Abdulla, Ugur

Professor & Head of Mathematics
Mathematical Sciences

I graduated with PhD degree in Mathematics in at the age of 22 from the Azerbaijan branch of the USSR Academy of Sciences in my hometown, Baku. I received my second doctorate degree, Doctor of Physical and Mathematical Sciences at Baku State University and became a full Professor of the Applied Mathematics Department at the age of 28. In 1995 the Royal Society of UK awarded me with Research Fellowships and I spent two years at the University of Nottingham under this fellowships. In 1998 Alexander von Humboldt Foundation awarded me with Humboldt Fellowships and I spent almost two years at the University of Paderborn. In 2000 I was awarded with Max-Planck Research Fellowships and I spent  two years at the Max-Planck Institute for Mathematics in the Sciences in Leipzig. Before taking faculty position at the University of Leipzig in 2003 Saxon State Ministry of Sciences and Fine Arts in Dresden, Germany awarded me with the degree Dr. rer. nat. habil. I joined Florida Institute of Technology in 2004. I was a Visiting Professor at the MIT Mathematics Department in Fall semester of 2012. Currently I am Professor of Mathematics and Head of the Department of Mathematical Sciences at the Florida Tech.  

Personal Overview

I am interested in recruiting strongly motivated graduate students who have a great passion for Mathematics. My personal view of mathematical research may be well expressed by the citation of Kolmogorov: The goal of a mathematician is not just to prove theorems, but to comprehend the open problem. If you are interested in pursuing PhD research under my supervision in one of the fields of my current research interests (oulined below), then you can apply to our PhD program in Applied Mathematics. The major goal and expectation would be to guide and train you to solve an open problem and to push the boundary of current knowledge in mathematics. 

Educational Background

M.S., Applied Mathematics Azerbaijan State University, Baku, USSR 1986
Ph.D., Mathematics Academy of Sciences, Baku, USSR 1988
High Dr.Sci., Mathematics & Physics, Baku State University, Baku, Azerbaijan 1994
Dr.rer.nat.habil., Saxon State Ministry of Sciences and Fine Arts, Dresden, Germany 2003 

Current Courses

I developed and regularly teach a new undergraduate course, MTH3210-Introduction to Partial Differential Equations and Applications, and three new graduate courses on Applied PDEs, Sobolev Spaces and Inverse Problems for PDEs under the general title MTH6100-Selected Topics on Nonlinear Analysis. I have also significantly modified and regularly teach undergraduate course MTH4201-Models in Applied Mathematics, and four graduate courses MTH5230-Partial Differential Equations, MTH5115-Functional Analysis, MTH5130-Theory of Complex Variables and MTH5420-Theory of Stochastic Processes.

Professional Experience

My expertise is in Partial Differential Equations and Dynamical Systems. My research contributions in PDEs are in various aspects of this fascinating field. I would divide them into four groups:

1. Potential Theory and Partial Differential Equations. 

Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results in XX century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs, and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. My recent research developments ([1-4]) precisely characterize the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extend the role of the Wiener test in classical analysis. The Wiener test at ∞ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at ∞ arises as a thinness criteria at ∞ in fine topology. In a probabilistic context, the Wiener test at ∞ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criteria for the removability of singularities and/or for unique continuation at the finite boundary point.

2. Qualitative Theory of Nonlinear PDEs. 

PDEs arising in a majority of real world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs, one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear degenerate/singular parabolic equation, so called nonlinear diffusion equation. In a series of publications ([5-9]) I developed a general theory of the nonlinear degenerate and/or singular parabolic equations in general non-smooth domains under the minimal regularity assumptions on the boundary. This development was motivated with numerous applications, including flow in porous media, heat conduction in a plasma, free boundary problems with singularities, etc. For example, primarily by applying this theory, I solved Barenblatt's problem about the short-time behaviour of interfaces for the reaction-diffusion equations [10,11]. 

3. Geometric Boundary Regularity Tests for PDEs and Asymptotic Laws for Wiener Processes.

One of the oldest problems in the theory of PDEs is the problem of finding geometric conditions on the boundary manifold for the regularity of the solution of the elliptic and parabolic PDEs. There is a deep connection between this problem and the delicate problem of asymptotics of the corresponding Wiener processes. In [14], I proved a geometric iterated logarithm test for the boundary regularity of the solution to the heat equation. In addition, I proved an exterior hyperbolic paraboloid condition for the boundary regularity, which is the parabolic analogy of the exterior cone condition for the Laplace equation. In fact, for the characteristic top boundary points of the symmetric rotational boundary surfaces, the necessary and sufficient condition for the regularity coincides with the well-known Kolmogorov-Petrovsky test for the local asymptotics of the multi-dimensional Brownian motion trajectories [15]. 

