New Problems with Old Roots

Afshin Goodarzi will present his Ph.D. in Mathematics at KTH Royal Institute of Technology in 2016. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Günter Ziegler at Freie Universität in Berlin, Germany.

Afshin Goodarzi’s project consists of three related components, which stem from important open questions in geometric combinatorics. Combinatorics, which has roots in antiquity, is a branch of mathematics that is concerned with counting the number of elements in a set. For example, one can count the number of ways six distinct numbers can be chosen from the first positive fifty integers. Another example of a combinatorial problem is counting the number of ways a map can be colored with a given number of colors, assuming that no two countries with a common border share the same color. 

Combinatorial methods are often used in other branches of mathematics, such as probability, statistics, or graph theory. One of the project’s goals is a generalization of several results in graph theory to higher geometric dimensions. Another goal, one related to the map-coloring problem, is finding the lowest dimension space that can contain certain geometric objects.

More generally, the objective of the project is to research how combinatorics can be applied to other branches of mathematics, most importantly to geometry, topology, and graph theory. The expectation is that the results will lead to deeper understanding of several open problems. Applications for the project’s research include optimization theory, computer science, and computer aided design.