Mathematicians Sweep NSF Grant Awards
Professor Lixin Shen has received a collaborative research grant for $183,400 to study image and signal processing; Professor Loredana Lanzani, $180,000 to explore harmonic analysis and partial differential equations; and Professor Graham Leuschke, $155,000 to investigate representation theory and non-commutative algebraic geometry.
“I am extremely proud of these professors, who embody our department’s commitment to research excellence,” says Uday Banerjee, professor and chair of mathematics. “Their cutting-edge work as teachers, scholars and administrators elevates the college, while raising the research profile of the University.”
Shen’s research focuses on algorithms for optimization problems that arise from a variety of applications, including parallel magnetic resonance imaging in medical imaging processing, as well as facial and fingerprint recognition in security identification systems. Such image/signal problems of practical importance are often modeled as large-scale optimization problems; therefore, he says it is essential to develop efficient computational algorithms to solve them.
An algorithm is a sequence of instructions, used to show how to perform a task.
“Image/signal processing problems of practical importance, such as incomplete data recovery, compressive sensing and matrix completion usually possess hierarchical structures or are represented in a multiscale analysis,” Shen says. “Multiscale analysis, however, is mainly used to sparsify [to scatter or disperse] the underlying image/signal in formulating the optimization problem, but it has not been fully exploited in the development of efficient algorithms. … I will make systematic use of the hierarchical structure in optimization problems of interest to solve them in an accurate and computationally efficient way.”
Like Shen, Lanzani is interested in harmonic analysis, but she also is an expert in partial differential equations and complex analysis. Her grant project deals with the study of so-called integral formulas in complex and harmonic analysis. These formulas are often used to recover information in large data sets that are difficult to reach
“For example, integral formulas may be used to determine the internal temperature of something, such as a tree, without poking holes in it,” she says. “After measuring the temperature at a surface level [e.g., the tree’s bark], one can use an integral formula to plot the value of the internal temperature.”
Lanzani says these formulas permeate pure and applied science, and help shed light on the study of heat transfer and celestial mechanics.
Leuschke, who is also associate chair for graduate affairs, works in commutative ring theory, which explains algebraic systems where addition, subtraction and multiplication are defined. Since every commutative ring has a corresponding geometric object, known as an algebraic variety, his work uses aspects of algebraic geometry, commutative algebra and representation theory. Combined with noncommutative algebra, this allows exploration of a new field called “non-commutative algebraic geometry.”
“I want to make sense of the geometric content of noncommutative rings and their representations,” says Leuschke, adding that his project will focus on various rings, categories and other abstract structures. “This kind of data is useful to mathematical physicists who work at the quantum level.”