Topics in the Calculus of Variations: Recent Advances and New Trends , May 20-25, 2018, The Banff International Research Station, Banff, Canada
2017 SIAM Conference on Analysis of Partial Differential Equations , December 9-12, 2017, Baltimore, Maryland, USA
I am an Assistant Professor of Mathematics at Ohio University. I received my Ph.D. in Applied Mathematics at Michigan State University under the direction of Prof. Keith Promislow. From 2011 to 2014 I was an RTG Postdoctoral Associate at the Center for Nonlinear Analysis at Carnegie Mellon University.
Research and Projects
Slow-Fast Dynamical Systems, Invariant Manifolds, and Calculus of Variations
Separation into different timescales, or more generally separation into ‘outer’ and ‘inner’ asymptotic expansions is ubiquitous in science and engineering. Many good engineers start with a simplified model (or zero-th order approximation in the asymptotic analysis language) that captures the dominant behavior, and in later design stages capture more and more precise dynamics that happen on faster timescales. More concretely, in fluid mechanics, the celebrated Prandtl ‘boundary layer theory’ is one famous example. In biochemistry and material science oftentimes ‘interfaces` are formed on fast ‘inner’ timescale and later evolve on a slower ‘outer’ timescale. Often such systems can be described as gradient flows associated with a functional dependent on a small parameter $\epsilon > 0$. Understanding the correct structure of the limiting problems as $\epsilon \rightarrow 0$ is one powerful technique of studying these systems. Here is some of my work in this area.
Related Publications (preprints hosted as part of Center for Nonlinear Analysis , Carnegie Mellon University, publications):
G. Hayrapetyan, M. Rinaldi, Slow motion for the 1D Swift-Hohenberg equation, Journal of Differential Equations, (2017). pdf
I. Fonseca, G. Hayrapetyan, G. Leoni, B. Zwicknagl Domain formation in membranes near the onset of instability, Journal of Nonlinear Science, (2016). pdf
G. Hayrapetyan, K. Promislow, Spectra of Functionalized Operators arising from hyper surfaces, Journal of Applied Mathematics and Physics (ZAMP), (2014). pdf
Open Source Cloud Controls and Simulation Toolbox
The Open Source Cloud Controls and Simulation Toolbox is a project consisting of
- CCST – a fast responsive online scientific console that can run computationally intensive numerical solvers, machine learning problems, and simulations ‘on the cloud’. This is a work in progress, so the current functionality is limited to certain orbital mechanics problems, some classic controls analysis, and some semantic segmentation problems.
- CCST Sim – a 3D OpenGL based desktop application for simulating dynamical systems, probabilistic filtering, and control algorithms.
Deep Learning, Semantic Segmentation and Autonomous Systems
Although ideas behind Deep Learning have been around for some time, it is only recently that this area of machine learning has seen enormous growth due to increase both of fast GPU based computational power and its success in tackling very complicated problems in image recognition. Here are some projects in this area: