Ibrahim Fatkullin
 Associate Professor, Mathematics
 Associate Professor, Applied Mathematics  GIDP
 Member of the Graduate Faculty
Contact
 (520) 6212246
 Mathematics, Rm. 602
 Tucson, AZ 85721
 ibrahim@math.arizona.edu
Bio
No activities entered.
Interests
Research
Applied and computational mathematics, mathematical physics, mathematical models related to liquid crystals and polymers.
Courses
202324 Courses

Independent Study
MATH 499 (Spring 2024) 
Thesis
MATH 910 (Spring 2024) 
Topics In Applied Math
MATH 577 (Spring 2024) 
Directed Research
MATH 492 (Fall 2023) 
Independent Study
MATH 599 (Fall 2023) 
Intro Ord Diff Equations
MATH 254 (Fall 2023) 
Thesis
MATH 910 (Fall 2023)
202223 Courses

Directed Research
DATA 492 (Spring 2023) 
Dissertation
MATH 920 (Spring 2023) 
Independent Study
MATH 499 (Spring 2023) 
Theory of Probability
MATH 464 (Spring 2023) 
Calculus II
MATH 129 (Fall 2022) 
Independent Study
MATH 499 (Fall 2022) 
Independent Study
MATH 599 (Fall 2022) 
Intro Ord Diff Equations
MATH 254 (Fall 2022)
202122 Courses

Honors Thesis
MATH 498H (Spring 2022) 
Independent Study
MATH 599 (Spring 2022) 
Intro to Linear Algebra
MATH 313 (Spring 2022) 
Theory of Probability
MATH 464 (Spring 2022) 
Honors Thesis
MATH 498H (Fall 2021) 
Independent Study
MATH 599 (Fall 2021) 
Theory of Probability
MATH 464 (Fall 2021)
202021 Courses

Theory of Probability
MATH 464 (Spring 2021) 
Thesis
MATH 910 (Spring 2021) 
Independent Study
MATH 599 (Fall 2020) 
Intro to Linear Algebra
MATH 313 (Fall 2020)
201920 Courses

Honors Independent Study
MATH 399H (Spring 2020) 
Intro Ord Diff Equations
MATH 254 (Fall 2019)
201819 Courses

Honors Thesis
MATH 498H (Spring 2019) 
Thesis
MATH 910 (Spring 2019) 
Calculus II
MATH 129 (Fall 2018) 
Honors Thesis
MATH 498H (Fall 2018) 
Intro Ord Diff Equations
MATH 254 (Fall 2018)
201718 Courses

Directed Research
MATH 492 (Spring 2018) 
Independent Study
MATH 599 (Spring 2018) 
Math Prin Numeric Anls
MATH 475B (Spring 2018) 
Anls Ord Diff Equations
MATH 355 (Fall 2017) 
Math Prin Numeric Anls
MATH 475A (Fall 2017)
201617 Courses

Intro Ord Diff Equations
MATH 254 (Summer I 2017) 
Dissertation
MATH 920 (Spring 2017) 
Math Prin Numeric Anls
MATH 475B (Spring 2017) 
Dissertation
MATH 920 (Fall 2016) 
Math Prin Numeric Anls
MATH 475A (Fall 2016) 
Theory of Probability
MATH 464 (Fall 2016)
201516 Courses

