'Sometimes, Finding Ways to Solve Problems Feels Like Trying to Cross Mountains without a Map'

Yulia Zaitseva became interested in mathematics in third grade, when her parents enrolled her in a math club, and she now holds a PhD. In this interview with the HSE Young Scientists project, she explains what an algebraic variety is, discusses operations beyond addition and multiplication, and shares her love for Kolomenskoye Park.

Why I Chose a Career in Science

I’ve been interested in mathematics since childhood. My parents took my sister and me to a math club when I was in third grade, and from then on mathematics has always been part of my life. Later, I joined many other clubs; we went to math camps in the Kostroma Region in summer and took part in math battles and olympiads.

In seventh grade, I enrolled in the mathematics track at School No. 179, and I am very grateful to my teachers there—they taught me not only knowledge and skills, but also ways of thinking about learning and life in general. At school, in our math workshop, we used a system based on 'flyers.' This meant that the teacher rarely presented material at the blackboard: all the necessary definitions were provided on a sheet of paper, and the students themselves derived the theorems from those definitions. In my view, this approach is quite close to real scientific work and helps you understand the subject and the objects you are studying on a much deeper level. After finishing school, I enrolled in the Faculty of Mechanics and Mathematics at Moscow State University using my diploma as a winner of the All-Russian Olympiad.

My academic supervisor, Prof. Ivan Arzhantsev—under whose guidance I completed my degree at MSU and defended my PhD dissertation—played a key role in my decision to pursue a career in science. His ongoing support and belief in my abilities have been truly invaluable. Prof. Arzhantsev is always ready to share interesting problems and ideas, offer help and guidance when something doesn’t work out, and share his life experience and practical advice.

The Focus of My Research

I work at the Laboratory on Algebraic Transformation Groups, where we focus on algebraic varieties and their automorphism groups.

An algebraic variety is the set of solutions to a system of polynomial equations. For example, a straight line or a circle can be viewed as algebraic varieties. An automorphism of a variety is, simply put, a symmetry—a mapping that has an inverse and sends the variety to itself. Since the objects we study are defined by polynomials, we are interested only in mappings that are also given by polynomials.

Let me offer an example of a question in this area. On a straight line, we can make shifts and stretches—transforming x into kx+b. If k is nonzero, this transformation is invertible, so these are automorphisms. The transformation group in this case is two-dimensional, determined by the two parameters k and b. If you take a plane instead of a line, the situation changes dramatically: there are many more transformations, forming an infinite-dimensional family. For example, you can leave the x-coordinate unchanged and add an arbitrary polynomial of x to the y-coordinate, which introduces infinitely many parameters.

Although the automorphism group of the plane is large, it is well studied: roughly speaking, it can be described in terms of shifts, stretches, and automorphisms involving the addition of a polynomial. But once we move to higher dimensions—for example, to three-dimensional space—the group becomes even larger, and we no longer have a description as complete as the one for the plane. However, we can describe important subgroups of this larger group, which allows us to begin understanding its overall structure.

What I Take Pride In

Perhaps one of the most interesting results is the classification of monoids in three-dimensional space. At school, we all learn addition and multiplication. But could we consider other operations? Of course we can—for example, we could multiply two numbers, cube the result, and then take the penultimate digit. However, no one really needs this operation, as it has no useful properties.

A monoid is a set equipped with an operation that satisfies a small number of natural properties. I have contributed to several articles on the classification of algebraic monoids under various constraints. In particular, we recently completed a classification of algebraic monoids on three-dimensional affine space.

Besides operations derived from addition and multiplication through simple combinations, there are several more unexpected ones that at first glance do not seem to satisfy all the required properties. Our latest paper has recently been reviewed by a reputable journal and, hopefully, will be published soon.

My Dream

To devote more time to science.

What I Do Besides Science

I teach algebra and linear algebra to first-year undergraduates at the Faculty of Computer Science. I also grade problems assigned to students in the math class at School No. 179. Teaching is an opportunity to pass on your knowledge and experience to the next generation. A well-conducted lesson creates a positive atmosphere and motivates productive work. They say that the best way to truly understand a topic is to teach it to someone else—and there is much truth in that.

If I Hadn't Become a Scientist

Sometimes, when struggling with a difficult problem in mathematics, I half-jokingly reassure myself that I could work as a proofreader for a mathematical journal, since I spot typos easily. However, it’s unlikely they would hire me without formal training in philology.

In school and during my first years at university, I enjoyed computer science. I also took part in various olympiads in this subject, including the All-Russian Olympiad. I believe that if a career in science hadn’t worked out for me, I could have pursued programming.

A Typical Day for Me

It depends on the day of the week. This semester, there are two weekdays when I go to HSE University to teach classes, hold consultations, and participate in a seminar at my laboratory. On two other weekdays, I travel to the school where I review problems assigned to students in a math club for grades five to seven. I dedicate one day entirely to science, trying not to be distracted by organisational matters. On weekends, I visit my parents and spend time with family.

Whether I Have Experienced Burnout

I’m not sure I have ever experienced true burnout. I am fortunate to be supported by my academic supervisor, colleagues, and family. When my mood approaches a breaking point, it helps to spend some time alone completing a simple work task. Another option is to distract myself with household chores, such as cleaning the kitchen stove.

My Interests Besides Science

As a child, I completed a choreography course at a children’s art school, where I studied ballet. We learned a variety of subjects, including classical and folk dance, and practiced extensively during rehearsals—some of which I skipped to attend math clubs. We performed at various venues, including school events, concerts for City Day celebrations, and other occasions. Once we even went to Serbia for an international competition.

Advice for Aspiring Scientists

When it comes to science, it seems to me that the most important factor in starting your scientific path is the choice of an academic supervisor and scientific school. When making this crucial decision, don’t hesitate to ask questions of senior students, potential supervisors, and their current students. To get a feel for the field and the team, you can attend scientific seminars of relevant groups, even if you don’t understand much at first.

It’s not always clear whether pursuing a career in science is the right choice. This path is not for everyone. You need to listen not only to others but also to your own instincts. While you’re at university, there are many opportunities to try your hand at research, so if you’re unsure, it’s worth experimenting. It’s generally easier to move from science to industry than the other way around.

My Favourite Place in Moscow

I have many favourite places in Moscow, but perhaps Kolomenskoye Park stands out. When I was a child, my parents, sister, and I lived nearby and went for walks there in every season. We admired the blooming wild roses in spring and rode a snowmobile in winter. One of my favourite spots is a viewpoint overlooking the Moskva River near the Dyakovsky apple orchard. It’s a peaceful and beautiful place.