A Love for Puzzles

Elena Cherkaev

Elena Cherkaev, professor and director of graduate studies for the Math Department, has always enjoyed solving logical paradoxes, and she believes mathematics is full of such intellectual challenges. She is also interested in linguistics, philosophy, and art history.

Originally from St. Petersburg, Russia, Cherkaev received a Ph.D. in mathematics from St. Petersburg State University. She studied measure and game theory for her master’s degree but later switched her focus to partial differential equations, optimization, and numerical methods. After graduation, she worked in the Academy of Sciences studying the Earth’s magnetic field and propagation of electromagnetic waves.

Director of Graduate Studies

Last spring Elena became the director of graduate studies for the department while continuing to teach and do research. “I have great faculty members and staff who help me to keep on track,” she said. She works with faculty and staff, including Paula Tooman, Karl Schwede, Kelly MacArthur, Maggie Cummings, Lajos Horvath, and others.

“Paula is an indispensable member of the department,” said Cherkaev. “She knows everything about the graduate program—she watches every deadline and keeps track of events, and she truly cares about each student.” Cherkaev also relies on Karl Schwede, who served as the director of the program before her. He created an electronic system to keep all the graduate records together and to track a student’s progress. With help from another professor, Schwede created a thesis-formatting template, so graduate students who use the template no longer have to send their thesis to the Thesis Office for format approval. Cherkaev noted the excellent graduate teaching workshops organized and run by Kelly MacArthur, the supervising and advising done by Maggie Cummings for the master’s degree in teaching, and the guidance Lajos Horvath provides for the master’s degree in statistics. “We have a strong graduate program,” said Cherkaev, “and I depend on these individuals and others to keep it running smoothly and effectively.”

As with any graduate studies program, the goal is to prepare students for their future professional life. “In recent years, there has been a growing interest and demand in industry for statisticians and data scientists, and some graduate students choose

nonacademic careers. We also have students who are great instructors—they enjoy teaching and want to pursue a teaching path,” said Cherkaev. “All of these careers are great professional paths, and I want to create an atmosphere that allows students to develop to the best of their abilities, so they are well prepared for their future.”

Research Projects

Cherkaev’s research encompasses a range of topics in applied mathematics that use partial differential equations, variational methods, and optimization. “The driving force behind any project is always an intellectual challenge, an interesting mathematical problem that I am trying to solve,” said Cherkaev. One of the topics she is working on is inverse problems and efficient numerical methods for their solution. An inverse problem arises whenever the measurements of the observed process contain only indirect information about the property of interest. An example can be an electrical tomography imaging or nondestructive testing to find the structure of a human’s body or the Earth’s subsurface from measurements of the responses due to the currents (or electromagnetic waves, or ultrasound sources) applied on the boundary. Using a variational approach and spectral decomposition in an imaging problem, Cherkaev developed a method of finding boundary excitations that bring the most valuable information about the structure and properties inside the body by concentrating the energy of the scattered field to the region of interest. This approach resulted in a model order reduction method that allowed an increase in the resolution and the ability to significantly speed up the computations.

Another focus for her research is mathematics related to the structure of heterogeneous materials or composites. Properties of natural and artificial composites depend on their finescale microstructure. For instance, mechanical properties and the ability of bone to withstand fracture depend on the structural organization of bone as a hierarchical composite. In many applications, we need to know the response of the composite to applied forces or electromagnetic fields. Cherkaev works on this homogenization problem and the related inverse homogenization when it is required to characterize parameters of the structure given the response of the medium. To solve the last problem, Cherkaev used a spectral decomposition developing a method for extraction of information about the micro-geometry of heterogeneous material from homogenized measurements of its response to the long-wavelength electromagnetic or acoustic field. “My work showed that microstructural statistics could be extracted from such data,” said Cherkaev. “This is important in many applications, for instance, for evaluation of the structure of bones, metamaterials, soil, sea ice, etc.” In collaboration with her former Ph.D. student Carlos Bonifasi-Lista, Cherkaev developed a method of characterizing bone micro-architecture that has a potential for osteoporosis monitoring. Her methods apply to modeling, optimization, and imaging of geophysical, biological, and artificial materials, as well as to high dimensional data sets. “The results obtained by applied mathematicians have many applications,” said Cherkaev, “because similar mathematical problems often describe different physical phenomena.”

Cherkaev finds teaching to be very rewarding and enjoys working with students on research projects. In her free time, she likes to attend the Utah Symphony, read The New Yorker, walk in Red Butte Garden, or hike in a canyon. She has worked on several research projects with her husband, Andrej Cherkaev, also a professor in the Math Department. They like to spend time together and enjoy visits from their grown children.