Shi Jin: Spring 2018:

- Physical Problems of different scales:
- * Basic physical equations in Quantum mechanics, classical mechanics, kinetic theory and hydrodynamics

- * The mathematical transition from microscopic to macroscopic scales:

- - The semiclassical limit of quantum mechanics

- - The Grad-Boltzmann limit to derive the Boltzmann equation

- - The fluid dynamical limit of kinetic equation

- Numerical Methods for Hyperbolic Systems of Conservation Laws and Compressible Flows
- * Mathematical Thoery of Hyperbolic Conservation Laws

- - Linear hyperbolic systems: characteristic method

- - Scalar nonlinear conservation laws: shocks, Rankine-Hugoniot jump condition, entropy condition, Riemann problem

- - Nonlinear systems of hyperbolic equations: riemann problems

* Numerical Methods for Hyperbolic Conservation Laws

- - stability and convergence theory of finite difference methods for linear problem

- - shock capturing Methods for nonlinear problems: Godunov and Roe methods, kinetic and relaxation methods

- - high resolution shock capturing methods: flux and slope limiters, TVD and ENO methods

- - multidimensional problems: operator splitting

- - source terms

- - Applications: gas dynamics, shallow-water, etc.

- Numerical Methods for Incompressible Flows
- * Introduction to Navier-Stokes equations

- - conservation laws and constitutive relation, scale, incompressible limit

- - various formulations: primitive variable, vorticity and impulse density

- * Numerical Methods

- - methods based on primitive varibale formulations: MAC scheme, projection methods, Poisson solver, Gauge method

- - methods based on vorticity formulation

- Level Set Methods for Interface Problems
- * Equations of Motions for Moving Interfaces

- - Theory of front evolution: formulation, total variation, weak solutions and entropy condition, curvature

- - The level set formulation

* Numerical Approximations to the Leves Set Equations

- - viscosity solutions, numerical schemes for Hamilton-Jacobi equations

- - high resolution schemes for the level set method

- - fast marching and fast sweeping methods

* Applications

- - various applications in geometry and physics

- Numerical Methods for Multiscale Problems
- * Domain decomposition method, heterogeneous multiscale methods, asymptotic-preserving methods, etc.

Fabian Waleffe, Fall 2016, Hydrodynamic instabilities, chaos and turbulence

Study of instabilities, bifurcations and exact coherent states primarily in natural convection (Rayleigh-Benard), Taylor-Couette flow and canonical shear flows (channels, pipes,...). Applied mathematical focus: the goal is to derive, explain and predict results that are observed in physical experiments from suitable governing equations using a combination of analysis, asymptotics and numerics.

Requires knowledge of differential equations, linear algebra, partial differential equations (equivalent of Math 320, 321, 322) and elementary mechanics (as in EMA 201, 202, Phys 247, 311,...). A first course in fluid mechanics is recommended (such as ME 363 or 563 or Math 705). Some Matlab (or python for scientific computing) experience will be necessary for homework and projects.

Amir Assadi, Spring 2017:

TOPOLOGICAL DATA ANALYSIS AND APPLICATIONS

Topology and geometry are branches of abstract mathematics with origins in the human perception of the physical world and the intuition formed to model space, time and mechanics. The major developments of the 20th century topology and geometry were abstract and far reaching generalization their origins and applications. The 21st century science and technology are driven by data and information sciences. Topology models data sets as abstract spaces, and offers powerful tools to meet the challenges problems of data science.

This course provides an introduction to topology and geometry from the viewpoint of applications to data science. As such, the lectures cover an overview of combinatorial, algebraic and differential topology in the context of computational problems arising in pattern recognition, feature extraction and their applications to brain research, systems biology and machine learning.

To benefit most from the lectures, the audience is encouraged to participate in the computation lab sessions that are designed for concrete applications in the above-mentioned subjects. Familiarity with a software package such as MATLAB or any high level programming language is needed for computational research, but not the theoretical aspects.

The course grade is based on a term paper that reports progress on a project in any of the application domains or pertinent theoretical research.

For Topological Data Analysis, see Afra Zomorodian's power points http://web.cse.ohio-state.edu/mlss09/mlss09_talks/3.june-WEN/zomorodian.pdf and

https://en.wikipedia.org/wiki/Computational_topology

Recommended Sources - Afra J. Zomorodian Topology for Computing More resources in the course syllabus.