Dr C P Anil Kumar, Assistant Professor of Mathematics, SIAS, Krea University authored a paper titled On Infinity Type Hyperplane Arrangements and Convex Positive Bijections, published in the Indian Journal of Pure and Applied Mathematics (IJPA).
About the Publication
Consider a hyperplane arrangement, i.e., a finite collection of hyperplanes, in an n-dimensional Euclidean space. We say that an arrangement is generic if there are at least n hyperplanes and a codimension k intersection is formed by exactly k many hyperplanes. This is an interesting arrangement – for example, for a fixed m ≥ n, a generic arrangement of m hyperplanes in an n-dimensional Euclidean space has the maximum number of regions. Two arrangements are (combinatorially) isomorphic (or equivalent) if there is an ambient linear homeomorphism which preserves incidences between strata and the polyhedral structure of each stratum. In general, it is an interesting (and difficult) problem to understand the isomorphism classes of arrangements.
In this paper, the author introduces and defines a new type of hyperplane arrangements called infinity type hyperplane arrangements. The concept of normal systems (which is a collection of normal directions to these hyperplanes) are used to uniquely identify such generic arrangements. The main theorem states that the equivalence of two infinity type generic hyperplane arrangements is equivalent to the isomorphism of corresponding normal systems. The author also considers the case of all generic arrangements and proves that equivalence of arrangements implies isomorphism of normal systems. Further, a counter example is provided to show that the converse need not be true.
Read the article here.