This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Current students' projects

Background: Discrete Calculus. An Introduction.

## Physics simulations and discrete exterior calculus

What questions do you plan to answer?

- What parts of calculus, ODEs, PDEs, differential geometry, etc have discrete counterparts?
- When do the latter approximate the former?
- When they do, how well?
- What phenomena discrete in nature can be modeled by discrete calculus?
- Can continuous, discrete, and mixed phenomena be described by a single theory?

Is this of interest to the research community?

- The issues are of broad interest in these fields, as well as physics, computer science, etc.

What methods will you use? What will the student's role be?

- The methods come from algebraic topology and adjacent fields developed throughout 20th century.
- The methods have to be elaborated and then specialized to a range of applications.
- The students will progressively
- create simple code to illustrate the methods;
- create simple simulations for well known applications;
- write tutorials for the use of the code;
- devise more advanced and computationally intensive models.

How computationally intensive is the research?

- The output will be computer programs and mathematics illustrating each other;
- A lot of initial models can be implemented with Excel, progressing to Mathematica, MATLAB, and higher order languages;
- Realistic models will require computational power beyond a desktop.

Narrowly, the goal of this project has been to create simulations of heat transfer, wave propagation, liquid flow etc, based on a modern view of calculus. This approach is based on differential forms. Normally, one deals with partial differential equations, i.e., equations with respect to derivatives of the quantities involved, and then "samples" or "discretizes" these PDEs via finite differences to create a simulation. Instead, we look at the derivations of these PDEs and, based on the physics, represent each quantity as a discrete differential form of appropriate degree determined by its nature. These equations are ready-made simulations (similar to cellular automata) of the processes. The main advantage of this bottom-up approach is that the laws of physics (conservation of energy, mass, etc.) are satisfied *exactly* rather than approximately, as is the case with discretization of PDEs.

## Plans

- Learn some discrete calculus.
- Design simulations based on the physics of the processes:
- Design models outside physics:
- Develop and code tools of discrete exterior calculus, starting with Excel:
- cubical complexes (also simplicial complexes, cell complexes),
- the boundary operator,
- duals of these complexes,
- discrete differential forms,
- the exterior derivative of discrete forms,
- the Hodge star operator for forms,
- the Laplacian.

- Study convergence of the discrete to the continuous.

## Past participants and their contributions

From Computational science training: 2012 and before:

- Zachary Ahlers
- Christopher Means
- Jack Goodman
- Joseph Snyder
*The topology of data* - James Molchanoff
*3D image analysis* - Misha Dowd
*Evaluating image-to-image search* - Julie Lang
*Modeling with discrete exterior calculus*