Institute For Advanced Study

BASIC RESEARCH IN MATHEMATICS

RESEARCH IN MATHEMATICS

RESEARCH IN MATHEMATICS

RESEARCH IN MATHEMATICS

PROBLEMS IN THEORETICAL PHYSICS

PROBLEMS IN PARTICLE THEORY

INSTITUTE FOR ADVANCED STUDY, SCHOOL OF MATHEMATICS -THE SCHOOL OF MATHEMATICS AT THE INSTITUTE FOR ADVANCED STUDY (IAS) SEEKS TO ADVANCE BASIC RESEARCH IN MATHEMATICS AND TO CREATE AN ENVIRONMENT THAT FOSTERS MATHEMATICAL TALENT. THIS AWARD SUPPORTS BETWEEN 8 AND 12 POSTDOCTORAL FELLOWS, AND BETWEEN 1 AND 2 MID-CAREER MATHEMATICIANS ANNUALLY (WITH A LARGER COHORT FOR THE FIRST YEAR OF THE AWARD), AND A SUMMER PROGRAM FOR SMALL GROUP COLLABORATIONS. THESE RESEARCHERS FORM PART OF THE GROUP OF APPROXIMATELY 70 MATHEMATICIANS WHO VISIT THE SCHOOL TO PURSUE THEIR RESEARCH, UNHAMPERED BY ADMINISTRATIVE DUTIES. APPROXIMATELY ONE-THIRD OF THESE MATHEMATICIANS ARE PART OF A SPECIAL PROGRAM FOCUSED ON A ROTATING TOPIC OF CURRENT INTEREST, AND THE REMAINING ARE UNRESTRICTED IN FIELD. THE SCHOOL ACTIVELY CURATES AN ENVIRONMENT WHERE SCHOLARS INTERACT IN A WAY THAT IS NOT NARROWLY CONSTRAINED BY FIELD BOUNDARIES. A LONG-RUNNING MENTORING PROGRAM ENSURES THAT POSTDOCTORAL MEMBERS REALIZE THEIR FULL SCIENTIFIC POTENTIAL; EACH SUCH MEMBER IS TYPICALLY ASSIGNED BOTH A MID-CAREER AND A MORE SENIOR MENTOR. THE SCHOOL ALSO HOSTS NUMEROUS WEEKLY SEMINARS ACROSS ALL FIELDS OF MATHEMATICS, SOME JOINTLY WITH PRINCETON UNIVERSITY, AS WELL AS REGULAR BROADLY ACCESSIBLE COLLOQUIA, AND WEEK-LONG SPECIALIZED WORKSHOPS EACH TERM ASSOCIATED WITH THE SPECIAL PROGRAM. RECENT TOPICS FOR THIS SPECIAL PROGRAM HAVE BEEN ALGEBRAIC AND GEOMETRIC COMBINATORICS (2024-2025), P-ADIC ARITHMETIC GEOMETRY (2023-2024), AND DYNAMICS, ADDITIVE NUMBER THEORY AND ALGEBRAIC GEOMETRY (2022-2023). THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.