Internship
MATH 593 (Summer I 2016) 
Dissertation
MATH 920 (Spring 2016) 
Independent Study
MATH 499 (Spring 2016) 
Math Prin Numeric Anls
MATH 475B (Spring 2016)
Scholarly Contributions
Journals/Publications
 Xue, J., Sethuraman, S., & Fatkullin, I. (2017). Hydrodynamics for a class of dynamic Young tableau. Have submitted to Elec. J. Probab..More infoDifferent hydrodynamic limits for the shape of a class of random dynamic Young tableau are found in different scales depending on the type of interactions.
 Xue, J., Xue, J., Sethuraman, S., Sethuraman, S., Fatkullin, I., & Fatkullin, I. (2019). On Hydrodynamic Limits of Young Diagrams. Elec. J. Probab. (accepted subject to minor revision), 43pgs.More infoWe consider a family of stochastic models of evolving twodimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding ‘dynamical’ limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.
 Fatkullin, I., & Xue, J. (2021). Limit shapes for Gibbs partitions of sets. Journal of Statistical Physics, 183(22), 23.
 Fatkullin, I., & Xue, J. (2020). Limit Shapes for Gibbs Partitions of Sets. Journal of Statistical Physics. doi:10.1007/s10955021027568More infoThis study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets, which lead to the sums of Poisson random variables. Under mild monotonicity assumptions, we study all possible scenarios arising from different asymptotic behaviors of the energy, and also compute local limit shape profiles for cases in which the limit shape is a step function.
 Xue, J., Sethuraman, S., & Fatkullin, I. (2020). On Hydrodynamic Limits of Young Diagrams. Elec. J. Probab., 25, Paper no. 58, 44pgs. doi:doi:10.1214/20EJP455More infoWe consider a family of stochastic models of evolving twodimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding ‘dynamical’ limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.
 Fatkullin, I., & Slastikov, V. (2017). Limit shapes for Gibbs ensembles of partitions. Journal of Statistical Physics, 172(6), 1545.
 Fatkullin, I., Sethuraman, S., & Xue, J. (2018). On Hydrodynamic Limits of Young Diagrams. Electronic Journal of Probability, 25(58), 144.More infoDifferent hydrodynamic limits for the shape of a class of random dynamic Young tableau are found in different scales depending on the type of interactions.
 Slastikov, V., & Fatkullin, I. (2018). Limit shapes for Gibbs ensembles of partitions. Journal of Statistical Physics, 172(6), 15451563. doi:10.1007/s1095501821177More infoWe explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulationfragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.
 Zhelezov, G., & Fatkullin, I. (2018). Coalescing particle systems and applications to nonlinear Fokker–Planck equations. Communications in Mathematical Sciences, 16(2), 463490. doi:10.4310/cms.2018.v16.n2.a8
 Fatkullin, I., & Slastikov, V. (2015). DIFFUSIVE TRANSPORT IN TWODIMENSIONAL NEMATICS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMSSERIES S, 8(2), 323340.More infoWe discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the DoiSmoluchowski kinetic equation and vortex dynamics in twodimensional systems. We also discuss moment closures for the kinetic equation and Landaude Gennestype free energy dissipation.
 Slastikov, V., & Fatkullin, I. (2014). Diffusive transport in twodimensional nematics. Discrete and Continuous Dynamical Systems  Series S, 8(2), 323340. doi:10.3934/dcdss.2015.8.323More infoWe discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the DoiSmoluchowski kinetic equation and vortex dynamics in twodimensional systems. We also discuss moment closures for the kinetic equation and Landaude Gennestype free energy dissipation.
 Fatkullin, I. (2013). A study of blowups in the KellerSegel model of chemotaxis. Nonlinearity, 26(1), 8194.More infoAbstract: We study the KellerSegel model of chemotaxis and develop a composite particlegrid numerical method with adaptive time stepping which allows us to resolve and propagate singular solutions. We compare the numerical findings (in two dimensions) with analytical predictions regarding formation and interaction of singularities obtained through analysis of the stochastic differential equations associated with the model. © 2013 IOP Publishing Ltd & London Mathematical Society.
 Fatkullin, I., & Slastikov, V. (2015). Diffusive transport in twodimensional nematics. Discrete and Continuous Dynamical Systems S, 8(2), 323340.
