Nonlinear partial differential equations (PDEs) are difficult mathematical problems to study.
Notice it didn’t say "solve."
Ph.D. students researching nonlinear PDEs aren't looking for solutions. They're analyzing the mathematical properties in each equation. They study the classical problems in the existence and uniqueness of solutions and their properties such as regularity, complexity, stability, long-time behavior, absorbing sets and attractors.
Mihaela Ignatova, a 4th-year mathematics Ph.D. student in USC College, researches nonlinear parabolic PDEs, inspired by the Kuramoto-Sivashinsky equation, an important model in combustion and flame propagation. For this type of self-generating chaos system, rigorous mathematical analysis is crucial for better understanding of the model and provides the proper guideline for numerical computations.
“This research gets harder and harder to accomplish,” Ignatova said. “This is nothing like the mathematics I was doing five years ago. After earning my Ph.D. next year, I'd like to continue in the field of nonlinear PDEs and hope to find an academic position.”
Ignatova’s perseverance and all around great work has earned her a 2010-11 USC Women in Science and Engineering (WiSE) Merit Fellowship for current doctoral students. She was among four USC students to receive a stipend of $5,000 each, intended to help keep outstanding Ph.D. candidates on the road to careers in science and engineering.
“The WiSE Merit Fellowship has helped to keep me motivated,” said Ignatova, whose research in PDEs is used by scientists to model a broad range of physical systems, from gravitation to fluid dynamics. “It shows I’m moving in the right direction.”
Imagine constantly having numbers, letters and Greek symbols occupy your brain at once. This kind of research demands high levels of concentration.
Ignatova thinks about a problem, writes it down and begins conducting her meticulous analysis. But sometimes it doesn’t end there. If she can find the time, she at times ponders an equation during walks. She often ends her evenings thinking about math, analyzing in her mind.
“Sometimes I need to watch a movie just to distract myself,” she said.
Ignatova has been proficient in mathematics since her girlhood in Sofia, Bulgaria. At six, her grandfather began giving her word problems. Learning chess and backgammon at a young age taught her about strategy. Her elementary school focused on mathematics and she attended the Sofia High School of Mathematics. She chose USC on a recommendation by her adviser at Sofia University, where she earned her bachelor’s and master’s in mathematics.
Her adviser in the College, Igor Kukavica, professor of mathematics, described Ignatova as “motivated, talented and hard working.” She already has a paper accepted for publication in Advances in Differential Equations and another nearly ready for submission.
“In her accepted paper, she obtained a beautiful result describing the complexity of solutions of higher order PDEs with non-analytic coefficients,” Kukavica said. “Previously, only the analytic coefficients had been treated.”
Another College WiSE Merit Fellowship winner is Whitney Behr, a 4th-year Ph.D. candidate in earth sciences.
“It gives me some extra confidence,” Behr said of the fellowship. “It means I’m on the right track and may be competitive when I go looking for a job.”
Behr researches structural geology. She measures and analyzes present-day rock geometries to glean information into its history of deformation or strain. Understanding a rock’s stress field can give insight about its geologic past and structural evolution, enabling scientists to see patterns of rock deformation due to tectonic plates.
“This can have direct implications to earthquake research,” Behr said. “Right now, we’re trying to understand how fast the San Andres Fault is moving.”
Raised in Tujunga in the far northern reaches of Los Angeles, Behr plays classical guitar and began her academic career as a music major at Pasadena City College. Uncertain about what she would do with a music degree, she accepted an opportunity to study in Cal State Northridge’s geological sciences department. After earning her bachelor’s in geological sciences, she chose USC College over UCLA and UCSB, partly because of the monetary incentives USC offers.
Behr’s adviser, John Platt, professor of earth sciences in the College, considers Behr the best student he has advised of his 26 Ph.D. candidates in the past three decades. She has two papers (one in which she was principal author) soon to appear in the Bulletin of the Geological Society of America, one (also senior authored) submitted to Earth and Planetary Science Letters, and one (co‑authored) submitted to Geology.
“She has extraordinary energy, enthusiasm and professionalism in her approach to her work,” Platt said. “During her first two years as a graduate student, she astonished me by her ability to set up and organize a complex research project, manage people many years her senior, and write convincing and professional research proposals (she raised $70,000 in research funds for the project), all while handling a teaching assistant job and heavy graduate course load.”
Behr’s field geology research has taken her to Spain, China, Italy, and she is now headed for Greece.
“My parents never went to college and none of my siblings have so far,” she said. “It feels great to be on an academic track.”
The two Ph.D. students who earned 2010-11 merit awards from the USC Viterbi School of Engineering are Sara Abedi of the Sonny Astani Department of Civil and Environmental Engineering, and Joyita Dutta of the Ming Hsieh Department of Electrical Engineering.
The WiSE College Committee also offers $5,000 awards as an extra incentive to prospective Ph.D. students in the sciences.