IAS/PARK CITY MATHEMATICS INSTITUTE

IAS/PARK CITY MATHEMATICS INSTITUTE

PROBLEMS IN PARTICLE THEORY

AF: LARGE: THEORY OF COMPUTATION - PUSHING THE STATE-OF-THE-ART

IAS/PARK CITY MATHEMATICS INSTITUTE

CDI TYPE II: PSEUDORANDOMNESS

NSF-SU2C: CANCER CONVERGENCE EDUCATION NETWORK

PARK CITY MATHEMATICS INSTITUTE -THE PARK CITY MATHEMATICS INSTITUTE (PCMI) IS AN ANNUAL SUMMER EVENT, HELD IN PARK CITY, UTAH, WHICH PROVIDES A VERTICALLY INTEGRATED SET OF PROGRAMS, EACH ONE FOCUSED ON STUDENTS, FACULTY OR RESEARCHERS IN MATHEMATICS AT A DIFFERENT CAREER STAGE. PCMI IS AN OUTREACH PROGRAM OF THE PRINCETON INSTITUTE FOR ADVANCED STUDY, AND HAS BEEN HELD ANNUALLY SINCE 1991. EACH YEAR THE PROGRAM CENTERS AROUND A RESEARCH THEME IN THE MATHEMATICAL SCIENCES, CHOSEN TO REFLECT THE MOST ACTIVE AREAS IN MATHEMATICS WHERE NEW ADVANCES ARE MOST LIKELY TO BE STIMULATED BY THE TYPE OF INTERACTION THAT PCMI FACILITATES. THE CONSTITUENT SUBPROGRAMS INCLUDE: THE CORE GRADUATE SUMMER SCHOOL, A SET OF 8 MINICOURSES AIMED AT GRADUATE STUDENTS, BOTH MORE ADVANCED AND THOSE JUST STARTING OUT; AN ACCOMPANYING RESEARCH PROGRAM, WHICH IS AN OPPORTUNITY FOR RESEARCHERS AT ALL LEVELS, RANGING FROM POSTDOCS TO THE MOST DISTINGUISHED LEADERS IN THE FIELD, TO INTERACT AND COLLABORATE; AN UNDERGRADUATE SUMMER SCHOOL, WHICH INCLUDES BOTH A LECTURE COURSE AND A SMALL GROUP RESEARCH EXPERIENCE; THE UNDERGRADUATE FACULTY PROGRAM, WHICH BRINGS TOGETHER A SMALL GROUP OF FACULTY FROM UNDERGRADUATE-FACING INSTITUTIONS WHO WISH TO REENGAGE WITH THE RESEARCH ACTIVITIES IN THEIR FIELD; FINALLY, A SMALL WORKSHOP CENTERED ON A VARIETY OF TOPICS RELATED TO DIVERSITY AND INCLUSION IN THE MATHEMATICS CLASSROOM AND THE GENERAL MATHEMATICS COMMUNITY. EACH GROUP IS OCCUPIED DURING THE THREE WEEKS OF THE PROGRAM WITH THEIR SPECIFIC ACTIVITIES. PCMI PROVIDES A SPACE WHERE INTERACTION BETWEEN THESE DIFFERENT GROUPS CAN HAPPEN EASILY AND NATURALLY. THESE INTERACTIONS LEAD TO NEW RESEARCH COLLABORATIONS, NEW MENTORING OPPORTUNITIES, AND THE POSSIBILITY FOR THE YOUNGER PARTICIPANTS TO LEARN MORE ABOUT BECOMING PART OF THE MATHEMATICAL COMMUNITY. A CENTRAL GOAL OF PCMI IS TO BRING TOGETHER A GROUP OF STUDENTS AND MORE SENIOR MATHEMATICIANS, DIVERSE IN AS MANY DIMENSIONS AS POSSIBLE, TO ENGAGE IN HIGH LEVEL MATHEMATICAL INTERACTIONS, WITH AN UNDERLYING EMPHASIS ON PEDAGOGICAL EXCELLENCE AND THE IMPORTANCE OF THE INTERRELATIONSHIP OF THE MEMBERS OF THIS BROAD MATHEMATICAL COMMUNITY. THE MATHEMATICAL FOCUS IN A PCMI SUMMER SESSION IS CONCENTRATED ON AREAS OF RECOGNIZED IMPORTANCE WITH A HIGH LEVEL OF ACTIVITY. PROGRAMS ARE TYPICALLY SELECTED ABOUT EIGHTEEN MONTHS IN ADVANCE, WITH A SPECIALLY DESIGNATED GROUP OF ORGANIZERS, THEMSELVES LEADING SPECIALISTS IN THE FIELD, TO GUIDE THE INTELLECTUAL COHERENCE OF THE SESSION. UPCOMING PROGRAMS ARE: MOTIVIC HOMOTOPY THEORY (2024) AND EXTREMAL AND PROBABILISTIC COMBINATORICS (2025). SOME OF THE RECENT PROGRAMS INCLUDE: QUANTUM COMPUTATION (2023), NUMBER THEORY INFORMED BY COMPUTATION (2022), QUANTUM FIELD THEORY AND MANIFOLD INVARIANTS (2019) AND RANDOM MATRIX THEORY (2018). ATTENDEES HAVE INCLUDED A NUMBER OF RECENT FIELDS MEDALISTS AND OTHER OF THE MOST DISTINGUISHED RESEARCHERS. THE OUTCOMES OF EACH OF THESE SESSIONS INCLUDES A VOLUME OF LECTURE NOTES, PUBLISHED BY THE AMS, AND A PROFESSIONALLY PRODUCED VIDEO ARCHIVE OF THE GRADUATE MINI-COURSE LECTURES. THE LONG HISTORY OF THE PROGRAM, AND THE VARIETY OF FIELDS WHICH IT HAS COVERED, MEANS THAT VERY MANY MATHEMATICIANS IN THE US AND ABROAD HAVE ATTENDED PCMI AS STUDENTS OR RESEARCHERS AT SOME POINT IN THEIR CAREERS, AND MANY REPORT THAT THE EXPERIENCE WAS PIVOTAL IN THE DEVELOPMENT OF THEIR RESEARCH PROGRAM. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.

THEORETICAL PHYSICS -THIS AWARD FUNDS THE RESEARCH ACTIVITIES OF PROFESSOR EDWARD WITTEN AND A GROUP OF POSTDOCTORAL RESEARCHERS AT THE INSTITUTE FOR ADVANCED STUDY IN PRINCETON. THIS PROJECT IS DEVOTED TO THE STUDY OF ELEMENTARY PARTICLE PHYSICS, INCLUDING ITS CONNECTIONS TO OTHER AREAS OF STUDY RANGING FROM ASTROPHYSICS AND COSMOLOGY TO CONDENSED-MATTER PHYSICS AND GEOMETRY. CURRENT AND PLANNED WORK OF THE RESEARCHERS SUPPORTED BY THIS GRANT COVERS A VERY WIDE RANGE OF TOPICS, INCLUDING THE CONNECTIONS BETWEEN GAUGE THEORY (THE MODERN LANGUAGE OF THE PHYSICS GOVERNING THE ELEMENTARY PARTICLES), STRING THEORY (A SPECULATIVE PROPOSAL FOR UNIFYING THE FUNDAMENTAL FORCES OF NATURE), AND GEOMETRY. THESE RESEARCHERS WILL ALSO STUDY THE POSSIBILITIES FOR NEW MODELS OF PARTICLE PHYSICS THAT MIGHT BE RELEVANT AT CURRENT AND FUTURE PARTICLE-PHYSICS EXPERIMENTS; NEW METHODS OF COMPUTING AND STUDYING THE WAYS IN WHICH ELEMENTARY PARTICLES SCATTER OFF EACH OTHER; NEW EXPERIMENTAL HINTS OF THE COSMIC DARK MATTER WHICH MAKES UP MOST OF THE MATTER IN THE UNIVERSE; AND MORE. RESEARCH IN THESE AREAS THUS ADVANCES THE NATIONAL INTEREST BY PROMOTING THE PROGRESS OF SCIENCE IN ONE OF ITS MOST FUNDAMENTAL DIRECTIONS: THE DISCOVERY AND UNDERSTANDING OF NEW PHYSICAL LAWS. THIS PROJECT WILL ALSO HAVE SIGNIFICANT BROADER IMPACTS. THE POSTDOCTORAL FELLOWS SUPPORTED BY THIS PROJECT WILL ALSO GAIN EXPERIENCE WORKING AT A HIGH LEVEL ON THESE EXCITING SCIENTIFIC PROBLEMS, WHICH HAVE ALL ATTRACTED GREAT INTEREST NATIONALLY AND INTERNATIONALLY. MORE TECHNICALLY, PROFESSOR WITTEN'S RECENT AND ONGOING WORK HAS INVOLVED OPERATOR ALGEBRAS THAT GIVE A NEW PERSPECTIVE ON BLACK-HOLE ENTROPY AND ON THE NATURE OF DE SITTER SPACE IN QUANTUM THEORY. THE POSTDOCTORAL FELLOWS SUPPORTED BY THIS PROJECT HAVE A WIDE RANGE OF INTERESTS AND WORK WITH EACH OTHER ON FRONTIER TOPICS IN THEORETICAL HIGH-ENERGY PHYSICS AS WELL AS WITH PROFESSOR WITTEN AND OTHER LOCAL FACULTY. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.

