This award provides support for participants in a workshop on Geometric Analysis on Euclidean and Homogeneous Spaces at Tufts University, January 8-9, 2012. The workshop will encompass the related fields of integral geometry, harmonic analysis, differential geometry, representation theory, microlocal analysis and partial differential equations. These subjects find applications in areas as diverse as inverse problems, tomography, and signal and data analysis, all of which will be represented at the workshop. The workshop will be organized so that researchers and graduate students in the different fields will have ample opportunities to interact with the hope of developing collaborations. This range of fields is not typically represented together at workshops, and cross-fertilization between these fields is an important goal of the workshop. The workshop is preceded by the annual Joint Mathematics Meeting in Boston and uses the momentum of this national event to bring together well-known experts, young researchers, and graduate students. Women, graduate students, postdoctoral fellows, and young faculty will have the highest priority for support. The PIs plan to disseminate the research through proceedings that will include works by researchers from both the special session and the workshop.
The interface between the topics of this conference has lead to fruitful research and applications to medicine, industry, and science in the past. This meeting will provide the opportunity to push this positive development forward by bringing together, engaging, and motivating graduate students and young emerging researchers as well as more established experts. In particular graduate students are involved in the workshop through participation in a poster session. This allows the students to present their work and interact with each other and established experts in an informal setting. Supporting a diverse group of graduate students and beginning researchers is the highest funding priority.
occurred on January 8-9, 2012 at Tufts University and over 45 researchers and 18 graduate students participated. Participants were in a range of areas related to geometric analysis including representation theory, harmonic analysis on homogeneous spaces, integral geometry, tomography, and sampling. Participants learned about others research and new collaborations were developed along with collaborations on new problems. Because of the broad nature of the workshop, mathematicians were able to learn about research in a range of fields, and overview talks were given by preeminent mathematicians in the fields. The workshop had a markedly international character, as eighteen of the forty-four participating researchers came from institutions abroad: Israel, Denmark, Sweden, Australia, India, Japan, the Netherlands, and France. Attracting young researchers was an important goal, and we actively recruited students and young faculty. Graduate students presented their research at a popular, well-attended poster session, and they discussed their research informally with professional mathematicians in a relaxed atmosphere. The conference represented fields including harmonic analysis, representation theory, and tomography. Harmonic analysis studies how structures (such as functions or differential operators on certain spaces) can be expressed as superpositions of simpler or more fundamental components. It has applications in mathematics, physics, and engineering, and draws upon mathematics such as representation theory, differential geometry, and partial differential equations. Representation theory is behind important transforms such as the wavelet, shearlet, and short time Fourier transform. Understanding the sampling of these transforms leads to new results in the theory of function spaces as well as applications in reconstruction. The applied field of tomography allows scientists to image objects using indirect data. It is based, in large part, on the pure mathematics represented at the workshop. Mathematics behind imaging was discussed at the workshop including radar, ultrasound, and near cloaking (making objects almost invisible to certain radiation). A speaker also presented a tomographic method for detecting hidden nuclear materials. Hybrid tomographic methods were presented at the conference; they allow doctors to image the body without harmful radiation. The workshop was a unique opportunity to gather international experts in these disparate fields to disseminate information and exchange ideas.
