The undergraduate research done for credit comes from the Senior project, required for Discrete Math majors. More and more, applied (and other!) majors are pursing research for credit. Here is a list of how's been doing what recently.,

Since the Senior Project is only required for the smaller Discrete Math program, at this time, we do not have a long record of these projects. Below is the current list of those students who have been or are going through the project.Fall 2003:ACE Lab Seminar HYND,RYAN CHARLES MATH MATH4801McCuan LOPEZ,ROBERTO EDUARDO MATH MATH4801McCuanFrancisco, Kovalev, Nichols and Warmbrand are working on combinatorial games. We are currently look at Stephen Barnes's proof of the periodicity of Chomp. We will be looking at some other mirere games later this term

FRANCISCO,WILLIAM KENNETH MATH MATH4803Morley KOVALEV,VICTOR SERGEYEVICH CMPE MATH4803Morley NICHOLS,CATHERINE MICHELLE MATH MATH4803Morley WARMBRAND,CASEY M DMTH MATH4803Morley STIMPSON,ANDREW JAY PHYSS MATH4803Heil Continuing Summer 2003 REUs in Small World Networks CALLAGHAN,THOMAS SHIELDS DMTH MATH4999MUCHA WARMBRAND,CASEY M DMTH MATH4999MUCHA LUDERS,BRANDON DOUGLAS AE S MATH4803Lacey Compuational Number Theory Senior Projects: WELLS,JONATHAN MESHAD DMTH MATH4080 LIU WILSON,JOSHUA SCOT DMTH MATH4080Morley Simulated Annelin POWELL,MATTHEW ALEXANDER DMTH MATH4080Yu WARMBRAND,CASEY M DMTH MATH4090WANG Fractal Geometry Note: About 25 people are signed up for the one hour Putnam Class.Summer 2003:BELL,WILLIAM NATHANIEL CS/DMATH MATH4804MUCHA Computer Graphics TSUJI,TOMONORI CS/DMATH MATH4080Shonkwieler Math Biology WARMBRAND,CASEY M DMTH MATH4080WANG Fractal Geometry Spring 2003: ACE Lab Seminar BENT,STEPHANIE MARSHA MATH MATH4801McCuan HYND,RYAN CHARLES MATH MATH4801McCuan LOPEZ,ROBERTO EDUARDO MATH MATH4801McCuan PALAGHITA,TUDOR IOAN AE MATH4801McCuan SAXTON,CARINA RENATA MATH MATH4801McCuan BELL,WILLIAM NATHANIEL CS/DMTH MATH4804MUCHA Computer Graphics CHOE,JANG WON DMTH MATH4090MORELY Combinatorial Games TSUJI,TOMONORI CS MATH4090Shonkwieler Math Biology OGILVIE,ROBERT JAMES CS/DMTH MATH4090Trotter SKOOG,DAVID WILLIAM CS/DMTH MATH4090Trotter JAEHN,MATTHEW EDWARD DMTH MATH4090YU OK,KWANG HO DMTH MATH4090YUFall 2002:John McCuan's ACE Lab Demonstration Seminar ELMS,JEFFREY DYLAN CMPE MATH4801McCuan HYND,RYAN CHARLES MATH MATH4801McCuan LOPEZ,ROBERTO EDUARDO MATH MATH4801McCuan Workaround for a Graduate Course PELLEGRINI,RUSSELL MATH MATH4803Landsberg MWF 0905-0955

Student | Topic | Sponsor |

## 2003-04 | ||

Casey Warmbrand, Eric Rahm (?) | Fractals | Professor Wang |

## 2002-03 | ||

Tomori Tsuji | Math Biology Modelling | Professor Shonkweiler |

Nathan Bell | Computer Graphics | Professor Mucha |

David Skoog and Jim Ogilvie | Topics in Combinatorics | Professor Trotter |

## 2001-02 | ||

Patty Pichardo, Claire Conner, Michael Tiffany, Tye Howard (CS) |
Topics in Cryptography | Professor Lacey |