THEORETICAL PHYSICS -THIS AWARD FUNDS THE RESEARCH ACTIVITIES OF PROFESSOR EDWARD WITTEN AND A GROUP OF POSTDOCTORAL RESEARCHERS AT THE INSTITUTE FOR ADVANCED STUDY IN PRINCETON. THE PROJECT IS DEVOTED TO THE STUDY OF ELEMENTARY PARTICLE PHYSICS, INCLUDING ITS CONNECTIONS TO OTHER AREAS OF STUDY RANGING FROM ASTROPHYSICS AND COSMOLOGY TO CONDENSED-MATTER PHYSICS AND GEOMETRY. CURRENT AND PLANNED WORK OF THE RESEARCHERS SUPPORTED BY THIS GRANT COVERS A VERY WIDE RANGE OF TOPICS, INCLUDING THE CONNECTIONS BETWEEN GAUGE THEORY (THE MODERN LANGUAGE OF THE PHYSICS GOVERNING THE ELEMENTARY PARTICLES), STRING THEORY (A SPECULATIVE PROPOSAL FOR UNIFYING THE FUNDAMENTAL FORCES OF NATURE), AND GEOMETRY. THESE RESEARCHERS WILL ALSO STUDY THE POSSIBILITIES FOR NEW MODELS OF PARTICLE PHYSICS THAT MIGHT BE RELEVANT AT CURRENT AND FUTURE PARTICLE-PHYSICS EXPERIMENTS; NEW METHODS OF COMPUTING AND STUDYING THE WAYS IN WHICH ELEMENTARY PARTICLES SCATTER OFF EACH OTHER; NEW EXPERIMENTAL HINTS OF THE COSMIC DARK MATTER WHICH MAKES UP MOST OF THE MATTER IN THE UNIVERSE; AND MORE. RESEARCH IN THESE AREAS THUS ADVANCES THE NATIONAL INTEREST BY PROMOTING THE PROGRESS OF SCIENCE IN ONE OF ITS MOST FUNDAMENTAL DIRECTIONS: THE DISCOVERY AND UNDERSTANDING OF NEW PHYSICAL LAWS. THIS PROJECT WILL ALSO HAVE SIGNIFICANT BROADER IMPACTS. THE POSTDOCTORAL FELLOWS SUPPORTED BY THIS PROJECT WILL ALSO GAIN EXPERIENCE WORKING AT A HIGH LEVEL ON THESE EXCITING SCIENTIFIC PROBLEMS, WHICH HAVE ALL ATTRACTED GREAT INTEREST NATIONALLY AND INTERNATIONALLY. MORE TECHNICALLY, PROFESSOR WITTEN'S RECENT AND ONGOING WORK HAS INVOLVED OPERATOR ALGEBRAS THAT GIVE A NEW PERSPECTIVE ON BLACK-HOLE ENTROPY AND ON THE NATURE OF DE SITTER SPACE IN QUANTUM THEORY. THE POSTDOCTORAL FELLOWS SUPPORTED BY THIS PROJECT HAVE A WIDE RANGE OF INTERESTS AND WORK WITH EACH OTHER ON FRONTIER TOPICS IN THEORETICAL HIGH-ENERGY PHYSICS AS WELL AS WITH PROFESSOR WITTEN AND OTHER LOCAL FACULTY. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.

AF: MEDIUM: THEORY OF COMPUTATION - NEW ALGORITHMIC AND HARDNESS TECHNIQUES

THEORETICAL PHYSICS

THEORETICAL PHYSICS

THEORETICAL PHYSICS

THEORETICAL PHYSICS

PROGRAM FOR WOMEN AND MATHEMATICS

POSTDOCTORAL PROGRAM IN THEORETICAL AND OBSERVATIONAL ASTROPHYSICS

TOPOLOGICAL MICROSTRUTCTURE EVOLITION

THEORETICAL PHYSICS

INSTITUTE FOR ADVANCED STUDY/PARK CITY MATHEMATICS INSTITUTE

WOMEN AND MATHEMATICS PROGRAM

COLLABORATIVE RESEARCH: AT THE INTERFACE OF LHC THEORY AND EXPERIMENT

GALOIS REPRESENTATIONS AND MODULAR FORMS

THE ASTROPHYSICS OF HYDROGEN AND HELIUM REIONIZATION

MATHEMATICS FOR TEACHING: A PROBLEM-BASED RESOURCE FOR TEACHERS

IAS/PARK CITY MATHEMATICS INSTITUTE UNDERGRADUATE SUMMER SCHOOL 9

STELLAR DYNAMICS IN THE CENTERS OF GALAXIESWE PROPOSE TO DEVELOP AND EMPLOY NOVEL NUMERICAL ALGORITHMS TO INVESTIGATE THE DYNAMICAL EVOLUTION OF ST

PROBING FUNDAMENTAL PHYSICS ON COSMOLOGICAL SCALES

ACCRETION DISKS ARE ENCOUNTERED IN A WIDE VARIETY OF ASTROPHYSICAL SETTINGS: COMPACT BINARIES, ACTIVE GALACTIC NUCLEI, STAR AND PLANET FORMATION. WHE

LARGE SCALE STRUCTURE IN THE MILDLY NONLINEAR REGIME

SIGNPOSTS OF PLANETARY SYSTEMS AROUND EVOLVED STARS: A NEW PERSPECTIVE ON EXTRASOLAR PLANETS

ASPECTS OF HARMONIC ANALYSIS AND HAMILTONIAN PDES

PROBING FUNDAMENTAL PHYSICS ON COSMOLOGICAL SCALES

NSF-BSF: NEW DATA ANALYSIS TECHNIQUES FOR A NEW ERA IN GW ASTROPHYSICS -SINCE THE FIRST DETECTION OF GRAVITATIONAL WAVES BY NSF'S LIGO OBSERVATORY, GRAVITATIONAL WAVE ASTROPHYSICS HAS BECOME ONE OF THE MOST DYNAMIC FIELDS OF PHYSICS AND ASTRONOMY. IN THE NEXT FEW YEARS THE LIGO/VIRGO/KAGRA (LVK) COLLABORATION WILL CONTINUE TO IMPROVE THE SENSITIVITY OF THEIR OBSERVATIONS, DISCOVER NEW BLACK HOLE AND NEUTRON STAR MERGER EVENTS AND MAKE THEIR DATA PUBLIC. THIS AWARD AIMS TO IMPROVE THE TECHNIQUES USED TO ANALYZE GRAVITATIONAL WAVE DATA AND MAKE ASTROPHYSICAL INFERENCES USING IT. THE IMPROVED ANALYSIS PIPELINES WILL BE APPLIED TO UPCOMING DATA IN ORDER TO LEARN ABOUT THE ASTROPHYSICAL PROCESSES THAT CREATED THESE SOURCES OF GRAVITATIONAL WAVES AND STUDY THE GRAVITATIONAL DYNAMICS IN A VERY EXTREME REGIME. ALTHOUGH WE CAN SEE THESE MERGERS OCCURRING ACROSS OUR UNIVERSE WE STILL DO NOT UNDERSTAND HOW THESE BINARY SYSTEMS FORMED AND QUESTIONS REMAIN ABOUT MANY PROPERTIES OF THE BLACK HOLES AND NEUTRON STARS IN THEM. THE RESEARCH PRODUCED UNDER THIS AWARD WILL BE DISSEMINATED WIDELY ACROSS THE SCIENTIFIC COMMUNITY AND EFFORTS WILL BE MADE TO MAKE IT ACCESSIBLE TO NON-SCIENTISTS THROUGH PUBLIC LECTURES AND OTHER OUTREACH ACTIVITIES. THIS AWARD AIMS TO IMPROVE THE TECHNIQUES USED TO ANALYZE GRAVITATIONAL WAVE DATA FROM THE LVK COLLABORATION AND MAKE ASTROPHYSICAL INFERENCES USING IT. THE IMPROVED ANALYSIS PIPELINES WILL BE APPLIED TO UPCOMING DATA IN ORDER TO LEARN ABOUT THE ASTROPHYSICAL PROCESSES THAT CREATED THE GRAVITATIONAL WAVE SOURCES AND STUDY THE GRAVITATIONAL DYNAMICS IN A VERY EXTREME REGIME. IN PARTICULAR, THIS AWARD WILL PROVIDE AN INDEPENDENT ANALYSIS AND ASTROPHYSICAL INTERPRETATION OF THE LVK DATA: 1. THE TEAM WILL USE THE PARAMETER ESTIMATION CODE COGWHEEL WHICH IS EXTREMELY EFFICIENT DUE TO THE IMPLEMENTATION OF A NEW SET OF COORDINATES AS WELL AS THE RELATIVE BINNING ALGORITHM EXTENDED FOR HIGHER MODES TO DO A VARIETY OF NOVEL STUDIES THAT ITS EFFICIENCY ENABLES. 2. THEY WILL COMBINE THE RESULTS OF THEIR PARAMETER STUDY, INCLUDING FROM EVENTS FOUND BY THEIR INDEPENDENT PIPELINE TO SET CONSTRAINTS ON THE PROPERTIES OF THE POPULATION OF GW SOURCES. 3. THEY WILL MAKE SEVERAL IMPROVEMENTS TO THEIR INDEPENDENT SEARCH PIPELINE RANGING FROM TECHNICAL ONES RELATED TO THEIR TREATMENT OF THE NOISE AND OF SO-CALLED GLITCHES TO ADDING THE CAPABILITY OF INCLUDING HIGHER MODES AND PRECESSION AS PART OF THE SEARCH. 4. THEY WILL WORK ON IMPROVING H(T) AT LOW FREQUENCY BY USING INFORMATION FROM THE SO-CALLED AUXILIARY CHANNELS. LIGO RECENTLY RELEASED THE AUXILIARY CHANNELS AROUND GW170814 AND THEY INTEND TO INVESTIGATE DIFFERENT TECHNIQUES TO IMPROVE THE LOW FREQUENCY ?CONTROL NOISE?. 5. THEY WILL AUGMENT THEIR CURRENT EFFORTS, WHICH ARE MAINLY CENTERED AROUND DATA ANALYSIS, WITH SOME MORE THEORETICAL WORK RELATED TO TIDAL DISSIPATION AROUND BLACK HOLES AS WELL AS EFFORTS TO IMPROVE EXISTING WAVEFORMS. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.

