Master's Degree Projects


My long term interests are to learn how to design software architectures and mathematical models that allow me to model high level function that has been abstracted from the known details of biological information processing. Of course, this sort of interest covers a lot of ground, so I have potential Master's level projects in many areas. I have many Ph.D level projects too, but that is another story!

The basic research areas are discussed below and as you can see they are both very exciting and very daunting as they combine so many disciplines. What I do for a Master's project is to work with each of you who is interested in finding a reasonable subset of any of these things that is doable within the approximately 18 month MS time frame. And of course, make sure it is fun! areas.

  • Mathematical Things
  • I am very interested in exploring some ideas in algebraic topology such as CW complexes. I think there is a connection to graphs as well to the ligand - receptor model discussed below in psychiatric disease.
  • Differential geometry and its relationship with physics.
  • Passing from classical physics to a quantum physics generally involves mapping the variable for position ``x'' and the variable for momentun ``p = mv'' - to self adjoint hermetian operators on a complex Hilbert space. The eigenvalues of such operators are real and form a discrete countably infinite sequence. Physicists interpret the self-adjoint hermetian operators as quantities whose observed values are these eiqenvalues. Thus things like position and momentun can only occur in discrete packets and that matches what they see in experiments. This process of mapping variables to operators is called ``quantization'' and it is full of interesting master's level projets which have a nice bit of analysis in them.
  • A new book on introductory string theory came out and it is nicely accessible for a variety of master's level projects that mix some physics and mathematics.
  • Mathematical Teaching
  • I have been reading some papers about brain circuits that appear to be critical for counting and even perhaps for algebra. Interesting master projects could be developed from a combination of reading in this area (these are all new papers -- 2003 and 2004) and connecting them to mathematical pedagogy.
  • Genomic Models
  • There are special types of gene circuits called regulatory circuits which are essentially multiple input - multiple output control devices. This would look something like this: [N input slots] --genes-- [M output slots] where different combinations of inputs both spatially and temporally determine a reshuffling of how the gene sequence is used to create a protein. The M output slots indicate that there may be many such possible proteins from one such regulatory unit. It would be interesting to model this process in a variety of ways such a graphs, systems of equations, boolean networks and so forth. There is a lot of data we could use to verify the models also.
  • The flip side of the above is to consider data from some source and try to fit a regulatory model to it. Currently, I have seen interesting approaches using tools from computational algebra. So there are nice master's projects in here that would use both analysis and algebra.
  • Psychiatric Disease Models: there are very few models of psychiatric disease such as schizophrenia and depression. The current literature suggests that these diseases are probably caused by a combination of miswiring in the brain during development and problems with the dopamine - serotonin pathways which modulate our behavior. While sounding complicated, there are many things we can do to build interesting models that are not too complicated that may shed insight.
  • A low level approach is to look at how neurotransmitters like dopamine and serotonin bind to specialized molecules called receptors on the surface of cells. Current theory for such interactions comprises ligand - receptor interaction models, comes from about 1996 and should be updated. I have some ideas about interesting graph theory models that might be useful.
  • Models of serotonin and dopamine pathways tied to what we know about how these pathways are disrupted in the brains of diseased individuals. Their abnormal brain chemistry gives us powerful clues as to how we should shape our models for a healty individual. It also gives us the potential for building a model which exhibits diseased and healthy behavior for various parameter sets. This could be very useful in developing a better understanding of these diseases.
  • Modeling Biological Information Processing: small scale to large scale.
  • Models of how the action potential of an excitable nerve cell can be used for extracting information about the inputs to the cell. The cell can be interfaced with a sensor which reads a waveform induced through nonlinear coupling or the action potential can be read directly. In either case, the underlying mathematics is that systems of ordinary differential equations. I am using these models for the classification and recognition of toxins.
