American Institute Of Mathematics

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.ONE IN-PERSON WORKSHOPS WAS HELD IN 2021 INVOLVING APPROXIMATELY 21 PARTICIPANTS.THREE IN-PERSON SQUARES WORKSHOPS WERE HELD IN 2021 INVOLVING APPROXIMATELY 12 PARTICIPANTS.VIRTUAL WORKSHOPS WERE HELD IN 2021 INVOLVING APPROXIMATELY 93 PARTICIPANTS.ARC'S (COMMUNITIES) VIRTUAL WORKSHOPS WERE HELD INVOLVING 96 PARTICIPANTS.

DIRECTED GRANTS:THE FOLLOWING DIRECTED GRANTS WERE ACTIVE DURING 2021:1. 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 2021 BY THE PI.2. 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 2021 BY THE PI.3. COLLABORATIVE RESEARCH: UTMOST: UNDERGRADUATE TEACHING IN MATHEMATICS WITH OPEN SOFTWARE AND TEXTBOOKSUTMOST 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?X 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 2021 BY THE PI.

DIRECTED GRANTS (CONT):4. FRG: AVERAGES OF L-FUNCTIONS AND ARITHMETIC STRATIFICATIONSOME OF THE MOST DIFFICULT CHALLENGES IN ALL OF MATHEMATICS, SUCH AS THE REIMANN HYPOTHESIS AND THE BIRCH AND SWINNERTON-DYER CONJECTURE, ARE NATURALLY PHRASED IN TERMS OF L-FUNCTIONS. THESE FUNCTIONS ENCODE INFORMATION SUCH AS HOW MANY PRIMES THERE ARE UP TO A GIVEN MAGNITUDE, OR THE FREQUENCY OF RATIONAL NUMBER SOLUTIONS TO CERTAIN EQUATIONS, OR THE DISTRIBUTION OF SPECIAL POINTS ON A SURFACE, ALL OF WHICH ARE IMPORTANT IN NUMBER THEORY. L-FUNCTIONS ARE OFTEN STUDIED IN COLLECTIONS CALLED FAMILIES. IN THIS PROJECT WE WILL USE A NEW APPROACH CALLED "STRATIFICATION" TO STUDY THE DISTRIBUTION OF THE VALUES OF L-FUNCTIONS IN FAMILIES.IN RECENT YEARS RESEARCHERS HAVE FOUND VERY PRECISE CONJECTURES ABOUT THE STATISTICS OF VALUES AND ZEROES OF THE REIMANN ZETA FUNCTION AND OTHER FAMILIES OF L-FUNCTIONS. FOR LOW ORDER MOMENTS THESE CONJECTURES FOLLOW FROM PRECISE KNOWLEDGE OR CONJECTURES ABOUT CORRELATIONS OF GENERALIZED DIVISOR FUNCTIONS. BUT FOR HIGHER MOMENTS THIS LINKAGE HAS BEEN MISSING. THE MAIN GOAL OF THIS PROJECT IS TO COMPLETE THIS PICTURE AND PROVE THAT THE MOMENT CONJECTURES FOR FAMILIES OF L-FUNCTIONS FOLLOW FROM KNOWLEDGE OF DIVISOR CORRELATIONS, WHICH IS EQUIVALENT TO COUNTING POINTS IN SPECIFIED REGIONS OF CERTAIN VARIETIES. WE WILL ALSO INVESTIGATE THE SAME SCENARIO BUT FOR AVERAGES OF RATIOS OF L-FUNCTIONS IN FAMILIES WITH THE DIVISOR CORRELATIONS REPLACED BY THE GENERAL HARDY-LITTLEWOOD CONJECTURES ABOUT PRIME TUPLES. FOR THE REIMANN ZETA-FUNCTION THE PROJECT WILL FOLLOW THE METHOD OUTLINED IN RECENT WORK BY TWO OF THE PRINCIPAL INVESTIGATORS. FOR OTHER FAMILIES OF L-FUNCTIONS ANOTHER INNOVATION IS REQUIRED. THIS PROJECT WILL BE INFORMED BY MANIN'S IDEAS FOR COUNTING RATIONAL POINTS ON VARIETIES BY IDENTIFYING THE STRATIFIED SUBVARIETIES THAT PLAY A ROLE AND COUNTING THE POINTS ON THESE. WE WILL ALSO INVESTIGATE THE EXPONENTIAL SUMS THAT NATURALLY ARISE IN COUNTING POINTS ON THESE SUBVARIETIES. PROGRESS CONTINUED THROUGH 2021 BY THE PI.5. MATH FESTIVALS & MATH OUTREACH PROJECTSVARIOUS ACTIVITIES AND PROJECTS THAT SUPPORT LOCALLY ORGANIZED EVENTS THAT INSPIRE K-12 STUDENTS TO EXPORE MATHEMATICS THROUGH COLLABORATIVE, CREATIVE PROBLEM-SOLVING.6. APPLICATIONS OF NUMBER THEORY TO THE QUANTUM GATES MODELBOTH THE RESEARCH AND BROADER ACTIVITIES IN THIS AWARD INCLUDE CURRENT DEVELOPMENTS IN NUMBER THEORY AND THEIR APPLICATIONS IN QUANTUM COMPUTING AND QUANTUM CHAOS. IN TERMS OF PRACTICAL APPLICATIONS OF THIS PROJECT, THE PI EXPECTS THAT HIS REFINED VERSION OF ROSS AND SELLINGER ALGORITHM WILL BE USED IF A PHYSICAL QUANTUM COMPUTER IS BUILT.THE HOPE IS THAT QUANTUM COMPUTERS WILL EVENTUALLY BE ABLE TO EFFICIENTLY SIMULATE QUANTUM PHYSICS AND STUDY MANY (IMPORTANT) COMPUTATIONALLY DIFFICULT PROBLEMS INACCESSIBLE TO MODERN-DAY COMPUTERS. THERE ARE MODELS SUCH AS QUANTUM GATES MODEL THAT GIVE THEORETICAL CONSTRUCTIONS OF EFFICIENT CIRCUITS TO BE USED IN QUANTUM COMPUTERS. THIS MODEL IS CONNECTED TO THE STUDY OF INTEGRAL SOLUTIONS TO DIOPHANTINE EQUATIONS, AN ANCIENT SUBJECT OF INTEREST TO MATHEMATICIANS. A QUESTION OF INTEREST TO BOTH QUANTUM COMPUTER SCIENTISTS AS WELL AS MATHEMATICIANS IS THE OPTIMAL APPROXIMATION OF REAL SOLUTIONS OF SPECIAL DIOPHANTINE EQUATIONS BY INTEGRAL SOLUTIONS. THE PI HAS PROVED NEW (OPTIMAL) RESULTS IN THIS DIRECTION. FURTHERMORE, HE HAS PROVED THAT THIS TASK IS COMPUTATIONALLY HARD (NP-COMPLETE) FOR GENERIC INPUTS. PROGRESS CONTINUED THROUGH 2021 BY THE PI.