WE PROPOSE TO INVESTIGATE THE DYNAMICS; STABILITY AND EVOLUTION OF THE STELLAR SYSTEMS FOUND AROUND

WE PROPOSE TO INVESTIGATE THE LONG-TERM EVOLUTION OF PLANETARY SYSTEMS FROM AFTER THE INITIAL STAGE OF PLANET FORMATION LASTING A FEW MYR UNTIL THE

RECONSTRUCTING ANCIENT HISTORY THROUGH SQUEEZE DIGITIZATION AT THE INSTITUTE FOR ADVANCED STUDY

HARMONIC ANALYSIS AND APPLICATIONS

CONTACT AND SYMPLECTIC STRUCTURES AND HOLOMORPHIC CURVES

PROBING FUNDAMENTAL PHYSICS ON COSMOLOGICAL SCALES

THE ZAYDI MANUSCRIPT TRADITION: A DIGITAL REPOSITORY AND PORTAL

PROBING FUNDAMENTAL PHYSICS ON COSMOLOGICAL SCALES

DYNAMICS IN GALAXY CENTERS

PROBING FUNDAMENTAL PHYSICS ON COSMOLOGICAL SCALES -THIS AWARD FUNDS THE RESEARCH ACTIVITIES OF PROFESSOR MATIAS ZALDARRIAGA AT THE INSTITUTE FOR ADVANCED STUDY, IN PRINCETON, NEW JERSEY. COSMOLOGY, THE SCIENCE DEVOTED TO UNDERSTANDING THE HISTORY OF OUR UNIVERSE AS WELL AS THE PHYSICAL LAWS THAT GOVERN ITS EVOLUTION, IS EXPERIENCING A REVOLUTION THANKS TO THE HEALTHY INTERPLAY BETWEEN ADVANCES IN FUNDAMENTAL PHYSICS AND ASTROPHYSICS AND THE AVAILABILITY OF A VAST AMOUNT OF NEW DATA. IN THE NEXT DECADES LARGE SURVEYS OF THE LOCATION OF GALAXIES WILL PLAY A CENTRAL ROLE AS SOURCES OF NEW INFORMATION. THE WORK OF PROFESSOR ZALDARRIAGA AIMS TO IMPROVE OUR THEORETICAL UNDERSTANDING OF THE PHYSICAL PROCESSES RESPONSIBLE FOR THE DEVELOPMENT OF THE STRUCTURE THAT THESE SURVEYS MAP. THESE SURVEYS WILL ALSO HELP TO DEVELOP NEW MODELING AND ANALYSIS TOOLS FOR SUCH SURVEYS AND IMPROVE COSMOLOGICAL CONSTRAINTS DERIVED FROM THESE OBSERVATIONAL STUDIES. RESEARCH IN THIS AREA THUS ADVANCES THE NATIONAL INTEREST BY PROMOTING THE PROGRESS OF ONE OF THE MOST FUNDAMENTAL AREAS OF SCIENCE: THE DISCOVERY AND CHARACTERIZATION OF THE PHYSICAL LAWS THAT DESCRIBE THE HISTORY OF OUR UNIVERSE. THE FUNDED WORK IS ALSO EXPECTED TO HAVE SIGNIFICANT BROADER IMPACTS. ASTRONOMY AND COSMOLOGY ARE SUBJECTS OF BROAD APPEAL TO THE PUBLIC. PREVIOUS EFFORTS BY THE PI TO DISSEMINATE THE RESEARCH IN A FORM ACCESSIBLE TO NON-SCIENTISTS WILL BE CONTINUED AND EXPANDED. THESE INCLUDE PERIODIC PUBLIC LECTURES BOTH FOR ADULTS AND IN PRIMARY AND MIDDLE SCHOOLS, INTERVIEWS IN THE MEDIA, INCLUDING THOSE TARGETED TO HISPANIC AUDIENCES, AND A LECTURE SERIES AT A SENIOR CITIZENS CENTER. THE FUNDED RESEARCH WILL ALSO CONTRIBUTE TO THE EDUCATION OF STUDENTS AND POSTDOCTORAL FELLOWS. MORE TECHNICALLY, THE WORK OF PROFESSOR ZALDARRIAGA WILL COMBINE TWO STATE-OF-THE-ART TECHNIQUES: ANALYTICAL EFFECTIVE FIELD THEORY IDEAS THAT PROVIDE THEORETICAL CONTROL WITH COMPUTATIONAL, AND STATISTICAL TECHNIQUES THAT CONFRONT THEORY WITH OBSERVATIONS IN AN EFFICIENT AND STATISTICALLY PRINCIPLED WAY. THE AIM IS TO USE THESE NEW TECHNIQUES TO HARNESS PREVIOUSLY INACCESSIBLE INFORMATION ABOUT COSMOLOGICAL INITIAL CONDITIONS AND THE NON-LINEAR PHYSICS THAT RELATES THEM TO THE OBSERVATIONS, AND THEREBY PROVIDE A UNIQUE AND POTENTIALLY TRANSFORMATIVE CONTRIBUTION TO THE FIELD. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.