## 2000-01 | ||

Taeksee Oh | Artifical Intelligence | Professor Bellinfante |

** Nathan Bell's Senior Project Description**

When asked about the single most difficult shot in "Shrek", producerJeffrey Katzenberg replied "It's the pouring of milk into a glass."It is both tedious and time-consuming to animate natural phenomenamanually. Moreover, given people's familiarity with naturalphenomena, it is unlikely that a manual animation would be convincing,in a photorealistic sense. For these reasons, physics-based modelsare highly desirable in computer graphics simulations of natural phenomena.

My present research, in collaboration with Peter J. Mucha in Mathematics, aims to produce a computationally efficient,physics-based model of granular media for computer graphics. The computational efficiency of our model is due to the use of differentmethods for different physical scales; in computational physics terminology, our approach is multiscale. Our objective is a realistic animation of a sand hour glass.

At the smallest physical scale we use a microscopic model of particle interactions. Our approach uses molecular dynamics techniques, a well-established practice in Computational Physics simulations of granular media. Three different forces govern particle interactions:

The sand hour glass illustrates both of these properties. The sand though an hourg lass flows at a constant rate due to the frictional load-bearing of the walls. The sand at the base forms a heap with a slope equal to the angle of repose. To provide a visually convincing animation, it is important for a model of granular media to exhibit these characteristics.

Presently, we have a microscopic model capable of modeling the hour glass or a landslide, among other systems. The model isimplemented in the C++ language and rendered using the OpenGL graphics library. Our implementation uses techniques and practices from Computational Physics, Applied Mathematics, and Computer Science. Among these techniques are:

Particles are modeled as spheres which interact with boundaries.Boundaries can be finite planes as well as curves rotated about anaxis. These objects are sufficient to accurately model the hour glass. Microscopic models are appropriate for systems of thousands or tens ofthousands of particles. However, such models are too computationally expensive to apply to systems of hundreds of thousands or millions ofparticles. Our focus now is the development of a multiscale algorithm capable of communicating the essential physical interactions between the microscopic simulation and a macroscopic model of the "bulk" ofthe system. At any time instant, the vast majority of sand particles in the hourglass are stagnant. From an animation standpoint, the detailed microscopic interactions in the bulk are unimportant to therealism of the simulation. This observation will allow us to significantly reduce the computational cost of the simulation by using the microscopic model for particles near the surface of the volume andusing an appropriate macroscopic model for the bulk.

As in any suchmultiscale approach, the key questions to be answered are in the interface between the microscopic and macroscopic models. Answering these questions in a satisfactory manner will enable us to animate hour glass systems with a far greater number of particles, since the vast majority of the particles will be suitably approximated by the macroscopic model, the remaining limiting factor being the microscopic molecular dynamics simulation of the far smaller number of particlesin active motion near the surfaces.

Georgia Tech requires Discrete Math majors to complete a two-semestersenior project. Initially, the prospect of two semesters of researchwas daunting, as I have had little previous research experience. In late August I approached Peter Mucha with a research idea. From my initial idea, simulating marbles flowing through a funnel; grew our present objective, simulating granular media. This topic fits particularly well with Prof. Mucha's other research interests such as efficient numerical methods for simulating sedimenting suspensions.S ince August, I have been exposed to a variety of subjects with which I had no prior experience. Topics such as numerical methods forsolving ODE and molecular dynamics were previously unknown to me. Using these techniques in my research has led me to pursue relevant courses. In short, this experience has piqued my interest in research.

Source code and Images of the project can be found at: http://www.schwanz.org/~nathan/ The most recent images are here:http://www.schwanz.org/~nathan/screenshots/misc/stress/ (colored by stress)and here:http://www.schwanz.org/~nathan/screenshots/v0.1.8/ (colored by velocity)