Georgia Institute of Technology College of Sciences

In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology.  More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions.  The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.

Traditional spacecraft trajectory optimization approaches focus on automatizing solution generation by capturing the solution space analytically, or numerically, in a single or few instances. However, critical human-computer interactions within optimization processes are almost always disregarded, and they are not well understood. In fact, human intervention spans across the entire optimization process, from the formulation of a problem that lands on known solution schemes, to the identification of an initial guess within the algorithm basin of convergence, to tuning the algorithm hyper-parameters, investigating anomalies, and parsing large databases of optimal solutions to gain insight. Vision-based interaction with sets of multi-dimensional information mitigates the complexity of several applications in astrodynamics. For example, visual-based processes are key to understanding solution space topology for orbit mechanics (e.g., Poincare’ maps), formulating higher quality initial trajectory guesses for early mission design studies, and investigating six-degree-of-freedom (6DOF) dynamics for proximity operations. The capillary diffusion of visual-based data interaction processes throughout astrodynamics has motivated the creation of virtual reality (VR) technology to facilitate scientific discovery since the advent of modern computers. The recent appearance of small, portable, and affordable devices may be a tipping point to advance astrodynamics applications via VR technology. Nonetheless, the tangible benefits for adoption of virtual reality frameworks are not yet fully characterized in the context of astrodynamics applications. What new opportunities virtual reality opens for astrodynamics? What applications benefits from virtual reality frameworks? To answer these and similar questions, our work focuses on a programmatic early assessment and exploration of VR technology for astrodynamics applications. The assessment is constructed by a review of VR literature with elements that are external to the astrodynamics community to facilitate cross-pollination of ideas. Next, the Johnson-Lindenstrauss lemma, together with a set of simplifying assumptions, is employed to analytically capture the value of projecting higher-dimensional information to a given lower dimensional space. Finally, two astrodynamics applications are presented to display solutions that are primarily enabled by virtual reality technology.

Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know  $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. In this setting, we can obtain conditions on $\Omega, A, l_i$ that allow the stable reconstruction of $f$. Dynamical Sampling is closely related to frame theory and has applications to wireless sensor networks among other areas. In this talk, we will discuss the Dynamical Sampling problem, its motivation, related problems inspired by it, current/future work, and open problems. 

The seminar will be held on Zoom and can be found at the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09