AT LEAST 20% OF EXTRASOLAR PLANETS ARE KNOWN TO RESIDE IN BINARIES. SOME OF THEM, SUCH AS ALPHA CEN AND GAMMA CEPHEI, ARE IN SMALL SEPARATION BINARIE

BETWEEN THE DISK AND THE STAR: BOUNDARY LAYERS IN ASTROPHYSICAL ACCRETION DISKS

H-PRINCIPLE AND FLUID DYNAMICS

SYMPLECTIC GEOMETRY AND DYNAMICS

STELLAR DYNAMICS NEAR BLACK HOLES

TAS::57 3600::TAS 'THE UNIMATH LIBRARY FOR TYPE THEORY AND PROGRAMMING LANGUAGES'

FELLOWSHIPS IN THE SCHOOL OF HISTORICAL STUDIES

MIN-MAX PROBLEMS FOR FAMILIES OF CYCLES IN RIEMANNIAN MANIFOLDS

GRANTS-R1-CAAXX- INST. FOR A.S

UNIVALENT FOUNDATIONS OF MATHEMATICS

CONFERENCE: WOMEN+ AND MATHEMATICS PROGRAM -THIS AWARD SUPPORTS TWO ITERATIONS OF THE WOMEN+ AND MATHEMATICS (W+AM) PROGRAM AT THE INSTITUTE FOR ADVANCED STUDY (IAS). THE PROGRAM WAS CREATED IN 1994 WITH THE SUPPORT OF THE NATIONAL SCIENCE FOUNDATION, THE INSTITUTE FOR ADVANCED STUDY, AND PRINCETON UNIVERSITY. THE OVERARCHING MISSION OF W+AM IS TO RECRUIT AND RETAIN MORE WOMEN IN MATHEMATICS IN ORDER TO COUNTER THE INITIAL IMBALANCE IN THE NUMBERS OF MEN AND WOMEN ENTERING MATHEMATICS TRAINING AS WELL AS THE HIGHER ATTRITION RATE OF FEMALE MATHEMATICIANS COMPARED TO THEIR MALE COUNTERPARTS AT EVERY CRITICAL TRANSITION STAGE IN MATHEMATICAL CAREERS. THE MOST CONCENTRATED PART OF THE W+AM PROGRAM IS THE WEEK-LONG ANNUAL PROGRAM IN MAY, WHICH IS ORGANIZED AROUND A MATHEMATICAL TOPIC OF CURRENT RESEARCH INTEREST. FOR EACH OF THE UPCOMING TWO PROGRAMS, THERE WILL BE APPROXIMATELY 32 PARTICIPANTS, RANGING FROM UNDERGRADUATES TO POSTDOCTORAL SCHOLARS, IN ADDITION TO SENIOR MATHEMATICIANS, SUCH AS LECTURERS AND SEMINAR SPEAKERS, WHO WILL SPEND A WEEK AS PART OF THE IAS MATHEMATICAL COMMUNITY SURROUNDED BY OTHER MATHEMATICIANS, MANY IN NEARBY RESEARCH AREAS. IN MAY 2024, THE PROGRAM WILL BE ON SYMMETRY AND ARITHMETIC, AND THE TWO LECTURE SERIES WILL BE A GLIMPSE INTO THE LANGLANDS PROGRAM BY ANA CARAIANI OF IMPERIAL COLLEGE LONDON AND DELIGNE-LUSZTIG THEORY: EXAMPLES AND APPLICATIONS BY CHARLOTTE CHAN OF THE UNIVERSITY OF MICHIGAN. THE 2025 PROGRAM WILL BE IN THE AREA OF ALGEBRAIC AND GEOMETRIC COMBINATORICS. IN ADDITION TO THE TWO LECTURE COURSES, THE SCIENTIFIC PROGRAM IS ROUNDED OUT BY PROBLEM SESSIONS FOR EACH COURSE LED BY POSTDOCTORAL SCHOLARS, RESEARCH SEMINARS, AND COLLOQUIA. THERE ARE ALSO PANEL DISCUSSIONS AND OTHER META-MATHEMATICAL AND SOCIAL ACTIVITIES ARRANGED FOR AND BY THE PARTICIPANTS. MORE INFORMATION IS AVAILABLE AT THE PROGRAM WEBPAGE HTTPS://WWW.IAS.EDU/MATH/WAM. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.

TOPICS IN DYNAMICS, AUCTIONS AND VOTING

THE UNDERGRADUATE PROGRAM OF THE IAS/PARK CITY MATHEMATICS INSTITUTE (PCMI)

THE UNDERGRADUATE PROGRAM OF THE IAS/PARK CITY MATHEMATICS INSTITUTE

TPSE MATH DEPARTMENT CHAIRS CONFERENCE

TOPICS IN DYNAMICS AND POLITICAL ECONOMY

COLLABORATIVE RESEARCH: TOWARDS PETASCALE COSMOLOGICAL SIMULATIONS

DEVELOPING OCR FOR SQUEEZES: UNLOCKING THE TEXT OF GREEK INSCRIPTIONS USING THE KRATEROS DATABASE [OPTICAL CHARACTER RECOGNITION (OCR) HAS LONG BEEN APPLIED WITH INCREASING LEVELS OF ACCURACY TO PRINTED TEXTS. MORE RECENTLY, SPURRED BY ADVANCES IN NEURAL NETWORKS AND INCREASES IN PROCESSING CAPABILITY, OCR HAS ALSO BEEN APPLIED SUCCESSFULLY TO MANUSCRIPT TEXTS, ALLOWING FOR READABLE AND SEARCHABLE EDITIONS TO BE CREATED FOR HUGE CORPORA OF TEXTS. THIS PROJECT WILL USE THE INSTITUTE FOR ADVANCED STUDY?S KRATEROS DATABASE, A DIGITAL COLLECTION FEATURING THOUSANDS OF IMAGES OF EPIGRAPHIC SQUEEZES?PAPER NEGATIVES OF INSCRIPTIONS?TO DEVELOP OCR FOR GREEK INSCRIPTIONS. THE PROJECT?S AIMS ARE TO ACCELERATE THE PUBLICATION OF NEW ANALYSES OF HISTORIC GREEK TEXTS, DRAW CRITICAL ATTENTION TO PROBLEM SPOTS IN EXISTING TRANSCRIPTIONS, AND IDENTIFY INDIVIDUAL STONECUTTERS? HANDWRITING. THIS INNOVATIVE APPROACH TO HUMANITIES RESEARCH WILL LAY THE GROUNDWORK FOR BREAKTHROUGH DISCOVERIES THAT CAN TRANSFORM OUR UNDERSTANDING OF ANCIENT HISTORY.]

