‘The Joy of Science Lies in the Euphoria of Learning’

For Elena Nozdrinova, mathematics is her life's work and a realm where she discovers universal order and harmony. In her interview with the HSE Young Scientists project, she speaks about dynamical systems, the Nizhny Novgorod scientific school, and favourite pastimes that help her grow.

How I started in mathematical science

Although I have always felt that mathematics is my life's calling, my journey in mathematical science has been long and challenging. I began my career as a teacher at my alma mater, the pedagogical university. Teaching brings me joy, because I see academia as a space where one can engage in stimulating scientific discussion and connect with bright-eyed, highly motivated individuals who have come here in pursuit of knowledge and self-actualisation. In addition to my role as a teacher, serving in parallel as a manager at my own university allowed me to gain experience in administrative work that has proven invaluable later on.

My teaching career continued at RANEPA IPACS in Nizhny Novgorod, and during my time there, I felt the need to grow as a researcher in my field—in particular, by defending a PhD thesis, which is undoubtedly a marker of professional achievement for a scientist. It was also a natural progression in my career as a teacher. Given the profile of the university where I was teaching, I had the option of writing a thesis in economics and subsequently defending it at Lobachevsky State University of Nizhny Novgorod. However, as I said, mathematics was simultaneously my life’s calling and the field in which I wanted to apply my knowledge at the time. Therefore, I decided to make every effort to earn a PhD in my chosen field. I began by contacting my former teacher at the pedagogical university, Evgeny Zhuzhoma. By then, he was working at HSE University, where a doctoral programme in mathematics was just about to be launched. He recommended me as a doctoral student to the prominent, internationally recognised scientist Olga Pochinka, who later became my permanent supervisor and mentor. Olga and I still recall our first meeting with warm feelings and smiles. I had graduated from university a few years prior and was nearing the due date for the birth of my daughter. I told Olga about my passion for mathematics and my readiness to make every possible effort to be admitted to the doctoral programme in this field. Perhaps Olga saw a sparkle in my eyes and recognised my dedication to hard work, because she agreed to coach me for admission to the programme.

I still recall going over to her home and spending entire days studying with her, while her daughter was looking after my baby. I remember taking walks with the baby in a stroller while holding on to my study materials—I kept studying all the time

My efforts were rewarded: I was admitted to the doctoral programme, successfully completed it, and in 2021, defended my thesis entitled 'On classes of stable isotopic connectivity of surfaced gradient-like diffeomorphisms'.

I wish to emphasise that Olga's contribution to my development as a scientist and my professional success has been immense. She has incredible energy. She radiates this energy, generating a plethora of new ideas, and always supports her colleagues' initiatives. Her attitude is both fascinating and contagious. Being around her, one simply cannot help but also become active and determined. Currently, I have my own students whom I supervise and consistently provide guidance to, never leaving them alone to grapple with unresolved scientific problems. By doing so, I feel that I am carrying on the tradition established by my academic supervisor.

My research focus, or what I am working on

I study dynamical systems. A dynamical system refers to any object or process that mimics real-world physical, chemical, biological, or economic phenomena. It is defined by its initial state and the governing law that guides the system's transition from the initial to some other state.

Mathematically, such a law can be expressed through a set of differential equations. However, it is seldom possible to express the solution of a differential equation explicitly by using a formula. Modern computer calculations can certainly provide an approximate solution to differential equations within a finite time interval but do not allow us to understand the overall behaviour of phase trajectories. Therefore, the methods of qualitative analysis of differential equations, which are my current research focus, assume a crucial role.

The answer to the question of what modes of behaviour can be observed in a given system can be obtained from the so-called ‘phase portrait’ of the system, which represents the totality of its trajectories depicted in phase space. A few key trajectories are then identified, which determine the system's characteristics. These include equilibrium points corresponding to stationary modes of the system, as well as closed trajectories (limit cycles) corresponding to periodic oscillation modes. A mode's stability can be assessed by observing the behaviour of neighbouring trajectories: a stable equilibrium or cycle attracts all nearby trajectories, whereas an unstable one repels at least some of them. Thus, a phase space broken down into trajectories creates a portrait of the dynamical system.

The practical application and prospects of my work

Since dynamical systems can actually describe all processes unfolding in the world around us, the potential scope of their application is enormous. The study of dynamical systems has multiple distinct branches and specialised areas of focus, and numerous topological methods have been employed to interpret them (topology is the modern geometry that examines objects with respect to their continuous deformations), since dynamical systems can be defined on various types of surfaces. However, the practical application of this aspect of fundamental mathematics relies on its theoretical elaboration in the first place, which is precisely what I focus on. I study structurally stable dynamical systems which remain unchanged in their qualitative properties with small variations, ie additions to the equation. My professional interests do not include the practical application of dynamical systems. The theoretical focus allows me to examine the object in its pure form as an embodiment of harmony and to appreciate its beauty. This approach, perhaps, best reflects the joy of pursuing science and experiencing euphoria from the process of learning. Once you feel the satisfaction of understanding something or solving a problem, you know what it means to experience true bliss in its purest form and simply cannot resist continuing this journey, because you want to do even more. This drives you to seek out unusual materials, intricate facts, novel subjects, and challenging problems.

