Multidimensional data is ubiquitous in the application, e.g., images and videos. I will introduce some of my previous and current works related to this topic.
1) Lattice metric space and its applications. Lattice and superlattice patterns are found in material sciences, nonlinear optics and sampling designs. We propose a lattice metric space based on modular group theory and
metric geometry, which provides a visually consistent measure of dissimilarity among lattice patterns. We apply this framework to superlattice separation and grain defect detection.
2) We briefly introduce two current projects. First, we propose new algorithms for automatic PDE modeling, which drastically improves the efficiency and the robustness against additive noise. Second, we introduce a new model for surface reconstruction from point cloud data (PCD) and provide an ADMM type fast algorithm.
We study whether all stationary solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported or has some decay at infinity. Our main results are the following:
(1) On the one hand, we are able to show that for any non-negative smooth stationary vorticity that is compactly supported (or has certain decay as |x|->infty), it must be radially symmetric up to a translation.
(2) On the other hand, if we allow vorticity to change sign, then by applying bifurcation arguments to sign-changing radial patches, we are able to show that there exists a compactly-supported, sign-changing smooth stationary vorticity that is non-radial.
We have also obtained some symmetry results for uniformly-rotating solutions for 2D Euler equation, as well as stationary/rotating solutions for the SQG equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques. This is a joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.
Chun-Hung will discuss his employment experience as an ACO alummus. The conversations will take place over coffee.
Mathapalooza! is simultaneously a Julia Robinson Mathematics Festival and an event of the Atlanta Science Festival. There will be puzzles and games, a magic show by Matt Baker, mathematically themed courtroom skits by GT Club Math, a presentation about math and dance by Manuela Manetta, a presentation about math and music by David Borthwick, and a gallery of mathematical art curated by Elisabetta Matsumoto. It is free, and we anticipate engaging hundreds of members of the public in the wonders of mathematics. More info at https://mathematics-in-motion.org/about/Be there or B^2 !
Georgia Tech is leading the way in Creating the Next in higher education.In this talk I will present (1) My vision for ACO and (2) how my research relates naturally to ACO both where the A,C,O fields are going, and my own specific interests
Understanding the structure of RNA is a problem of significant interest to biochemists. Thermodynamic energy functions are often key to this pursuit, but it is well-established that these energy functions do not perform well when applied to longer RNA sequences. This work specifically investigates the branching properties of RNA secondary structures, viewed as plane trees. By employing Markov chain Monte Carlo techniques, we sample from the probability distributions determined by these thermodynamic energy functions. We also investigate some of the challenges in employing Markov chain Monte Carlo, in particular the existence of local energy minima in transition graphs. This talk will give background, share preliminary results, and discuss future avenues of investigation.
This is a SCMB MathBioSys Seminar posted on behalf of Melissa Kemp (GT BME)
Constriction of blood vessels in the extremities due to traumatic injury to halt excessive blood loss or resulting from pathologic occlusion can cause considerable damage to the surrounding tissues with significant morbidity and mortality. Optimal healing of damaged tissue relies on the precise balance of pro-inflammatory and pro-healing processes of innate inflammation. In this talk, we will present a discrete multiscale mathematical model that spans the tissue and intracellular scales, and captures the consequences of targeting various regulatory components. We take advantage of the canalization properties of some of the functions, which is a type of hierarchical clustering of the inputs, and use it as control to steer the system away from a faulty attractor and understand better the regulatory relations that govern the system dynamics.EDIT: CANCELLED
Advisors: Turgay Uzer and Cristel Chandre
Thirty years after the demonstration of
the production of high laser harmonics through nonlinear laser-gas
interaction, high harmonic generation (HHG) is being used to probe
molecular dynamics in real time and is realizing its
technological potential as a tabletop source of attosecond pulses in the
XUV to soft X-ray range. Despite experimental progress, theoretical
efforts have been stymied by the excessive computational cost of
first-principles simulations and the difficulty of
systematically deriving reduced models for the non-perturbative,
multiscale interaction of an intense laser pulse with a macroscopic gas
of atoms. In this thesis, we
investigate first-principles reduced models for HHG using
classical mechanics. On the microscopic level, we examine the
recollision process---the laser-driven collision of an ionized electron
with its parent ion---that drives HHG. Using nonlinear dynamics, we
elucidate the indispensable role played by the ionic
potential during recollisions in the strong-field limit. On the
macroscopic level, we show that the intense laser-gas interaction can be
cast as a classical field theory. Borrowing a technique from plasma
physics, we systematically derive a hierarchy of
reduced Hamiltonian models for the self-consistent interaction between
the laser and the atoms during pulse propagation. The reduced models
can accommodate either classical or quantum electron dynamics, and in
both cases, simulations over experimentally-relevant
propagation distances are feasible. We build a classical model based on
these simulations which agrees quantitatively with the quantum model
for the propagation of the dominant components of the laser field.
Subsequently, we use the classical model to trace
the coherent buildup of harmonic radiation to its origin in phase
space. In a simplified geometry, we show that the anomalously high
frequency radiation seen in simulations results from the delicate
interplay between electron trapping and higher energy recollisions
brought on by propagation effects.
Oral Comprehensive Exam
The purpose of this work is approximation of generic Hamiltonian dynamical systems by those with a finite number of islands. In this work, we will consider a Lemon billiard as our Hamiltonian dynamical system apparently with an infinitely many islands. Then, we try to construct a Hamiltonian dynamical system by deforming the boundary of our lemon billiard to have a finite number of islands which are the same or sub-islands of our original system. Moreover, we want to show elsewhere in the phase space of the constructed billiard is a chaotic sea. In this way, we will have a dynamical system which preserves some properties of our lemon billiards while it has much simpler structure.
Cristobal Guzman will discuss his employment experience as an ACO alummus. The conversations will take place over coffee.