FELLOWSHIP PROGRAMS IN HISTORICAL STUDIES

COLLABORATIVE RESEARCH: NEW DECOUPLINGS AND APPLICATIONS

THE GEOMETRY OF MODULI SPACES OF SHEAVES

TRANSLATIONAL TILINGS AND ORTHOGONAL SYSTEMS OF EXPONENTIALS -THIS PROJECT CONCERNS THE STRUCTURE OF TWO IMPORTANT AND CLOSELY RELATED MATHEMATICAL OBJECTS: TRANSLATIONAL TILINGS AND ORTHOGONAL SYSTEMS. A TRANSLATIONAL TILING IS A COVERING OF A SPACE USING TRANSLATED COPIES OF CERTAIN BUILDING BLOCKS, CALLED THE ?TILES?, WITHOUT OVERLAPS. THE FOCUS OF THIS PROJECT IS ON THE MAJOR QUESTION IN THE AREA: WHICH ARE THE POSSIBLE WAYS THAT A SPACE CAN BE TILED. THE STRUCTURE OF TILINGS HAS BEEN EXTENSIVELY STUDIED OVER THE YEARS BY MANY MATHEMATICIANS; AMONG THE MOST FAMOUS AND SIGNIFICANT CONTRIBUTORS ARE THE MATHEMATICAL PHYSICIST ROGER PENROSE AND THE AMATEUR MATHEMATICIANS ROBERT AMMANN AND MAURITS CORNELIS ESCHER. THIS MATHEMATICAL STUDY HAS FOUND MANY IMPORTANT APPLICATIONS TO OTHER SCIENCES. IN PARTICULAR, IT LED TO ONE OF THE MOST SIGNIFICANT BREAKTHROUGHS IN PHYSICAL SCIENCE: THE DISCOVERY OF NATURAL QUASICRYSTALS, PHYSICAL SOLIDS WHOSE ATOMS? ARRANGEMENT IS NOT PERIODIC. SINCE APERIODIC TILINGS (I.E., PAVINGS OF SPACE WHICH CANNOT BE ?DETERMINED? BY HOW THEY LOOK IN AN ARBITRARILY LARGE BUT BOUNDED PART OF THE SPACE) SERVE AS THE MATHEMATICAL MODELS OF QUASICRYSTALS, THE DISCOVERY OF THOSE APERIODIC FORMS IN NATURE BY DAN SHECHTMAN (FOR WHICH HE WON THE NOBEL PRIZE), WAS BASED ON THE MATHEMATICAL DISCOVERY OF APERIODIC TILINGS. ORTHOGONAL SYSTEMS ARE OF IMPORTANCE TO SEVERAL BRANCHES OF MATHEMATICS, INCLUDING NUMBER THEORY, ALGEBRAIC GEOMETRY AND ANALYSIS. ONE OF THEIR ADVANTAGEOUS PROPERTIES IS THAT THEY CAN FACILITATE AN AMENABLE DECOMPOSITION OF A FUNCTION (E.G., DESCRIBING LIGHT OR SOUND WAVES) INTO MORE MANAGEABLE PIECES WITH A HIGH LEVEL OF MUTUAL INDEPENDENCE. STUDYING THESE PIECES SEPARATELY CAN YIELD INSIGHT INTO THE PHYSICAL PROPERTIES OF A SYSTEM SUCH AS DOMINANT FREQUENCIES. THE RESEARCH CARRIED OUT (IN CONJUNCTION WITH COLLABORATORS) CONSISTS OF REVEALING CONNECTIONS TO A RANGE OF DIFFERENT AREAS OF MATHEMATICS AND BUILDING ON PAST RESULTS IN THOSE AREAS, AS WELL AS DEVELOPING NOVEL METHODS AND TECHNIQUES. THE FIRST PART OF THIS PROJECT WILL BE DEVOTED TO THE STUDY OF THE STRUCTURE OF TRANSLATIONAL TILINGS. THE MAIN GOAL IS SOLVING THE WELL-KNOWN PERIODIC TILING CONJECTURE, WHICH ASSERTS THAT ANY BOUNDED MEASURABLE TILE OF THE EUCLIDEAN SPACE MUST ADMIT AT LEAST ONE PERIODIC TILING. THIS CONJECTURE IS KNOWN TO HOLD FOR TILINGS OF THE REAL LINE, AND THERE ARE SOME PARTIAL RESULTS TOWARDS IT IN HIGHER DIMENSIONS. HOWEVER, THE PERIODIC TILING CONJECTURE HAS NOT BEEN SETTLED YET IN DIMENSIONS TWO AND HIGHER. OVER TIME IT HAS BECOME APPARENT THAT IN MANY RESPECTS TRANSLATIONAL TILES ?BEHAVE LIKE? DOMAINS WHOSE HILBERT SPACE (I.E., THE SPACE OF QUADRATICALLY INTEGRABLE FUNCTIONS THAT ARE SUPPORTED ON THE DOMAIN) ADMITS AN ORTHOGONAL BASIS OF EXPONENTIAL FUNCTIONS. THUS, A SECOND AREA OF FOCUS IS ON THE STRUCTURE OF FREQUENCY SETS OF ORTHOGONAL SYSTEMS OF EXPONENTIALS IN TWO DIFFERENT SETTINGS: IN TIME-FREQUENCY SPACES (GABOR BASES) AND IN HILBERT SPACES OF CERTAIN DOMAINS. THIS PART OF THE PROJECT AIMS TO FIND NEW APPROACHES TO THE LATTER MENTIONED STUDIES OF ORTHOGONAL SYSTEMS, BUILDING NEW BRIDGES BETWEEN NUMBER THEORY, ALGEBRAIC GEOMETRY AND FOURIER ANALYSIS. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.

