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If I have seen further than others, it is because I stand on the shoulders of giants. -Isaac Newton
Ongoing
Sloan Business School (MIT) and Professor Tavneet Suri
As shown nearly a decade ago by Marx, Stoker and Suri, sattelite imaging and modern computation techniques can be quite effective at understanding economic traits of certain areas devoid of rigorous census taking. Roofs of houses in slums are largely uniform in material. Quantifying the amount and color of light they reflect can determine their age and quality. This information in aggregate can serve as an indicator for investment in housing in slum areas of the countries in question. This turns out to be quite a potent (and one of few) markers for understanding this kind of economic development. In this team I am spearheading the technological efforts and am in charge of data processing and algorithm development. We are just getting started as of October 2021.
https://arxiv.org/abs/2108.13565
Summer 2021
With support from Northeastern University's math department and RTG grant
A configurations is a set of points and lines where each line is incident to k and each point is incident to n lines. For the our work we considered configurations where n = k. Representations of configurations can come in two main flavors. The first is a combinatorial one, merely a table mapping incidences of points and lines. The second is a geometric one, the actual realization of an incidence structure in the Euclidean plane. Interestingly not all combinatorial configurations can be realized as geometric ones. This is called the realization problem and most studies in configurations attempt to understand which combinatorial configurations are realizable. This problem is still very much open in its general form.
In our work we sought to achieve two things. The first was to investigate how the lense of group theory can help understand when combinatorial configurations might be realizable. If they are we also use these tools to infer their structure. To achieve this, we relied upon classic notions of isomorphisms of configurations and proved how the set of all isomorphisms form a group. As it turns out, there exists a bijective mapping between combinatorial configurations and bipartite graphs with some specific properties (these are called Levi graphs). It can be shown then that the isomorphism group of a combinatorial configuration is the same as the isomorphism group of its graph. This turned out to be critical, as determining the isomorphism group of a configuration is mostly intractable, but isomorphism groups of graphs is a much more deeply studied (and somewhat computationally solved) problem. With these tools we investigated cases where combinatorial configurations with groups of certain kind (usually cyclic groups of non prime order) yielded geometric configurations realizations whose "geometric" isomorphism group was a cyclic subgroup of the combinatorial one. This turned out to give us a fascinating intuition and examples. To illustrate, the combinatorial configuration with isomorphism group c9 has 3 geometric realizations. Two of these have isomorphism groups c3 (containing 3, 3 point orbits) with 3 way rotational symmetry. The last has isomoprhism group c9 and no apparent rotational symmetry. We provided general conjectures on combinatorial configurations with cyclic isomorphism groups and how the prime factorization of their order paired with the fundamental theorem of finite Abelian groups imply the existence geometric realizations with characteristics as in the mentioned example. Generally this turned out to be quite intractable to prove, for arbitrary n, given our time frame and firepower. I hope to investigate these conjectures in the future.
The second aspect of our research was to dive deeply into a specific brand of combinatorial configurations, called cyclic configurations. A cyclic configuration can be thought of as a configuration table where there exists a hamiltonian walk through incidences where the path of the walk goes always goes from point i to point i + x for some fixed x. It can be shown that configurations of this form have cyclic isomorphism groups, and hence the name. We studied when for a given number of points n which values of coprime x yielded valid configurations. We generated theorems on the valid values of x for arbitrary n. Interestingly we showed that when plotting valid points, that these values always fall on the perimeter or bisecting lines of a triangle whose position and size was a function of n. We provided further results on the nature of the configuration when x is taken to be the centroid of said triangle.
Our paper is also published on arXiv, and the link along with a download of the paper and presentation can be found here.