH WITH UNDERGRADUATE STUDENTS AT THEIR HOME INSTITUTIONS, IN SOME CASES USING THE PROBLEMS PRESENTED, (C) TO INVOLVE SOME PARTICIPANTS IN LONG-TERM RESEARCH COLLABORATIONS, AND (D)TO ESTABLISH A NETWORK OF FACULTY AT PRIMARILY UNDERGRADUATE INSTITUTIONS TOGETHER WITH FACULTY AT RESEARCH UNIVERSITIES WHO SUPPORT COLLABORATION AND UNDERGRADUATE RESEARCH. PROGRESS CONTINUED THROUGH 2019 BY THE PI.2. ANALYTICAL THEORY OF L-FUNCTIONSTHIS 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 ANALYTICAL THEORY OF L-FUNCTIONS, WHICH ARE FUNDAMENTAL OBJECTS IN NUMBER THEORY THAT ENCODE ARITHMETIC INFORMATION. PROGRESS CONTINUED THROUGH 2019 BY THE PI.3. COLLABORATIVE RESEARCH: AIM & ICERM RESEARCH EXPERIENCE FOR UNDERGRADUATE FACULTY (REUF)ENABLING MORE AMERICANS TO EARN UNDERGRADUATE 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 NEED, THE AMERICAN INSTITUTE OF MATHEMATICS AND THE INSTITUTE FOR COMPUTATIONAL AND EXPERIMENTAL RESEARCH IN MATHEMATICS WILL CONDUCT A SERIES OF FOUR ANNUAL RESEARCH EXPERIENCES FOR UNDERGRADUATE FACULTY (REUF) WORKSHOPS DURING THE SUMMERS OF 2016, 2017, 2018 AND 2019. PROGRESS CONTINUED THROUGH 2019 BY THE PI.4. COLLABORATIVE RESEARCH: UTMOST: UNDERGRADUATE TEACHING IN MATHEMATICS WITH OPEN SOFTWARE AND TEXTBOOKSTHE 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 WITH 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. PROGRESS CONTINUED THROUGH 2019 BY THE PI.UTMOST 3.0 WILL ADDRESS CHALLENGES IN THE UNDERGRADUATE STEM CURRICULUM, PARTICULARLY THE NEED TO PROMOTE STUDENT LEARNING AND THE DEVELOPMENT OF MATHEMATICAL AND COMPUTATIONAL SKILLS. SPECIFICALLY, THIS PROJECT FOCUSES ON OPEN SOURCE MATHEMATICS TEXT BOOKS THAT ARE AVAILABLE IN FREE ONLINE VERSIONS. IT SEEKS TO UNDERSTAND TWO INTERRELATED QUESTIONS: "HOW DO INSTRUCTORS AND STUDENTS USE TEXTBOOKS?" AND "HOW CAN WE DEVELOP TEXTBOOKS THAT BETTER SUPPORT TEACHING AND LEARNING." TO ANSWER THESE QUESTIONS, THE PROJECT WILL COMPLETE A COMPREHENSIVE EDUCATIONAL RESEARCH STUDY THAT INCLUDES GATHERING DATA FROM MULTIPLE CLASSROOMS IN A VARIETY OF SETTINGS, AND SUBSEQUENT ANALYSIS OF THAT DATA. THE DEVELOPMENT PORTION OF THE PROJECT WILL FOCUS ON PRETEXT, A PUBLISHING SYSTEM DESIGNED TO ENCOURAGE THE CREATION OF FREE, OPEN SOURCE TEXTBOOKS. THE PROJECT WILL EXAMINE THE USE OF EXISTING BOOKS CREATED WITH PRETEXT, AND FROM THESE OBSERVATIONS FURTHER DEVELOP THE PRETEXT PLATFORM, SO THAT OPEN SOURCE TEXTBOOKS CAN HAVE INCREASED EFFECTIVENESS. THE INTEGRATION OF RESEARCH AND DEVELOPMENT ACTIVITIES IS DESIGNED TO CREATEA CONTINUOUS CYCLE OF INNOVATION BETWEEN TE RESEARCH AND DEVELOPMENT ACTIVITIES. THE EDUCATION RESEARCH COMPONENT OF THIS PROJECT WILL STUDY 49 COURSES TAUGHT AT TWO-YEAR COLLEGES, AND FOUR-YEAR COLLEGES AND UNIVERSITIES. THESE COURSES INCLUDE FIRST-YEAR CALCULUS, SECOND-YEAR CALCULUS, SECOND-YEAR LINNEAR ALGEBRA, AND UPPER-DIVISION ABSTRACT ALGEBRA. THE RESEARCH STUDY WILL INVESTIGATE THE WORK OF INSTRUCTORS IN PLANNING AND TEACHING LESSONS DRAWN FROM AN ONLINE TEXTBOOK AND THE WORK OF STUDENTS AS THEY USE THE SAME TEXTBOOKS TO LEARN THE MATERIAL. IN ADDITION, THE STUDY WILL CONTRAST SUCH WORK WITH THE WORK THAT INSTRUCTORS AND STUDENTS DO WHEN USING LESS DYNAMIC RESOURCES, E.G. A PDF OR BOUND COPY OF THE SAME MATERIAL. PRETEXT IS A NEW AUTHORING PLATFORM THAT ENABLES AUTHORS TO EASILY FASHION A TEXTBOOK FOR BOTH PRINT (STATIC) AND ONLINE (DYNAMIC) FORMATS, INCLUDING BOTH COMPUTATIONAL AND INTERACTIVE COMPONENTS IN THE ONLINE VERSION. THE PROJECT WILL CONTINUE THE DEVELOPMENT OF PRETEXT AND THE TECHNICAL UNDERPINNINGS TO CREATE HIGH QUALITY ONLINE VERSIONS OF TEXTBOOKS, WHILE ONLY REQUIRING AUTHORS TO CONCENTRATE ON THEIR CONTENT. THE ACCESSIBILITY FEATURES FOR READERS WITH DISABILITIES WILL BE FURTHER IMPROVED. THE INHERENT TECHNICAL STRUCTURE OF ONLINE PRETEXT BOOKS WILL ALLOW AUTOMATED COLLECTION OF STUDENT TEXTBOOK USAGE DATA, ORGANIZED BY INDIVIDUAL READER, WITH RESOLUTION TO THE MINUTE AND AT THE LEVEL OF INDIVIDUAL COMPONENTS, SUCH AS VIEWING A VIDEO. THIS INTERPLAY BETWEEN RESEARCH AND DEVELOPMENT ACTIVITIES WILL PRODUCE LARGE AMOUNTS OF HIGH-QUALITY DATA ABOUT STUDENTS' USE OF THEIR TEXTBOOKS. THE OPEN TEXTBOOK INITIATIVE AT THE AMERICAN INSTITUTE OF MATHEMATICS WILL CONTINUE AND EXPAND ITS LEADERSHIP IN VETTING AND RECOMMENDING QUALITY FREE AND LOW-COST TEXTBOOKS, AND HELP OTHER STEM DISCIPLINES TO ADOPT ITS SUCCESSFUL EVALUATION CRITERIA. WORKSHOPS FOR INSTRUCTORS, AUTHORS, AND SOFTWARE DEVELOPERS WILL BLEND DISSEMINATION, PROFESSIONAL DEVELOPMENT, EDITORIAL REVIEW, AND SOFTWARE DEVELOPMENT. A NOVEL FEATURE OF THESE WORKSHOPS WILL BE TEACHING TEST-SITE INSTRUCTORS HOW TO CONTRIBUTE TO THE IMPROVEMENT OF THE OPEN SOURCE TEXTBOOKSTHAT THEY ARE USING IN THEIR COURSES. PROGRESS CONTINUED THROUGH 2019 BY THE PI.

INVESTIGATING THE IMPACT OF MATH TEACHERS' CIRCLES ON MATHEMATICAL KNOWLEDGE FOR TEACHING AND CLASSROOM PRACTICE:THE MATH TEACHERS' CIRCLE 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 MATHEMATICAL TEACHERS LEARN BY PARTICIPATING IN THE WORKSHOPS AND HOW TEACHERS USE WHAT THEY HAVE LEARNED IN THEIR MATHEMATICAL TEACHING. PROGRESS CONTINUED THROUGH 2019 BY THE PI.

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