Dejan Slepčev
Assistant Professor of Mathematical Sciences
Research Interests
Slepčev studies nonlinear partial differential equations and some of their diverse applications. He is particularly interested in analyzing the mathematical descriptions of a variety of dynamic phenomena, such as behavior of liquids on a small scale, mass transport, and mechanisms that underlie aggregation in biological systems.
Professional Background
Ph.D. Mathematics, University of Texas at Austin, 2002.
M.A. Mathematics, University of Wisconsin at Madison, 2000.
B.Sc. Mathematics, University of Novi Sad, Serbia, 1995.
After earning his doctorate from the University of Texas at Austin, Slepčev did postdoctoral work at the University of Toronto. In 2004, he joined the University of California at Los Angeles as assistant adjunct professor and assistant researcher. He joined the Carnegie Mellon faculty in 2006.
What are nonlinear partial differential equations?
Partial differential equations involve unknown functions of several variables and their partial derivatives. They provide a quantitative description for many systems that extend continuously in space or both in space and time. Wave propagation, fluid flow, heat conduction, elastic properties of materials, phase transition, chemical reactions with diffusion, gravitation (via general relativity), material transport in physical and biological systems, and many other phenomena are described using partial differential equations.
Why is it important to study the structure and properties of nonlinear partial differential equations?
Understanding the mathematical properties of equations provides insights into the systems they model. When equations are proposed to describe a phenomenon, mathematicians investigate their properties: are they internally coherent? do they have a solution? is the solution unique? For systems that change over time, we investigate the equations that model them to understand how the system will evolve, why certain phenomena appear, and what governs the patterns that we see. Even systems that are guided by a few simple laws can develop rich behavior. Learning about the equations can, in the end, give us the power to predict the system’s behavior and, when possible, control it in an optimal way. Furthermore, by finding structural similarities between equations that describe phenomena in different fields of science, we can advance the transfer of knowledge between the fields.
What specific types of equations do you study?
One of my interests is studying thin-film equations — from their mathematical structure to their applications. Thin-film equations model very thin layers of fluids on a solid substrate. Such situations are common in nature: they occur whenever a fluid coats a solid, but they are also relevant to novel applications such as microfluidic devices. Fluids on a small scale behave very differently from macroscopic fluids that we are used to dealing with in our everyday lives. The inertial effects become negligible, surface tension becomes the dominant driving force, and intermolecular forces create interesting phenomena. I am specifically interested in the process of coarsening in thin-film equations.
What is coarsening?
Thin, nearly uniform layers of some fluids can destabilize under the effects of intermolecular forces. For example, if you spread a thin layer of water over a nonstick skillet, the layer will not stay uniform — soon it will break and dry spots will emerge. In some fluids this process, called dewetting, leads to the formation of droplets connected by an ultra-thin layer of fluid. This structure coarsens over time. Larger droplets grow at the expense of smaller droplets, which eventually vanish. The average distances between droplets grow and their size grows while their number is decreasing. This interesting phenomenon persists for a long time and exhibits selfsimilarity — the pattern of the droplets early on is similar to what one sees much later, only the early droplets are small and not that far from each other, while the droplets later on are comparatively large and further from each other.
What can we learn by studying coarsening?
By studying thin-film equations that model the coarsening process, we can understand why and how it is occurring and make quantitative predictions. Since various coarsening processes appear on small length-scales (in our case the height of the droplets is on the order of 10-1000 nanometers), understanding how patterns spontaneously appear and evolve could impact how patterned surfaces are manufactured.
How can nonlinear partial differential equations be used to model aggregation in biological systems?
Populations of individuals aggregate into large groups — herds of zebras, flocks of birds and swarms of locusts. How do the decisions of individuals lead to the formation of these groups? How do they determine the properties of the groups? Simple rules that model individuals’ decisions can lead to complex aggregate behavior. Interestingly, such behavior on a large-scale can be similar to how fluids behave on a small scale. I am collaborating with mathematicians at UCLA to study large populations using continuum models for population density. The equations we study explain how phenomena analogous to surface tension occur at the group boundary.
What drew you to Carnegie Mellon’s Department of Mathematical Sciences?
My primary attraction to Carnegie Mellon was the colleagues that I now have the opportunity to interact with. Additionally, I am very excited to be a part of the Center for Nonlinear Analysis, one of the top places in the field. Some of the best mathematicians are here and are studying equations related to my interests. I am also looking forward to benefiting from the ongoing interaction with other science and engineering departments here at Carnegie Mellon.
Amy Pavlak
January 2007
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