Mathematics and Natural Sciences

Hayk Harutyunyan, PhD


Nonlinear Nanophotonics for Emerging Integrated Quantum Systems.

Photons and entangled multiphoton states, owing to their weak interaction with the environment, serve as major carriers of physical signals and quantum information providing links to and from disparate quantum systems. Exquisite control, generation, and detection of visible and infrared photons are therefore of critical importance for future quantum systems. This is where nonlinear photonics - the science of controlling light by light - can provide an unmatched potential. Nonlinear photonics is capable of controlling quantum and classical light flow in unprecedented ways across time and space and over a broad frequency range. However, relatively weak overall efficiency of existing nonlinear systems hinders efficient control of quantum light. We envisage that the full potential of nonlinear photonics for manipulation of quantum fields can be accessed by controlling electronic responses at the fundamental atomistic scale. In this project, we aim to examine emergence of new and strong nonlinear responses in nascent material platforms, such as atomically thin materials and nanophotonic cavities. We then build on these newly discovered materials and functions to explore pathways that enhance quantum nonlinear interactions and enable conceptually new ways of precise control of quantum states.

Carl (Ji) Yang, PhD


Training Powerful Graph Neural Network Models for Brain Network Analysis with Limited Supervision

Mapping the connectome of the human brain as a networked system has become one of the most pervasive paradigms in neuroscience. Understanding what these brain networks are, how they develop, deteriorate, and vary across individuals will provide a range of insights from disease diagnosis, to understanding the neural basis of behavior and cognition, and even repairing/augmenting the brain functions. The recent success of graph neural networks provides a promising approach to the modeling of brain networks involving complex interactions among brain regions. However, these powerful neural network models are not specifically designed for brain network data, and their effective training often requires significant amounts of labeled data, which are hard to guarantee in most neurological studies. The goal of this project is to develop holistic brain network data-oriented graph neural network architectures (Thrust 1) and their label-efficient training strategies with transfer learning (Thrust 2), with targeted clinical applications of rs-fMRI-based mental disorder analysis across the large-scale public ABCD study of NIH and relatively smaller local AURORA study of Emory. Together, techniques developed in this project will constitute a mature toolkit for many research areas where deep learning on brain network data is of interest.

Liana Yepremyan, PhD


Extremal problems in Latin squares and more

An n × n Latin square is an n × n square filled with n different symbols, each of which occurs exactly once in each row and column. This definition might remind the reader of a commonly known puzzle called ‘sudoku’ which is a 9 × 9 Latin square with some additional constraints. The study of Latin squares is very old; it dates back to the 1700s to the work of the famous mathematician Leonard Euler. Latin squares have connections to various fields in mathematics. For example, they are multiplication tables of quasigroups from algebra. They are also used as error-correcting codes when, for example, one wishes to transmit data via a noisy channel, such as power lines. No matter how one fills a 3 × 3 Latin square, it is always possible to pick three cells, no two of which share a row, column or symbol. Such a collection of cells is called a transversal. In contrast, it is not possible to find a transversal in a 2 × 2 Latin square. A famous conjecture from the 1960s (due to Ryser) asserts that for odd n, it is always possible to find a transversal in an n × n Latin square. This is the main question I propose to study. I chose it not only because of its intrinsic beauty but also because it has ties to many other combinatorial and algebraic problems, some of which I intend to study as well. I believe that progress on this question would lead to the development of new methods in combinatorics and algebra.