  • Models of cortex that have been influenced by some nice building blocks due to Stephen Grossberg and colleagues. There is a lot of evidence that at birth, cortical tissue is fairly uniform. Such cortical tissue is called isocortex. Once exposed to environmental input, it is reshaped into the specialized cortical tissue we see in auditory (sound) and visual (photons) cortex. Somehow, outputs from specialized cortex are combined in a process called information fusion in other cortical areas loosely collected into a group called the associative cortex. Grossberg has come up with very nice computational circuits that may underly isocortex processing. I am building models using these ideas for a variety of purposes. The basic tools are simple ordinary differential equation systems, graph theory and agent based modeling.
  • The outputs from specialized cortical models group together in a coordinated fashion for short spans of time in narrowly focused spatial groupings. I am interesting in models that can give insight into how these spatially and time modulated groupings arise and then how they combine to form local computational modules. This process may be able to explain the information fusion that goes on in the associative cortex.
  • The associative cortex models above are typically modulated by the neurotransmitters dopamine and serotonin. A model that uses this kind of modulation must use ideas from the brain circuitry we see in the brain stem area.
  • Immune System Models: Antigens are short pieces of foreign proteins which are found in the body after infection and injury. One part of the immune system creates a molecule which is kind of like a cradle. It is called a Major Histocompatibility Complex Class I molecule or MHCI for short. A short string of amino acids called a peptide is placed into this cradle to create a peptide MHCI complex, p-MHCI for short. Some of this stuff I am beginning to work with Jo Hoffacker on so she will be involved as well -- wow, two faculty members for the price of one!
  • Models of how the short peptide are created from foreign material could have many uses.
  • Flavivirus (dengue fever, west nile fever and yellow fever etc.) are known to increase the number of p-MHCI complexes that an infected cell puts on its surface. Hence, we would think that the infected cells become so visible to the immune system that our bodies soon clear the infection. However, the flavivirus pathogens exploit this increase in clever ways. It appears that this large increase in p-MHCI surface molecules leads to an immune system response that is very broadly tuned rather than being explicitly targeted to just the flavivirus infected cell. Thus, healthy cells are killed and the resulting collateral damage may be why 5 - 10% of flavivirus infections lead to serious health complications. So I am interested in modeling how this broadly tuned immun system response is created. Right now, the students in my MTHSC 974 class are thinking about these models using game theory as a mathematical starting point.
  • We can find a system of ordinary differential equations to model the host (that's us) - pathogen (flavivirus). Studying that is of great interest and uses pretty standard tools. However, we can also analyze the system using Jo's time scales approach which promises to give us some additional insights.
  • Each host - pathogen model is based on how that human builds her immune system. So a more interesting model is to use replace the human with a human population model and see what insights the new model can bring. Again, this can be standard sytems of ordinary differential equations or time scale systems. Either way, there are lots of interesting questions to ask giving rise to many reasonable master's level problems.
  • Agent Based Modeling: there are many models that don't fit well into a equation based framework. For example, in the flavivirus models discussed above, the next step is to ask how the mosquitos fit in. They harbor the virus for passage on to humans with a blood feeding. This can also be modeled with systems of differential equations giving a three pronged interaction: humans, flavivirus and mosquito. The big quesiton is however difficult to pose quantitatively. In the world, there are biogeographical zones in which mosquitoes with certain genetic signatures live and there are similar zones for human populations. How can we build a model that allows human trafficking between zones and mixing of mosquito populations and so forth? Models based on autonomous agents may be of help and I have been studying them for awhile
  • Use existing agent based modeling packages -- most in Java -- to build simple yet interesting models of portions of the immune system, information fusions and so forth.
  • Use a simulation package like SimPy to build simple models. We can ask and answer many questions using agents that are very hard to ask and get answers for with systems of equations.
  • Get involved in building an agent modeling package with me. This is based on python and uses asynchronous networking. It is full of interesting software ideas.


    • So to pull all of this together, I maintain interests in a wide variety of subfields from the the mathematical sciences, computer science and software engineering and biological threads such as neurobiology and neuroscience.



