‘I don’t need euphoria, I hope to maintain my enthusiasm’, says Liu

It was March 2020 and young Yingjian Liu had his bags packed in China and was on his way to Brazil, where he would begin his master’s degree at IMPA. The course began, but his arrival in Rio de Janeiro had to wait. With the pandemic, the plan was repeatedly postponed and the situation ended up becoming a family joke. “You leave China every March,” joked his uncle, in the face of failed attempts to travel. It wasn’t until March 2022 that Liu finally managed to land in Rio de Janeiro.

After completing his master’s degree, he went on to complete his doctorate at IMPA. And next Thursday (16), at 9:30 a.m., in room 232, Liu defends his thesis “Regularity of Lyapunov exponents in one-point spectra: the semi-simple case”. The research was supervised by IMPA’s director-general, Marcelo Viana, and will be broadcast live on YouTube from the institute.

With a degree in mathematics from Dalian University of Technology, Liu says he didn’t always have an easy time with the subject. “At first, I wasn’t good at basic arithmetic,” he recalls. The turning point came when he began to identify patterns. “I realized that mathematics was deeper and more interesting than I had imagined.”

While still in China, his career was already promising. After winning first prize at the National Mathematics Competition, he was guaranteed opportunities at institutions like Tsinghua and Peking University. But a conversation with colleagues sparked his curiosity about studying abroad. “I saw that IMPA is an institution with a very strong tradition in mathematics and I decided to apply.”

On the other side of the world, the start of his master’s degree brought an additional challenge: keeping up with activities with an inverted time zone. He studied at night and slept in the morning, in a routine marked by distance and the expectation of finally arriving in Brazil.

When he finally landed in Rio de Janeiro, he found the environment he was looking for: “IMPA is the ideal place to develop research in dynamical systems,” he says. He also highlights the city as an important part of the experience. “I really like the combination of the climate, the view and the people. Rio has an energy all of its own.”

His thesis is in the area of Dynamic Systems, a field that studies the evolution of systems over time – from physical phenomena to abstract models. The focus of the work is the so-called Lyapunov exponents, tools that measure the degree of stability or instability of a system.

“They are analogous to the eigenvalues of matrices, but in a dynamic context,” he explains. In simple terms, these exponents help to understand whether small disturbances tend to disappear or grow over time – something fundamental to understanding complex behaviors.

In his thesis, Liu investigates how these exponents vary when the probabilities associated with the system change. The main result shows that, under certain conditions, these variations occur in a controlled manner – a property known as log-Hölder continuity. The demonstration combines different mathematical techniques, including structural decompositions and probabilistic estimates.

With the completion of his doctorate, the next step is already set: to pursue an academic career, entering a post-doctoral program and working as a researcher at a university. “I don’t need euphoria, what I hope is to maintain my enthusiasm,” he summarizes about the path and expectations of life as a mathematician.