 Capponi, A., Shi, L., Fatkullin, I., & Capponi, A. (2011). Stochastic Filtering for Diffusion Processes With Level Crossings. IEEE Transactions on Automatic Control, 56(9), 22012206. doi:10.1109/tac.2011.2157404More infoWe provide a general framework for computing the state density of a noisy system given the sequence of hitting times of predefined thresholds. Our method relies on eigenfunction expansion corresponding to the FokkerPlanck operator of the diffusion process. For illustration, we present a particular example in which the state and the noise are onedimensional Gaussian processes and observations are generated when the magnitude of the observed signal is a multiple of some threshold value. We present numerical simulations confirming the convergence and the accuracy of the recovered density estimator. Applications of the filtering methodology will be illustrated.
 Fatkullin, I., Kovačič, G., & Eijnden, E. (2010). Reduced dynamics of stochastically perturbed gradient flows. Communications in Mathematical Sciences, 8(2), 439461.More infoAbstract: We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finitedimensional examples. Finally, we formally extrapolate the reduction technique to several infinitedimensional examples and derive equations of the stochastic kink motion in AllenCahntype systems. © 2010 International Press.
 Fatkullin, I., & Slastikov, V. (2009). Vortices in twodimensional nematics. Communications in Mathematical Sciences, 7(4), 917938.More infoAbstract: We study a twodimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended OnsagerMaierSaupe free energy may be decomposed into Landaude Gennestype and relative entropytype contributions. We then prove that in the high concentration limit the states of the system display characteristic vortexlike patterns and derive an asymptotic expansion for the free energy of the system. © 2009 International Press.
 Slastikov, V., & Fatkullin, I. (2009). VORTICES IN TWODIMENSIONAL NEMATICS ∗. Communications in Mathematical Sciences, 7(4), 917938. doi:10.4310/cms.2009.v7.n4.a6More infoWe study a twodimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended Onsager MaierSaupe free energy may be decomposed into Landaude Gennestype and relative entropytype contributions. We then prove that in the high concentration limit the states of the system display characteristic vortexlike patterns and derive an asymptotic expansion for the free energy of the system.
 Fatkullin, I., & Slastikov, V. (2008). On spatial variations of nematic ordering. Physica D: Nonlinear Phenomena, 237(20), 25772586.More infoAbstract: We present a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in densityfunctional theories, the suggested free energy is a functional of spatiallydependent orientation distribution, however, the nonlocal effects are taken into account via phenomenological elastic terms rather than by means of a direct paircorrelation function. In illustration of this approach we consider a simplified model with orientation parameter on a circle and reveal its relation to the complex GinzburgLandau theory. © 2008 Elsevier B.V. All rights reserved.
 Fatkullin, I., & Slastikov, V. (2005). Critical points of the Onsager functional on a sphere. Nonlinearity, 18(6), 25652580.More infoAbstract: We study Onsager's model of isotropicnematic phase transitions with orientation parameter on a sphere. We consider two interaction potentials: the antisymmetric (with respect to orientation inversion) dipolar potential and symmetric MaierSaupe potential. We prove the axial symmetry and derive explicit formulae for all critical points, thus obtaining their complete classification. Finally, we investigate their stability and construct the corresponding bifurcation diagrams. © 2005 IOP Publishing Ltd and London Mathematical Society.
 Slastikov, V., & Fatkullin, I. (2005). A Note on the Onsager Model of Nematic Phase Transitions. Communications in Mathematical Sciences, 3(1), 2126. doi:10.4310/cms.2005.v3.n1.a2More infoWe study Onsager's free energy functional for nematic liquid crystals with an orientation parameter on a unit circle. For a class of interaction potentials we obtain explicit expressions for all critical points, analyze their stability, and construct the corresponding bifurcation diagram. We also derive asymptotic expansions of the equilibrium density of orientations near the critical and zero temperatures.