4. Optimal Control and Inverse Problems for PDEs

In my recent paper [17], I developed a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperauture and free boundary. I employed optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of L2-norm declinations from the available measurement of the temperature on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In [17], I proved well-posedness in Sobolev spaces' framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control. I address the problems of convergence of the fully discretized family of optimal control problems, Frechet differentiability and iterative conjugate gradient methods in Hilbert spaces in a forthcoming paper.

 

Dynamical Systems and Chaos Theory

One of my current research interests is in Dynamical Systems and Chaos Theory. Jointly with my undergraduate research students Miss Almas Abdulla (MIT, Class of 2014) and Rashad Abdulla (UPenn, Class of 2017) we revealed a new law of the distribution of periodic orbits in chaotic regime for the one parameter family of discrete dynamical systems ([18]). We presented a new constructive proof of the result on the structure of minimal 2(2k+1)-orbits of the continuous endomorphisms on the real line. It is proved that there are four types of digraphs with accuracy up to inverse graphs. It is demonstrated that the first appearance, as the parameter increased, of the 2(2k+1)-periodic window within the chaotic regime in the bifurcation diagram of the one-parameter family of logistic type unimodal continuous endomorphisms is always a minimal 2(2k+1)-orbit with Type I digraph. 

 

Selected Publications

Potential Theory and Partial Differential Equations

1. Regularity of ∞ for the Elliptic Equations with Measurable Coefficients and Its Consequences, Discrete and Continuous Dynamical Systems - Series A, 32, 10(2012), 3379-3397.

2. Wiener's criterion at ∞ for the heat equation and its measure-theoretical counterpart, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 44-51.

3. Wiener's criterion at ∞ for the heat equation, Advances in Differential Equations, 13, 5-6(2008), 457-488.

4. Wiener's criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries, Nonlinear Analysis, 67, 2(2007), 563-578.

 Qualitative Theory of Nonlinear PDEs

5. Well-posedness of the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, Transactions of Amer. Math. Soc., 357, 1(2005), 247-265. 

6. Reaction-diffusion in nonsmooth and closed domains, Boundary Value Problems, Special issue: Harnack estimates, Positivity and Local Behaviour of Degenerate and Singular Parabolic Equations, Vol. 2007 (2007), 1-28.

7.On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, J. Math. Anal. Appl., 246, 2(2001), 384-403.

8. Reaction-diffusion in irregular domains, J. Differential Equations, 164, 2000, 321-354. 

9. Reaction-diffusion in a closed domain formed by irregular curves, J. Math. Anal. Appl., 246, 2000, 480-492.

10. Interface development and local solutions to reaction-diffusion equations, SIAM J. Math. Anal., 32, 2, 2000, 235-260. (together with J.R.King)

11. Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, 50, 4, 2002, 541-560. 

12. Instantaneous shrinking of the support of a solution of a nonlinear degenerate parabolic equation. (Russian) Mat. Zametki 63 (1998), no. 3, 323--331; translation in Math. Notes 63 (1998), no. 3-4, 285--292

13. On sharp local estimates for the support of solutions in problems for nonlinear parabolic equations. (Russian) Mat. Sb. 186 (1995), no. 8, 3--24; translation in Sb. Math. 186 (1995), no. 8, 1085--1106

 Geometric Boundary Regularity Tests for PDEs and Asymptotic Laws for Wiener Processes

14. First boundary value problem for the diffusion equation. I.Iterated logarithm test for the boundary regularity and solvability, SIAM J. Math. Anal., 34, 6, 2003, 1422-1434.

15. Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation, Boundary Value Problems, 2, 2005, 181-199.

16. Necessary and sufficient condition for the existence af a unique solution to the first boundary value problem for the diffusion equation in unbounded domains, Nonlinear Analysis, 64, 5(2006), 1012-1017.

Optimal Control and Inverse Problems for PDEs

17. On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I. Well-posedness and Convergence of the Method of Lines, Inverse Problems and Imaging, 7, 2(2013), 307-340.

Dynamical Systems and Chaos Theory

18. A.U.Abdulla, R.U.Abdulla, U.G.Abdulla, On the Minimal 2(2k+1)-orbits of the Continuous Endomorphisms on the Real Line with Application in Chaos Theory, Journal of Difference Equations and Applications, 19,9(2013),1395-1416..