COLLABORATIVE RESEARCH: MATHEMATICAL SCIENCES INSTITUTES DIVERSITY INITIATIVE

STRUCTURAL IMPROVEMENT OF THE HISTORICAL STUDIES AND SOCIAL SCIENCE LIBRARY AT THE INSTITUTE FOR ADVANCED STUDY

FUNCTORIALITY IN FUNCTION FIELDS

TPSE MATH CHAIRS +1 CONFERENCE #2

CONFERENCE: ANALYSIS AND BEYOND; PRINCETON; NEW JERSEY; MAY 21-24, 2016

UNIVERSALITY OF RANDOM MATRICES STATISTICS

CONFERENCE: VISIONS IN ARITHMETIC AND BEYOND -THIS AWARD PROVIDES FUNDING TO HELP DEFRAY THE EXPENSES OF PARTICIPANTS IN THE CONFERENCE VISIONS IN ARITHMETIC AND BEYOND (CONFERENCE WEBSITE HTTPS://WWW.IAS.EDU/MATH/EVENTS/VISIONS-IN-ARITHMETIC-AND-BEYOND ) TO BE HELD AT THE INSTITUTE FOR ADVANCED STUDY AND PRINCETON UNIVERSITY FROM JUNE 3 TO JUNE 7, 2024. THOSE SPEAKING AT THE MEETING INCLUDE THE LEADING RESEARCHERS ACROSS ARITHMETIC, ANALYSIS AND GEOMETRY. THE CONFERENCE WILL PROVIDE HIGH-LEVEL TALKS BY MATHEMATICIANS WHO ARE BOTH OUTSTANDING RESEARCHERS AND EXCELLENT SPEAKERS. THESE WILL SYNTHESIZE AND EXPOSE A BROAD RANGE OF RECENT ADVANCES IN NUMBER THEORY AS WELL AS RELATED DEVELOPMENTS IN ANALYSIS AND DYNAMICS. IN ADDITION TO THE TALKS BY LEADING RESEARCHERS THERE IS ALSO TIME ALLOTTED FOR A SESSION ON THE BEST PRACTICES FOR MENTORING GRADUATE STUDENTS AND POSTDOCS. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.

INTERACTIONS BETWEEN P-ADIC ARITHMETIC GEOMETRY AND COMMUTATIVE ALGEBRA

DERIVED CATEGORIES, HODGE THEORY, AND BIRATIONAL GEOMETRY

STRINGS 2014

AS PART OF THIS PROPOSAL THE PI WILL STUDY THEORETICAL QUESTIONS CONNECTED TO THE DESIGN CHOICES FOR

NEW APPLICATIONS OF FLOER HOMOLOGY TO SYMPLECTIC TOPOLOGY

HCC:SMALL:COLLABORATIVE RESEARCH: EXPLORING THE USE OF IMMERSIVE VIRTUAL REALITY TECHNOLOGIES FOR SCIENTIFIC RESEARCH, COMMUNICATION, AND OUTREACH

KNOTS AND SURFACES IN THREE-MANIFOLDS

UNITARY REPRESENTATIONS OF AFFINE HECKE ALGEBRAS AND REDUCTIVE P-ADIC GROUPS

FLOER HOMOLOGY AND LOW-DIMENSIONAL TOPOLOGY

FELLOWSHIPS IN THE SCHOOL OF HISTORICAL STUDIES

RANDOM SIMPLICIAL COMPLEXES

STRINGS 2014

WE PROPOSE TO INVESTIGATE THE BEHAVIOR OF STARS AND GAS IN THE CENTRAL REGIONS OF GALAXIES WHERE THE DYNAMICS IS DOMINATED BY A SUPERMASSIVE BLACK HOLE. THE SPECIFIC PROJECTS INCLUDE: (I) EXPLORE NOVEL ORBIT INTEGRATORS FOR THE GRAVITATIONAL N-BODY PROBLEM WHICH MAY ALLOW MUCH LARGER TIMESTEPS THAN THE CONVENTIONAL LEAPFROG OR HERMITE INTEGRATORS; (II) INVESTIGATE THE BEHAVIOR OF WARPEDACCRETION DISKS BELIEVED TO BE COMMON CONSTITUENTS OF ACTIVE GALACTIC NUCLEI WHERE THE SELF-GRAVITY OF THE DISK VISCOUS STRESSES AND THE LENSE-THIRRING EFFECT ALL PLAY IMPORTANT ROLES IN DETERMINING THE DISK SHAPE; (III) DEVELOP AN N-WIRE CODE WHICH AVERAGES OVER ORBITAL PHASE BY REPRESENTING EACH STAR AS AN ECCENTRIC WIRE AND USE THIS TO STUDY THE TIMESCALE AND OUTCOME OF RESONANT RELAXATION OF A STELLAR SYSTEM NEAR A BLACK HOLE THE SECULAR STABILITY OF SUCH SYSTEMS AND THE ECCENTRIC STELLAR DISK AT THE CENTER OF M31; (IV) IMPLEMENT AN N-DISK CODE WHICH REPRESENTS EACH STAR AS AN ANNULAR DISK BY AVERAGING OVER ORBITAL PHASE AND PERIAPSIS DIRECTION AND USE THIS TOMODEL THE COMPLEX STRUCTURE OF THE YOUNG MASSIVE STARS BETWEEN 0.1 AND 0.5 PC FROM THE GALACTIC CENTER---VARIOUSLY INTERPRETED AS ONE WARPED DISK TWO DISKS DISSOLVING DISKS ETC.---AS THE RESULT OF RESONANT RELAXATION BETWEEN A YOUNG STELLAR DISK AND THE SURROUNDING OLD STELLAR CLUSTER.