Applied Mathematics - University at Buffalo

Applied Mathematics at UB

The University at Buffalo Mathematics Department has a tradition of leading research in Applied Mathematics. Our applied group's interests include

biological and stochastic modeling
bifurcation and stability theory
dynamical systems
material science and crystal growth
nonlinear waves and optics
perturbation methods
population dynamics
scientific computation

Scientific computing plays an important role in many of these research projects. Several faculty have a network of SGI and Sun workstations with a 2-processor SGI Origin 200. The Math Department maintains a cluster of Sun workstations and PCs for general computing use and faculty have access to university-operated Sun and SGI timeshares. In addition, faculty have access to the Center for Computational Research (CCR) at UB which has cutting-edge supercomputers and visualization hardware.

The Applied Math group runs a weekly seminar series, with speakers from within the department and the unversity, as well as outside visitors. In some years, the seminars have been organized around a yearly topic. Example topics from past years include: Mathematical Finance, Thin Films, Phase Field Models, Stochastic Differential Equations, Crystallography.

In Spring 2006 the department hosted a workshop on Nonlinearity and Randomness in Complex Systems.

The Mathematics Department has a position opening for fall 2008 at the level of tenure-track assistant professor. We are seeking candidates in applied mathematics, with particular interest in modeling and simulation, scientific computing, applied probability and stochastic processes. The hire is part of a larger effort to build on UB's strategic strength in information and computing technology. The firm deadline for applications is December 1.

Faculty in Applied Mathematics

There are currently 11 faculty in the Applied Math group. Below are their areas of research, and several representative recent publications.

Gino Biondini - (Ph.D., University of Perugia, Italy) Nonlinear waves, integrable systems and their applications

The study of physical phenomena by means of mathematical models often leads to certain nonlinear partial differential equations which reveal a surprisingly rich mathematical structure. The study of these equations thus offers a unique combination of interesting mathematics and concrete physical/technological applications. Professor Biondini's research has two main goals: The first goal is to understand the properties of these equations and their solutions. This kind of research is usually called the study of "integrable systems", and requires a combination of techniques from different branches of mathematics. The second goal is to study the application of these nonlinear and/or stochastic systems to concrete physical situations, with the aim of obtaining results of practical usefulness. This often requires studying the combined effects of several kinds of perturbations which are often stochastic in nature, and can be done using exact methods, approximations (such as modeling, asymptotics and perturbative techniques), numerical methods (numerical modeling, Monte-Carlo simulations and variance reduction techniques) or combinations of all these approaches. Specific applications considered by Prof. Biondini are optical fiber communications, nonlinear optics and water waves.

Selected publications:

G Biondini, "Line soliton interactions of the Kadomtsev-Petviashvili equation", Phys. Rev. Lett. 99, 064103:1-4 (2007)

G Biondini, W L Kath and C R Menyuk, "Importance sampling for polarization-mode dispersion", J. Lightwave Technol. 14, 1201-1215 (2004). [Correction: J. Lightwave Technol. 24, 1065 (2006)]

J. Li, E. Spiller and G Biondini, "Noise-induced perturbations of dispersion-managed solitons", Phys. Rev. A 75, 053818:1--13 (2007)

G Biondini and Y. Kodama, "On the Whitham equations for the defocusing nonlinear Schroedinger equation with step initial data", J. Nonlin. Sci. 16, 435-481 (2006)

B. Prinari, M. J. Ablowitz and G. Biondini, "Inverse scattering transform for the vector nonlinear Schroedinger equation with nonvanishing boundary conditions", J. Math. Phys. 47, 063508:1--33 (2006)

M J Ablowitz, G Biondini and S Blair, "Nonlinear Schroedinger equations with mean terms in non-resonant multi-dimensional quadratic materials", Phys. Rev. E 63, 046605:1-15 (2001)

Cliff Bloom - (Ph.D., NYU) Partial differential equations, scattering theory and energy methods.

Bloom CO, High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries, Math Probl Eng 2, 333-365 (1996)
Bloom CO and Kazarinoff ND, Energy decay for hyperbolic systems of 2nd-order equations, J Math Anal Appl 132, 13-38 (1988)

Irwin Guttman - (Ph.D., Univ of Toronto) Statistical model selection procedures, design of experiments, Bayesian statistical inference

with M. Evans, Z. Gilula and T. Swartz, "Bayesian Analysis of Stochastically Ordered Distributions of Categorical Variables," Journal of the American Statistical Association, 92 (1997) 208-214.

"A Contrast of EB, modal and EM-algorithm estimates arising in the One-Way Analysis of Variance Situation," Statistics and Probability Letters, 28 (2000) 33-58.

with U. Menzefricke, "Posterior Distributions for Functions of Variance Components," Test, 12 (2003) 115-123.

with D. Pena and D. Redondas, "A Bayesian Approach for Predicting with Polynomial Regression of Unknown Degree," Technometries, 47 (2005) 23-33.

with M. J. Evans and T. Swartz, "Optimality and Computations for Relative Surprise Inferences," to be published in Canadian Journal of Statistics, 34 (2006).

