Group representations, complexity, and randomness

• Local randomness of compact groups: A compact group is a group of motions in a homogenous space. The simplest example is the group of rotations in space. These groups often appear in applied areas such as quantum mechanics. Mallahi-Karai and colleagues are applying the tools of harmonic analysis to study questions with a probabilistic flavor on such groups. For example, in their recent work, they have proven that if X and Y are two random rotations then their composition XY is more random than what is predicated by the well known Caught inequality. 
• Mathematical models of decision-making: Mallahi-Karai and colleagues propose a model of decision-making based on the Weiner process (also called ‘Brownian motion’, which is the random motion of particles). Previous scholars had applied this process to simple yes-or-no decisions, however Mallahi-Karai and colleagues broadened its scope to multiple-option decision-making. An example is if a person were trying to choose between three different mobiles phones. Such a decision-making process can be represented by a cube, which has six sides and each side corresponds to a choice. With the Weiner process, a point and direction are chosen randomly within the cube. The first wall that the moving ‘dot’ hits will be the choice (if ‘yes’, the decision has been made; if ‘no’, that wall disappears and the shape of the decision-making container changes into a four-sided object). The starting point is the ‘initial bias’, and with this model of decision-making, it is possible to predict the choice probabilities and mean response time of a given decision.

Some of Mallahi-Karai’s research was funded through a DFG grant, acquired jointly with Adele Diederich. During the 2019-2022 period, he supervised one master’s and nine bachelor’s theses.

• Locally random groups, (with A. Mohammadi and A. Salehi Golsefidy), Michigan Math. J. 72: 479-527, 2022.
• Decision with multiple alternatives: geometric models in higher dimensions — The disk Model (with Adele Diederich), Journal of Mathematical Psychology, Volume 100, 2021.
• Polynomial configurations in sets of positive upper density over local fields, Journal d’Analyse Mathématique 142 (2020), no. 1, 71–103.
• Decision with multiple alternatives: geometric models in higher dimensions — The Cube Model (with Adele Diederich), Journal of Mathematical Psychology, Volume 93, 2019.
• Kirillov’s orbit method and polynomiality of the faithful dimension of p-groups (jointly with Mohammad Bardestani and Hadi Salmasian), Compositio Mathematica,155, Issue 8, 2019, 1618-1654.