It is crucial, in my belief, to understand that theoretical science should always stay ahead of practical applications. A vivid illustration of this principle is graph theory, which originated nearly 300 years ago when the IT industry, computers, and the internet did not exist. This theory emerged and evolved as a distinct field long before finding its practical use. Today, graphs are taught in schools and have a wide range of applications. I believe that developments in the field of dynamical systems have the potential to replicate the success of graph theory in the future. I hope that my progress in science will extend beyond defending my thesis. I absolutely intend to pursue my research further and produce new results in my field, because thought never stands still; it is an ongoing process that needs to be captured and articulated in scientific papers, which can be read and appreciated by colleagues and future generations. Once you achieve results in one area, new objectives often come into sight, which may be related to a different field altogether. For example, when you understand how things works for diffeomorphisms, your thoughts may gravitate towards flows… Therefore, all the time, I feel an urge to explore something more in-depth and on a broader scale.

My dream

'Dreams become reality when thoughts turn into actions', according to Max Fry. I wish to become a valuable representative of my scientific school, so that my research may still be relevant and applicable even 200 or 300 years from now. I also dream that my passion for mathematics may become contagious, spreading to as many people as possible. I would love to share this passion with others by giving popular science talks to school pupils and nurturing a live interest in mathematics among my university students. I would like to ensure that future generations love mathematics as genuinely as I do.

Nizhny Novgorod Scientific Mathematical School

In pursuing my goal, I have consistently received tremendous support from my colleagues—representatives of the Nizhny Novgorod scientific mathematical school founded by Alexander Andronov in the 1930s at Lobachevsky University, with a focus on dynamical systems research. Alexander Andronov passed away at the age of 51, but his work has been carried on by his students and colleagues, including Evgenia Leontovich-Andronova, Nikolai Bautin, Dmitry Gudkov, Yuri Neymark, and others. Their efforts have resulted in the establishment of the classical model of a scientific school, which continues the tradition of conducting scientific seminars as a platform for lively interaction and discussion of recent literature and trends in mathematics. This interaction has fostered a particularly favourable environment for cooperation between students and mentors. I have been working closely with my academic supervisor; we exchange phone calls or meet up every day to discuss the study material. My colleagues and friends are always there for me. We have a very large community of mathematicians, making it possible not only to solve problems independently but also to turn to colleagues for support. Even if everyone is focusing on different fields of mathematics, discussion helps generate scientific ideas for all parties involved.

I genuinely believe that such interaction is essential. Being isolated and forced to solve every problem on your own would be far less effective and perhaps result in a feeling of depression rather than a positive outcome. Active communication with all strata of the scientific community, from the older generation to undergraduate students, lets us benefit from mutual learning. This is the reason why I am at HSE: I came here for the people. This university, I believe, provides a fantastic environment for scientists.

Scientists I would like to meet

The scientific community today is open to communication: if you ask a scientist questions related to their work, you are sure to receive answers. Communication with world-class scientists is truly gratifying, as you are consistently amazed by their memory and their way of thinking. An opportunity to engage with prominent scientists of the past would be even more exciting. In particular, I would be fascinated to meet Henri Poincaré, whose conjecture was proven by Grigori Perelman. This is the only one of the seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000 that has been solved.

What I think about burnout

Reflecting on the topic of burnout, I realise that I have never experienced it myself. I find my work truly energising. For me, the best form of rest is engaging in a different activity. I am not prone to procrastination—idling or wasting time is not my style at all.

Changing one’s activities is essential for productive work. One way to achieve this change is by doing things together with colleagues and students in an informal setting. For instance, my colleagues and I have been hiking and kayaking together for many years. Spending quality time with my daughter is an opportunity for bonding and sharing my knowledge. During our time together, we engage in hobbies from my childhood, such as embroidery and bead weaving. All of this helps me maintain a healthy balance in life and stay in good shape.

What I have recently read that interested me

For me, reading a book is the best way to relax and unwind. Among the most recent ones, I would mention We Are Our Brains: From the Womb to Alzheimer's by Dick Swaab. It explains the way neural connections form in the brain, the innate predispositions a person has from birth, and the hormones behind emotions such as love, sadness, and joy.

As for fiction, one book that stands out for me is The Goldfinch by Donna Tartt. I am particularly captivated by the author's style. It is crucial for me that a book is written in coherent and clear language, because I tend to binge-read, and reading a book becomes akin to watching a long movie. In addition, The Goldfinch contains numerous allusions to Russian classical literature and cultural phenomena, making it particularly interesting for Russian-speaking readers. It also explores parent-child relationships, which is extremely relevant when you are a parent yourself.

Advice for aspiring mathematicians

I am convinced that in science, there is no such thing as a stupid question. A common challenge for young scientists is that they are often afraid to ask questions because they assume that everyone around them already knows everything. But our scientific school teaches us that the most important thing in science is understanding exactly what you are doing. It is better to acquire a small part of the available information but truly comprehend, process, and integrate it, rather than, for example, hear an entire scientific report without having a coordinate system for understanding it, because some critical information was missed earlier.

My favourite location in Nizhny Novgorod

I have a genuine love for my hometown, and I take great pleasure in watching sunsets on the embankment, savouring coffee at local spots, meeting up with friends, and leisurely strolling along pedestrian streets. But the most cherished place for me is my grandparents' home, where I sleep more soundly than anywhere else, and there is always a delicious pastry waiting for me. It was there that I was first introduced to science. My grandfather, Boris Chubikov, was a Candidate of Sciences in Engineering, Lenin Prize awardee, and director of Alekseyev Central Hydrofoil Design Bureau. I often watched him working on technical drawings in his office. My grandfather's colleagues, including foreign scientists, often visited him in his home. I always enjoyed listening to him sharing the impressions from his travels—he visited more than 25 countries during his career—and also to his singing accompanied by the guitar, and his fascinating stories. He was always ready to satisfy my curiosity, and together we studied a vast number of books in search of answers to my questions.