Brian Hassard- (Ph.D., Cornell) Applied mathematics, bifurcation theory, precise numerical algorithms.

Du ZD, Hassard B, Precise computation of Hopf bifurcation and two applications, Dynamics of Continuous and Discrete Impulsive Systems A 8, 495-518 (2001).

Hassard B, An unusual Hopf bifurcation: The railway bogie, Int J Bifurcat Chaos 10 503-507 (2000)
Jeng JH, Hassard B, The critical wave number for the planar Benard problem is unique, Int J Nonlinear Mech 34 221-229 (1999)

Hassard BD, Counting roots of the characteristic equation for linear delay-differential systems, J Differ Equations 136 222-235 (1997)

Hassard B, Shiau LJ, A special point of Z(2)-codimension three Hopf bifurcation in the Hodgkin-Huxley model, Appl Math Lett 9 31-34 (1996)

Mikhail Khenner - (Ph.D., Universite de la Mediterranee Aix-Marseille II, France; Ph.D., Perm State University, Russia) Numerical methods for problems with interfaces and free boundaries, computational materials science, mathematical models for the crystal growth, hydrodynamic stability, pattern formation, flows in non-uniform fluid systems.

Dr. Khenner works in the area of (primarily computational) modeling in materials science and crystal growth. He developed PDE-based models of micro(nano)-scale crystal growth on patterned substrates, grain-boundary grooving by electromigration in polycrystalline thin films and recently, of morphological relaxation and pattern formation on surfaces of solid thin films, caused by pulsed laser irradiation.

Recent publications:

M. Khenner, Influence of Pulsed Laser Heating on Morphological Relaxation of Surface Ripple, Physical Review E 72 (2005) 011604
http://www.math.buffalo.edu/~mkhenner/oscill_relax.pdf

M. Khenner, R.J. Braun Numerical Simulation of Liquid Phase Electro-Epitaxial Selective Area Growth, Journal of Crystal Growth 279 (2005) 213-228
http://www.math.buffalo.edu/~mkhenner/LPEESAG.pdf

M. Khenner, Motion of Contact Line of a Crystal Over the Edge of Solid Mask in Epitaxial lateral Overgrowth, Computational Materials Science 32 (2005) 203-216
http://www.math.buffalo.edu/~mkhenner/OverTheEdge.pdf

M. Khenner, Computation of the Material Indicator Function Near the Contact Line (in Tryggvason's method), Journal of Computational Physics 200(1) (2004) 1-7
http://www.math.buffalo.edu/~mkhenner/Note_on_mat_indicator.pdf

M. Khenner, Enhancement of Epitaxial Lateral Overgrowth by Vapor-Phase Diffusion, International Journal of Engineering Science 42 (2004) 1439-1457
http://www.math.buffalo.edu/~mkhenner/ELO_by_vapor_diffusion.pdf

For more information, see http://www.math.buffalo.edu/~mkhenner/research.html

Avner Peleg - (Ph.D., The Hebrew University of Jerusalem, Israel) Applied mathematics, nonlinear waves, pattern formation, stochastic processes.

Dr. Peleg's research interests are in applied mathematics of optical communications (in optical fibers and in the atmosphere), and in mathematical modeling of dynamics of phase transitions (with applications in materials science). The primary research effort concerns
propagation of pulses of light in multichannel optical fiber communication systems. In multichannel transmission many pulse sequences propagate through the same optical fiber and as a result, collisions between pulses from different sequences (corresponding to different channels) are very frequent and can lead to severe limitation on system performance. Since state-of-the-art multichannel systems use more than 100 channels and since the pulse sequences are random, the task of obtaining an accurate description of the dynamics is a very challenging one. One nonlinear effect that is particularly important in these systems is called delayed Raman response. One of the goals of this research is to understand the ways in which the interplay between delayed Raman response and other random and nonlinear processes affects pulse propagation and transmission quality. Another goal is to find relations between the dynamical behavior of the optical pulses and the dynamics of coherent patterns in other fields, such as pattern formation and turbulence.

Other research topics include:
(1) Propagation of multiple laser beams in atmospheric turbulence;
(2) Modeling of interface-controlled relaxation dynamics of two-phase systems.

Recent publications:

A. Peleg, Intermittent dynamics, strong correlations, and bit-error-rate in multichannel optical fiber communication systems, Phys. Lett. A, Vol. 360, 533-538 (2007).

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J.V. Moloney, Optimized multi-emitter beams for free-space optical communications through turbulent atmosphere, Opt. Lett., Vol. 32, 885-887 (2007).

Y. Chung and A. Peleg, Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed Raman response, Nonlinearity, Vol. 18, 1555-1574 (2005).

J. Soneson and A. Peleg, Effect of quintic nonlinearity on soliton collisions in optical fibers, Physica D, Vol. 195, 123-140 (2004).

M. Conti, B. Meerson, A. Peleg, and P.V. Sasorov, Phase ordering with a global conservation law: Ostwald ripening and coalescence, Phys. Rev. E, Vol. 65, 046117 (2002).

