by John H. Tibbetts
Claudia Miller, professor in the Department of Mathematics, has been awarded a National Science Foundation grant for her project, “Homological approaches to differential forms, differential operators and transfer of algebra structures.”
Scholars have long used algebra to study unsolved questions from geometry, such as how investigators can find measures of singularity. A singularity is a place on a curve, surface, or higher dimensional space where it is not smooth, that is, it has a sharp point or crosses itself. Highly complex abstract structures are easier to understand and visualize in the smooth setting, and singularities present challenges and complexities.
The structural backbone given by algebraic geometry and commutative algebra can help investigators understand how to measure how extreme a singularity is or how far it is from being smooth. Miller will use homological methods to gain insight into these problems, helping investigators understand how to measure the extremity of a singularity, extracting information and presenting it as a visible mathematical object. She will deploy differential forms and operators to characterize smoothness and create new algebraic structures that yield not only invariants but also tools to discover new invariants. This work on singularities can lead to applications in computer vision and medical imaging, and has connections to string theory in physics.
For the Spring semester 2024, Miller was selected for a prestigious research professorship at the Simons Laufer Mathematical Sciences Institute (SLMath) in Berkeley, California. She will be on-site at the institute for a semester of talks and research activities. The intense program is reserved for distinguished mathematicians to collaborate on cutting-edge topics. She will participate in seminars and workshops, exchange ideas with other researchers in residence, and she will mentor postdoctoral fellows. The NSF grant will enable her to travel the semester beforehand and the summer after the program to work with colleagues at other universities, both in preparation for the intense program and to continue work begun there. It will also enable her to invite visitors to Syracuse to speak on their work, exchange new ideas, and provide exposure for the large group of Ph.D. students in algebra in the department.