 Fatkullin, I., & VandenEijnden, E. (2004). A computational strategy for multiscale systems with applications to Lorenz 96 model. Journal of Computational Physics, 200(2), 605638.More infoAbstract: Numerical schemes for systems with multiple spatiotemporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to Lorenz. Issues like consistency, accuracy, and efficiency are discussed in detail. The role of possible hidden slow variables as well as additional effects arising on the diffusive timescale are also investigated. As a byproduct we obtain a rather complete characterization of the effective dynamics in Lorenz model. © 2004 Elsevier Inc. All rights reserved.
 Fatkullin, I., & VandenEijnden, E. (2004). A computational strategy for multiscale systems with applications to Lorenz 96 model. Journal of Computational Physics. doi:10.1016/j.jcp.2004.04.013More infoNumerical schemes for systems with multiple spatiotemporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to Lorenz. Issues like consistency, accuracy, and efficiency are discussed in detail. The role of possible hidden slow variables as well as additional effects arising on the diffusive timescale are also investigated. As a byproduct we obtain a rather complete characterization of the effective dynamics in Lorenz model.
 Fatkullin, I., & VandenEijnden, E. (2003). Statistical Description of ContactInteracting Brownian Walkers on the Line. Journal of Statistical Physics, 112(12), 155163.More infoAbstract: The distribution of interval lengths between Brownian walkers on the line is investigated. The walkers are independent until collision; at collision, the left walker disappears, and the right walker survives with probability p. This problem arises in the context of diffusionlimited reactions and also in the scaling limit of the voter model. A systematic expansion in correlation between neighbor intervals gives a series of approximations of increasing accuracy for the probability density functions of interval lengths. The first approximation beyond mere statistical independence between successive intervals already gives excellent results, as established by comparison with direct numerical simulations.
 Fatkullin, I., & Hesthaven, J. S. (2001). Adaptive HighOrder FiniteDifference Method for Nonlinear Wave Problems. Journal of Scientific Computing, 16(1), 4767.More infoAbstract: We discuss a scheme for the numerical solution of onedimensional initial value problems exhibiting strongly localized solutions or finitetime singularities. To accurately and efficiently model such phenomena we present a full spacetime adaptive scheme, based on a variable order spatial finitedifference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity.
 Fatkullin, I., Kladko, K., Mitkov, I., & Bishop, A. R. (2001). Anomalous relaxation and selforganization in nonequilibrium processes. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 63(6 II), 067102/1067102/4.More infoAbstract: A model of dynamic selforganization is introduced. It is conjectured that the model can also be applied to stretchedexponential relaxation in many biological and physical systems. For demonstration purposes, the properties of selforganization in a simple model are studied.
Presentations
 Fatkullin, I. (2021, May). Gibbs measures on partitions: limit shapes and hydrodynamic limits. SIAM Dynamical Systems. online.
 Fatkullin, I. (2021, October). Gibbs measures on partitions and their evolutions. AMS Fall Western Virtual Sectional Meeting. online.
 Fatkullin, I. (2019, April). Limit shapes for Gibbs ensembles of partitions. Materials Research Society Spring Meeting. Phoenix, AZ: MRS.
 Fatkullin, I. (2019, December). Limit shapes for Gibbs ensembles of partitions. New Trends in the variational modeling and simulation of liquid crystals. Erwing Schroedinger Institute, Vienna, Austria: ESI.
 Fatkullin, I. (2019, May). Gibbs Ensembles of Partitions: from limit shapes to hydrodynamic limits. Applied and Computational Mathematics seminar. University of Edinburgh, UK: University of Edinburgh, INI.
 Fatkullin, I. (2019, Spring). Gibbs ensembles of partitions: from limit shapes to thermodynamic limits. Mathematical Design of New Materials. Isaac Newton Institute, Cambridge, UK: INI.
 Fatkullin, I. (2016, June). A model of aggregation and limit shapes of Young diagrams. Frontiers of Applied and Comp. Math.. NJIT.
 Fatkullin, I., & Slastikov, V. (2014, April). An aggregation model and discotic liquid crystals. AMS Western Spring Sectional Meeting. Albuquerque, NM: AMS.
 Fatkullin, I., & Slastikov, V. (2014, April). Diffusive transport in twodimensional nematics. AMS Western Spring Sectional Meeting. Albuquerque, NM: AMS.