Bruce Pitman - (Ph.D., Duke) Applied mathematics, scientific computing.

Professor Pitman works in two distinct areas. One is granular materials. The goal of this project is to understand the behavior of materials composed of particles which range in size from about 100 microns to 1 cm. Very basic questions about these materials are not understood, questions like: What is a good constitutive model for these materials? How does deformation proceed? What causes failure of a sample of material? What role does interstitial fluid play? What loads do these materials generate on storage vessels?

Another aspect of Pitman's work on granular materials involves large-scale geophysical mass flows -- for example debris flows, lava flows, mudslides. Together with colleagues in Geology, Geography and Engineering, Pitman has begun a modeling and computing program whose goal is a better understanding of these flows. These models and simulation capabilities, together with visualization and communication tools, will be used to assist scientists and public safety officials in hazard risk assessment.

The second of Pitman's research interests is renal hemodynamics. Together with colleagues in math and physiology, Pitman has been studying the dynamics of Tubuloglomerular feedback system (TGF). TGF acts to regulate blood flow into the nephrons, the primary functional unit in the kidney. It is in the nephrons that blood is concentrated into urine. Bringing together techniques of applied mathematics and advanced scientific computing, the models developed by this group offer insight into the bifurcation phenomenon and oscillatory flow measurements seen in experimental work.

Recent papers:

"A Model of Granular Flow over an Erodible Surface" Discrete and Continuous Dynamical Systems: Series B "Mathematical Modeling, Analysis and Computations" vol 3, number 4 (2003). This is a special issue dedicated to David G. Schaeffer's on the occasion of his 60th birthday. (pdf)

"Parallel Adaptive Numerical Simulation of Day Avalanches over Natural Terrain" A presentation of our simulation environment to solve the 'thin layer equations' modelling dry geophysical flows over terrain (pdf). This paper will appear in J. Volcanology and Geothermal Research.

"Kinematics of sand avalanches using particle-image velocimetry" J. Sediment Res. vol 71 (2001) p 355.

"The Mechanics of Particle-Fluid Flows at high Solids Volume Fraction" (postscript version, with figures). This paper presents a model of fluidized beds that incorporates solids-like stresses, and looks at the stability of slows. It also introduces a model for computing particle flows in a fluid, based on solving DEM-like forces and the Navier-Stokes equations. (IUTAM Speciality Conference on Segregation in Granular Materials, (2000) A. Rosato and D. Blackmore (eds.), Kluwer.

"Forces on Bins: The Effect of Random Friction" (postscript version, with figures). This paper studies the effect on stresses of a random component of friction, in models of granular material. First, the classic Janssen solution for stresses in a bin is reexamined; assumptions about the size of the random friction translate more-or-less directly into the size of stress fluctuations. Next, the equilibrium equations with random friction effects are solved numerically for a Mohr-Coulomb material. Here, the size of fluctuations in the coefficient of wall friction (i.e. a boundary condition) is seen to be the most significant contribution to stress fluctuations. (Phys. Rev. E, vol 57 (1998) p 3170)

Jim Reineck - (Ph.D., Wisconsin-Madison) Dynamical systems, Conley Index.

Mrozek M, Reineck JF, Srzednicki R, The Conley index over a base, T Am Math Soc 352 4171-4194 (2000)
Mrozek M, Reineck JF, Srzednicki R, The Conley index over the circle, J. Dynam. Differential Equations 12 385-409 (2000)

Mischaikow K, Mrozek M, Reineck JF, Singular index pairs, J. Dynam. Differential Equations 11 399-425 (1999)

Gedeon T, Kokubu H, Mischaikow K, Oka H, Reineck JF, The Conley index for fast-slow systems. I, One-dimensional slow variable, J. Dynam. Differential Equations 11 427-470 (1999)

Reineck JF, The connection matrix in Morse-Smale Flows. 2., T Am Math Soc 347 2097-2110 (1995)

John Ringland - (Ph.D., Texas-Austin) Applied mathematics, bifurcation theory, mathematical modeling, computational mathematics.

Professor Ringland works in the qualitative and quantitative analysis of deterministic and stochastic dynamical systems:

Mohammed-Awel J, Kopecky K, Ringland J, A situation in which a local nontoxic refuge promotes pest resistance to toxic crops, Theor. Pop. Biol. 71, 131-146 (2007)

Brucks K, Ringland J, Tresser C, An embedding of the Farey web in the parameter space of simple families of circle maps, Physica D 16, 142-162 (2002)

Brian Spencer - (Ph.D., Northwestern) Materials modelling, free boundary problems, instabilities and pattern formation.

Professor Spencer's research interests are in the applied mathematics of materials. The research combines physics-based mathematical models with asymptotic, analytic and numerical methods to describe growth processes, instabilities and microstructure formation in materials. Specific research programs include:

instabilities and pattern formation in strained alloy film deposition
formation of quantum dots in strained solid films
corner regularizations in crystal growth models