Project Grant 2453583

WEIGHTED L2 ESTIMATES AND PROBLEMS CONNECTED TO THE BOCHNER-RIESZ MEANS, THE RESTRICTION CONJECTURE, AND THE KAKEYA CONJECTURE -THIS PROJECT FOCUSES ON HARMONIC ANALYSIS, A PART OF MATHEMATICS THAT EXPLORES HOW DIFFERENT WAVES INTERACT?COMBINING OR CANCELING EACH OTHER OUT. A CENTRAL PROBLEM IN THIS FIELD IS THE FOURIER RESTRICTION CONJECTURE, WHICH SEEKS TO UNDERSTAND THE MINIMAL AMOUNT OF CANCELLATION THAT OCCURS WHEN A SUM OF WAVES IS CONFINED TO A CURVED SURFACE. WORK ON THESE PROBLEMS HAS LED TO THE DEVELOPMENT OF POWERFUL IDEAS AND TOOLS THAT HAVE PROVEN USEFUL IN OTHER AREAS OF MATHEMATICS, SUCH AS NUMBER THEORY AND DIFFERENTIAL EQUATIONS. A KEY OBJECTIVE OF THIS PROJECT IS TO DEEPEN THE STUDY OF SUCH PROBLEMS AND TO DEVELOP NEW MATHEMATICAL TOOLS AND INSIGHTS TO FURTHER OUR UNDERSTANDING OF WAVE BEHAVIOR. ADDITIONALLY, THIS PROJECT PROVIDES RESEARCH TRAINING OPPORTUNITIES FOR GRADUATE STUDENTS. THE PRINCIPAL INVESTIGATOR (PI) WILL WORK ON SEVERAL PROJECTS RELATED TO THE FOURIER RESTRICTION CONJECTURE. THE FIRST PART OF THE PROJECT IS CONCERNED WITH WEIGHTED L2 ESTIMATES AND THEIR APPLICATIONS TO LP PROBLEMS IN HARMONIC ANALYSIS. SPECIFIC EXAMPLES INCLUDE THE POINTWISE CONVERGENCE PROBLEM FOR THE PLANAR BOCHNER-RIESZ MEANS AND THE LP BEHAVIOR OF THE MAXIMAL SCHR?DINGER OPERATOR. THE PI WILL EXPLORE A RANGE OF WEIGHT FUNCTIONS MOTIVATED BY DIFFERENT APPLICATIONS. THE SECOND PART CENTERS ON INVESTIGATING THE FOURIER RESTRICTION CONJECTURE ITSELF, POTENTIALLY LEVERAGING THE TOOLS DEVELOPED IN THE FIRST PART. THE PI AIMS TO CREATE A NEW FRAMEWORK FOR ANALYZING THIS CONJECTURE. THE THIRD PART OF THE PROJECT FOCUSES ON IDENTIFYING AND UNDERSTANDING GEOMETRIC OBSTRUCTIONS RELATED TO THE FOURIER RESTRICTION CONJECTURE. IN PARTICULAR, THE KAKEYA MAXIMAL CONJECTURE IS KNOWN TO BE A MAJOR OBSTRUCTION. THE PI PLANS TO INTERPRET THE KAKEYA MAXIMAL CONJECTURE USING ALGEBRAIC METHODS, STUDY SPECIFIC EXAMPLES, AND ULTIMATELY EXPLORE NEW POTENTIAL OBSTRUCTIONS TO THE FOURIER RESTRICTION CONJECTURE. 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.