In April, we had the pleasure of welcoming Igor Velčić from the University of Zagreb (Faculty of Electrical Engineering and Computing) to our institute.

His research lies at the intersection of applied mathematics and continuum mechanics, with a particular focus on homogenization, dimension reduction, and multiscale modeling.

Together with him, we have prepared the interview below, in which he shares insights into his research, its applications, and his collaborations with Czech mathematicians.

How would you describe your research to someone who is not a mathematician?

My research lies within continuum mechanics, with a focus on homogenization and dimension reduction. Since the governing equations of nonlinear elasticity are often too complex to solve directly, the goal is to develop reliable simplified models for structures such as composites, plates, and rods.

These reduced models are useful not only for numerical simulations, but also for revealing essential physical mechanisms. In general, my work is concerned with deriving tractable models that accurately describe the behavior of thin and composite structures while retaining key features of the full continuum mechanical description.

What is homogenization in simple terms, and why is it important?

Homogenization theory is used to understand the behavior of composite materials. Combining different materials to obtain a product that inherits desirable properties from each component is a long-standing practice. Of particular interest are materials with fine-scale structures, where the resulting composite can exhibit new effective properties that are not simple averages of its constituents.

This requires mathematical tools that allow us to predict effective macroscopic behavior based on the properties of microscopic components, which is crucial for practical applications in materials science and engineering.

Can you tell us about your collaboration with Czech researchers and what you find interesting about it?

I have always admired the strong tradition of applied mathematics in the Czech Republic. It was a great honor to give a talk at the Nečas Seminar at Charles University. Over time, I have established contacts with several Czech mathematicians, including Martin Kružík, Tomáš Roubíček, and Šárka Nečasová, all of whom are internationally respected.

Among them, I have developed the closest scientific and personal connection with Martin Kružík, whose hospitality I greatly appreciate. Beyond the mathematical insights gained through these collaborations, I also feel that Croatia and Czechia share many cultural similarities, which makes my stays in Prague especially pleasant. All three colleagues have also visited Croatia, and I hope they left with equally positive impressions.

What kind of real-world applications are connected to your research?

Engineers have a strong interest in models for thin and composite structures, as reflected in the extensive literature on elastic plates, rods, and shells. Applications range from large-scale constructions such as bridges to biological systems like blood vessels.

A particularly illustrative example comes from medicine: stents. These elastic mesh-like structures are inserted into blood vessels to improve blood flow. Mathematical models can describe the behavior of such elastic networks, which are essentially composed of thin interconnected rods.

This naturally leads to questions of optimal design—how to configure these structures to achieve desired mechanical performance. In this setting, both accurate models for junctions of thin rods and effective optimization strategies play a crucial role.

What advice would you give to young researchers who are starting in applied mathematics?

It is always difficult to give advice to younger generations, since each era brings new challenges, tools, and forms of pressure. Today, many young people struggle with maintaining focus and engaging deeply with a topic, partly because the pace of life has accelerated.

At the same time, I am confident that there will always be talented individuals who can make good use of modern tools such as AI, while still knowing when to slow down and think carefully and deeply. Equally important is the willingness to invest sustained effort and not expect immediate results.

As the old Roman saying goes:

Per aspera ad astra.

We are grateful for his visit and inspiring discussion, and we look forward to further collaboration in the future.