Davis, Gary
Dr. Gary Davis
Research experience
PhD (Mathematics) Monash University, Melbourne, Australia
My research interests lie broadly in:
- Educational issues in Mathematics, Statistics, and Data Science.
- Analysis and modeling of spatial point processes
- Random walks with memory
- Structure of protein-protein interaction networks
Research projects
Theory and applications of geometric graph clustering. Clustering is a collection of algorithms for grouping a set of objects – for example, points in a plane – so that objects in the same cluster are more similar to each other than to those in other clusters. Geometric graph clustering is a form of clustering for points in some Euclidean space – in a plane, or 3-dimensional space, for example – in which we start by forming the complete graph on all points and then forming a geometric graph by only joining points if they are d or less apart, and slowly decreasing d until the graph disconnects. Because the algorithm is quadratic in the number of points, it is critical in applications to structure the data to reduce the algorithm to at least nlog(n) time, where n is the number of points.
Random walks with finite non-zero memory. Simulation, computation, and theoretical results on cover times and trapping probabilities and times for random walks with finite non-zero memory.
Educational questions arising from recent research on the human brain and mathematics. Recent experiments in neuroscience demonstrate the existence of a brain system that deals with mathematical thought and that is disjoint from the language areas of the brain. We provide an overview of these and related mathematically activated brain regions and raise a number of questions of interest to mathematics educators, including the nature of the brain’s functioning in proof production, the role of compressible mental units in mathematical thought, and the nature of attention and working memory.
The role of an ability to plan in student proof production. The planning role of the pre-frontal cortex has been demonstrated to be activated when professional mathematicians read mathematically oriented sentences. Yet the pre-frontal cortex of students in their late teens is, in general, not fully developed. Might this be a significant factor in student difficulty with proof production?
Potential research projects for students
(1) Random walks with memory
The integer lattice in the plane is the set of points p = (m, n) where m and n are integers (positive, negative or zero):
A random walk on the integer lattice is a finite sequence of points
p1 = (m1, n1), p2 = (m2, n2), … , pk = (mk, nk)
where each pi+1 is obtained from pi by moving up, down, left, or right one unit with probability ¼.
The memory of a random walk is the number of points before any given point that the random walk stores and cannot visit until they slip out of the memory store.
So, a random walk with memory 0 has no constraints – it can move up, down, left, or right with probability ¼ from any given point. A random walk with memory 1 (also called a no-backtracking random walk) is constrained to not visit the point is just previously visited. A random walk with infinite memory (also called a self-avoiding walk) gets trapped quickly, with probability 1.
A lot is known about random walks with memory 0, 1 and ∞. However for intermediate finite memory much less is known. For example, a random walk with memory 7 can get trapped:
Questions: Does a random walk of memory 7 get trapped with probability 1? If so, what is the average length of a random walk of memory 7 before it gets trapped with probability 1? What about random walks with other finite non-zero memories? What about random walks with memory on other lattices – triangular or hexagonal lattices, for example.
(2) Racial disparities in police killings data
The Washington Post keeps continually updated records of police killings in the US. These records include age, gender, race and the latitude-longitude where a police killing occurred. This applied statistics project is focused on on-going racial disparities in police killings such as the average age of people killed of different races, and the differing locations where people of different races are killed by police, utilizing such things as indicators of poverty in these locations
(3) Base 3/2 presentations of positive integers.
A 2<-3 exploding dots machine raises many questions about how positive integers may be reprinted in base 3/2. The article “Variants of Base 3 Over 2” (Journal of Integer Sequences, Vol. 23 (2020), Article 20.2.7) has many details and discussions about such representations and lists 8 open problems (also refer to the articles: “On Base 3/2 and its Sequences” and “Base 3/2 and Greedily Partitioned Sequences“).
Contact
email: gdavis@umassd.edu