Professor & Department Head, Mathematical Sciences

Welcome to the Department of Mathematical Sciences of the Florida Institute of Technology. Mission of the Department of Mathematical Sciences is to provide mathematical education with the goal of producing XXI century problem solvers, equipped with analytical tools and mathematical intuition to tackle a diverse range of contemporary problems; to push the boundary of knowledge in Mathematical Sciences and in emerging fields of Engineering and Natural Sciences through application of advanced methods of applied mathematics and the development of new mathematics capable of solving challenging problems of society.

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 or Operations Research. 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.

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

I am a Principal Investigator of the National Science Foundation funded REU Site on Partial Differential Equations and Dynamical Systems. Each summer from 2014 to 2016, selected undergraduate students participated in my summer REU Site (NSF award 1359074), which is designed to involve undergraduate students in innovative research in nonlinear partial differential equations, optimal control and inverse problems for systems with distributed parameters, and dynamical systems and chaos theory, while utilizing modern tools of mathematical and numerical analysis. Students had a great opportunity to pursue hands-on, original research on the frontier of modern mathematics. The goal of the REU Site is to equip students with intuitional and rigorous proof proof skills which allow them to tackle cutting-edge open problems in mathematics. See more details in http://math.reu.fit.edu

In Fall 2017 I teach new course MTH 6330 Calculus of Variations and Optimal Control Theory, which will cover classical topics of calculus of variations, such as Euler-Lagrange equation, critical points, semilinear elliptic PDEs, optimal control of ODEs and PDes, Pontryagin's maximum principle, differentiability in Banch spaces, gradient methods, regularization. Prerequisite is PDE 2 and Functional Analysis on the level of first seven chapters of Evans, PDEs and first five chapters of Yosida, Functional Analysis.

I also involved in team teaching of the new course MTH 3663 Biomath, where my topic is Dynamical Systems and Chaos Theory in Biological Modeling.

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.

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 research developments ([2-5]) 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, a problem about the short-time behavior of interfaces for the various reaction-diffusion equations was solved in [11-13].

**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 [17], 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 [18].

**4. Optimal Control and Inverse Problems for PDEs**

In [20,21] 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 temperature and free boundary. This research project is motivated by the bioengineering problem on the laser ablation of biological tissues. 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 L_{2}-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. Recent papers of my research group on Optimal Control and Inverse Problems is given in [20-25].

**5. Dynamical Systems and Chaos Theory**

This project relates to two topics of Dynamical Systems and Chaos Theory: 1. Classification of periodic orbits for the continuous endomorphisms in the interval. This topic originates from the celebrated result by Sharkovski (1964) on the coexistence of periodic orbits of continuous maps on the interval, which presents a surprisingly simple and elegant hierarchy of distribution of periodic orbits according to their periods; 2. Asymptotic behavior of parameter dependent continuous maps, chaos phenomena and universal transition from periodic to chaotic behavior for nonlinear maps. This topic originates from the pioneering work of Feigenbaum (1978) on the universal transition route from periodic to chaotic behavior through period doubling bifurcations for the logistic type unimodal maps.

Recently my DSCT research team solved an open problem on the classification of second minimal odd orbits of the continuous endomorphisms on the interval. It is proved that there are 4k−3 types of second minimal (2k+1)-orbits, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses. We then revealed a fascinating universal law of distribution of periodic orbits in chaotic regime for one-parameter family of unimodal continuous maps on the interval, and very deep connection between Sharkovski ordering and universality in chaos. Recent papers of my research team are [26-28].

I run Mathematical Blog http://math.reu.fit.edu/blog/ where I post series of video lectures on three topics: Nonlinear Partial Differential Equations, Optimal Control and Inverse Problems for PDEs, Dynamical Systems and Chaos Theory

**Potential Theory and Partial Differential Equations:**

1. U.G. Abdulla, Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and its Consequences, *Calculus of Variations and Partial Differential Equations*, 2018, to appear; arxiv:1601.04184.

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

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

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

5. U.G. Abdulla, 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**

6. U.G. Abdulla, 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.

7. U.G. Abdulla, 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.

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

9. U.G. Abdulla, Reaction-diffusion in irregular domains, *J. Differential Equations*, 164, 2000, 321-354.

10. U.G. Abdulla, Reaction-diffusion in a closed domain formed by irregular curves, *J. Math. Anal. Appl.*, 246, 2000, 480-492.

11. U.G. Abdulla & J.R. King, Interface development and local solutions to reaction-diffusion equations,* SIAM J. Math. Anal.*, 32, 2, 2000, 235-260.

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

13. U.G. Abdulla & R. Jeli, Evolution of interfaces for the nonlinear parabolic p-Laplacian type reaction-diffusion equations, European Journal of Applied Mathematics, 28, 5(2017), 827-853.

14. U.G. Abdulla, J. Du, A. Prinkey, Ch. Ondracek, S. Parimoo, Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption, *Mathematics and Computers in Simulation*, 153, November 2018, 59-82.

15. U.G. Abdullaev, 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

16. U.G. Abdullaev, 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**

17. U.G. Abdulla, 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.

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

19. U.G. Abdulla, 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**

20. U.G. Abdulla, 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.*

21. U.G. Abdulla, On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations: II. Convergence of the Method of Finite Differences, *Inverse Problems and Imaging*, Volume 10, Number 4(2016), 869-898.

22. U.G. Abdulla & J. Goldfarb Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems, *Inverse and Ill-posed Problems*, 26, 2(2018), 211-228.

23. U.G. Abdulla, E. Cosgrove, J. Goldfarb On the Frechet Differentiability in Optimal Control of Coefficients in Parabolic Free Boundary Problems, *Evolution Equations and Control Theory*, 6, 3(2017), 319-344.

24. U.G. Abdulla and B.Poggi Optimal Control of the Multiphase Stefan Problem, *Applied Mathematics & Optimization*, 2018, https://doi.org/10.1007/s00245-017-9472-7

25. U.G. Abdulla, V. Bukshtynov, A. Hagverdiyev Gradient method in Hilbert-Besov Spaces for the optimal Control of Parabolic Free Boundary Problems, *Journal of Computational and Applied Mathematics*, Volume 346, January 2019, 84-109.**5. Dynamical Systems and Chaos Theory**

26. 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.*

27. U.G. Abdulla, R.U. Abdulla, M.U. Abdulla & N.H. Iqbal Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos, International Journal of Bifurcation and Chaos, 27, 5(2017), 1-24. arxiv:1610.00814.

28. U.G. Abdulla, R.U. Abdulla, M.U. Abdulla & N.H. Iqbal Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering, submitted, arxiv:1701.02695