    I rank potential Master's Degree projects that I would be interested in directing in order of level of effort according to the following scale:
  • Rank One: A Rank One project requires a level of effort that is on the order of a three credit hour class with either substantial homework or project responsibilities. Typical examples are an assignment to read and understand a given paper or algorithm in the literature, work through the theory and perhaps implement it in computer code if appropriate. Projects at this ranking can also be essentially readings from the literature with a nice detailed report. The reports here will be on the order of 20 pages with references.
  • Rank Two: A Rank Two project requires a level of effort that is on the order of a four credit class with either substantial homework or project responsibilities. So this is about 25% more effort than a first rank project. This will involve more substantial writing and almost always using the readings to work on some sort of applied problem. This applied problem can be addressed theoretically (this would mean you would need to apply theorems from your readings to this problem and provide proofs of your conjectures) and/ or the writing of computer code to provide simulations of algorithms you have been studying. The reports here will be on the order of 40 to 50 pages with references. Many of these projects require you to do a lot of outside reading and effort and so it is possible to arrange for reading credit hours. At this ranking, I usually allow 1-2 credit hour of such credit. These projects can often become Rank 3 if your interest level becomes that strong. These projects could lead to a publication of some sort.
  • Rank Three: A Rank Three project requires a level of effort that is on the order of a six credit class with either substantial homework or project responsibilities. So this is about two 3 credit classes or about twice the level of effort for a rank one project. This will involve very substantial writing and using the readings to work on some sort of applied problem. This applied problem can be also addressed theoretically (as in Rank 2, this would mean you would need to apply theorems from your readings to this problem and provide proofs of your conjectures) and/ or the writing of computer code to provide simulations of algorithms you have been studying. The reports here will be on the order of 70 to 110 pages with references. These projects require you to do a lot of outside reading and effort and so it is standard to arrange for reading credit hours. At this ranking, I allow 3 credit hours of such credit. These type of projects probably can be published in an appropriate journal or conference.


    • A typical report will be done in LaTeX and in outline looks like this:

  • TitlePage
  • Title of Project
  • Your name and affiliation
  • Date
  • Abstract Page
  • Introduction: Here you introduce the problem. What you are doing for the project, why you are doing it and what the benefits are.
  • Background: Here you talk about your problem, introduce whatever science and math are necessary to understand what is going on. Sort of like a tutorial or a lecture to bright, yet untrained in this area, colleagues. This can be split over several sections if required.
  • Your Specific Task: What your task was and how you accomplished it. This can include theoretical development, algorithm development and software issues. This can be spread over several sections if necessary. If there is code development, subsections on the design of your code are very appropriate.
  • Your Results: Discuss your accomplishments.
  • Your Conclusions:
  • References:
  • Appendices: You usually place listings of computer code in the back of your report and use code fragments from these listings to illustrate your points in the main document.


    • Now I have created a few tex documents to help you along.
  • I use a collection of tex macros which are in the file setup.tex. If you wish to download the Latex source for this file file please click here!
  • The file MS_Project_Template.tex is an attempt to lay out the basic structure of the MS report. If you wish to download the Latex source for this file please click here!

    If you wish to browse the pdf version of this document please click here.

  • The file Notes.tex is another example file to help you with the basic structure of a Latex MS report. If you wish to download the Latex source for this file please click here!

    If you wish to browse the pdf version of this document please click here.

  • To create these files yourself, you use the Latex system on either a Windows or Linux host. There are lots of people who can help you get started. For these sample files, do this:
  • Download a few graphic files you need for these demos; please click here to download meta3.png.

    please click here to download voltage0-99.png.

  • type pdflatex MS_Project_Template to create the file pdflatex MS_Project_Template.pdf
  • type pdflatex Notes to create the file pdflatex Notes.pdf


    • Author: Dr. Peterson, Mathematical Sciences, Clemson University
    Last Updated: March 1, 2005
    petersj@clemson.edu