Ethics, Integrity, and Algorithms

Humanitarian Logistics/Humanitarian Mathematics

Sometimes it takes a mathematician, not just to solve mathematically posed problems, but even to see the fact that real life issues can be addressed via mathematical modeling and existing algorithmic techniques. Experts in the field of operations research (see "optimization consulting" below), which largely grew out of a compelling national interest to most efficiently manage the large scale logistical operations of WWII, have been slow to transfer this knowledge towards optimizing the effectiveness of such important work as large scale relief and development projects. Notable exceptions include the Center for Humanitarian Logistics at Georgia Tech and the Humanitarian Logistics research project at MIT which have begun to demonstrate and raise awareness of the benefits a technical computational background can bring to organizational short and long-term planning (pre-positioning of emergency supplies, efficient distribution of vaccines, etc.). Still, the efforts in such directions are relatively new and minuscule compared to the untapped potential.

Green Mathematics

The mathematical field of optimization, which I specialize in, is all about making large scale decisions for maximum efficiency --- precisely what environmentally sensitive operations management is looking for! More generally, any "green engineering" (as with other forms of engineering) relies on a wide variety of mathematical topics such as differential equations, image and signal processing, and probability and statistics. With a broad background in applied mathematics, I can model your sustainability, waste reduction, and energy efficiency goals in mathematical language and then provide the best solutions to guide your logistical planning and decision making. While my background as a graduate of a month-long backpacking course with the National Outdoor Leadership School and my training in wilderness survival may not come directly into play here, it certainly does help me to genuinely value the "green" objectives of "Green Mathematics"!

New Tools for Mathematics Education

Take any math textbook on the market, from elementary through graduate school level, and see how the steps for solving sample problems are taught. With near certainty I can guarantee that the actual mental process that the person who wrote the text went through in solving (and learning to solve) such problems himself was very different from what is reflected in the steps presented. Step-by-step rules are a much more refined and consumable product than the generally "messy", intuition based, non-linear process that any real human solver will go through. Enter metacognition, which is the process of "thinking about thinking", that is, being aware of your own cognitive processes as they actually occur and even consciously directing them as needed. Metacognition is hard work and for most of us is an exercise we haven't practiced; moreover, those who are most skilled in something (such as mathematics) often have the hardest time articulating that which comes so naturally for them. Therefore it is very rare for students to truly "get a window" into the brain activity of a proficient mathematician. Seeing this as a significant gap in the promotion of mathematical understanding, I am interested to produce resources, such as a video series, in which students could experience a mathematician in action, analogous to watching a professional athlete actually training for and playing his sport rather than just talking about how to play it.

Optimization Consulting

Everyone wants to optimize something, whether it be a tangible commodity or an abstract measure of performance. Businesses want to maximize profits and minimize cost. Production line managers want to minimize errors and waste. Automated sensing systems (such as those deployed in defense technologies) need to maximize the probability of producing a correct analysis of ambiguous data. Common optimization problems include: finding a shortest path (e.g. for a salesman to tour a group of cities, for a mail truck to complete all its deliveries, or for a bit to drill numerous tiny holes in an irregular pattern on a board), finding optimal matchings (e.g. of groups of buyers to groups of sellers in an auction-like environment), and routing and scheduling (e.g. airline flights in which the crew must always return to their home base). When decisions are complicated by competing factors that contribute to your ultimate goal in different ways, constraints on resources, and uncertainty about future events (e.g. supply costs or demand for your product), relying on the "best guess" judgment of human agents may result in unacceptable performance, or may even be impossible. Off-the-shelf software packages exist for many, but not all, applications and even existing software requires a high degree of expertise to even use (intelligent mathematical modeling is a prerequisite for input to such programs and for interpretation of output). My background includes a specialization in stochastic programming in which a sequence of decisions must be made with only partial knowledge (predictions rather than certainly) of the information that will ultimately influence the outcome of those decisions.

Scientific Computing and Software Development

While my fluency in C⁺⁺, Matlab, and various other programming and scripting languages is quite sufficient for most professional scientific computing needs, the greater strength that I have to offer a software team is in the area of algorithm development. It is one thing to be able to make a computer do what you want it to do, it is another thing to tell it the best thing to do! Efficient, elegant, or elaborate programming techniques can not make up for ineffective algorithms. Note that anyone who is unwilling to give due consideration to the ethical implications of their software project would not want to hire my services, because I do not compartmentalize a life of devotion to God and His commands as separate from any work I may do in the "secular" software realm.

Business as Mission

I am interested in pursuing partnerships with fellow servants of Christ to apply the above skills and services towards advancing the Kingdom of God, especially among the nations of the world unreached by the gospel of our Lord. Benefits of "Business as Mission" (BAM) include, but are not limited to: 1) financing missionaries and indigenous Christian workers, 2) obtaining legitimate access to nations that restrict Christian activity (though note that I do not support Christians acting as undercover secret agents), and 3) many opportunities to display "strange" Christian behavior (i.e. loving self-sacrifice) in a business world full of corruption/bribery, cheating, and a spirit of do-whatever-it-takes-to-stomp-the-other-guy all-consuming greed.

"The Matrix of a Determinant": Hyperbolic Polynomials, Hyperbolicity Cones, and Hyperbolic Programming

The explanation of this idea requires more mathematical background than the other points above and will only be of interest to a limited audience. Imagine a square matrix in which all of the entries are linear expressions of some set of variables. If you take the determinant of this matrix you end up with a multivariate polynomial. If the matrix was symmetric then it had all real eigenvalues, and so a particular property of this polynomial is that it has all real "roots" (in a certain "direction") for all choices of the independent variables. This is called a hyperbolic polynomial. Now, a question that could be asked (but apparently has not been examined very much in the existing mathematical literature) is whether the process can be reversed. That is, given a hyperbolic polynomial p, can we find a linearly parameterized matrix which has p as its determinant? By analogy to the very well-known process of finding the determinant of a matrix, I call this the problem of finding "the matrix of a determinant" (even though such language is not technically precise since, e.g., such a matrix would not be unique). My Ph.D. dissertation, while not fully answering the "matrix of a determinant" problem, does appear to provide the greatest known headway yet on the question. In particular, given an arbitrary hyperbolic polynomial p, I am able to construct a matrix whose determinant factors as p times a multiplicity of all of its derivatives (this forces the eigenvalues of the matrix to interleave with the roots of p so that in particular the matrix has all positive eigenvalues if and only if the polynomial does). In light of the immeasurable importance of matrix determinants and eigenvalues in so many and varied applications, I am seeking (with realistic expectation) to find practical applications for these results on the "matrix of a determinant" problem.

Ethics, Integrity, and Algorithms

Zachary Harris, Ph.D.

Independent Mathematics Scholar Dedicated to Exalting Jesus Christ in All Things

Professional Services

Qualifications

Relevant resources

Contact Form