François Monard, assistant professor of mathematics, has received a Faculty Early Career Development (CAREER) Award from the National Science Foundation (NSF) to support his work on theoretical and applied aspects of integral geometry.
Integral geometry is the mathematics behind a range of imaging techniques such as computerized tomography (CT) and other medical imaging technologies. Other applications include seismic imaging of Earth’s interior, imaging techniques for homeland security, and polarimetric neutron tomography for imaging magnetic fields inside materials.
“The basic problem is to reconstruct the internal features of something based on an understanding of how particles or waves propagate through it, and problems of integral geometry arise when the measurements take the form of cumulative values along a family of trajectories,” Monard said.
Applications such as CT scans are widely used and highly successful, but researchers continue to push the limits of these technologies. “Some applications should take into account that information often propagates along curves instead of straight lines, and addressing this issue is one of the main geometric motivations of my project”, Monard said.
Monard will pursue several avenues of investigation to address core questions in integral geometry. His approach combines deep theoretical tools in geometry and analysis with a concern for implementability to produce explicit answers. The project will also address practical issues such as noisy data, sampling, and uncertainty quantification, using methods from theoretical statistics and signal processing.
The funding for this project includes support for graduate students and a postdoctoral researcher, as well as undergraduate research opportunities. Monard also plans to host two regional workshops on integral geometry and its applications, aimed at graduate students working in this area at West Coast institutions.
The CAREER Awards are NSF's most prestigious awards in support of early-career faculty who have the potential to serve as academic role models in research and education and to lead advances in the mission of their department or organization. The award provides $425,000 over five years to support Monard’s research, education, and outreach activities.
Monard earned his Ph.D. in applied mathematics at Columbia University and joined the faculty at UC Santa Cruz in 2016.
