Georgia Institute of Technology College of Sciences

Zoom Link- https://gatech.zoom.us/j/97563537012?pwd=dlBVUVh2ZDNwdDRrajdQcDltMmRaUT09 (Meeting ID: 975 6353 7012 Passcode: 525012)

In this talk I will discuss the conjecture that every 3 manifolds can be smoothly embedded in symplectic 4 manifolds. I will give some motivation on why is this an interesting conjecture. As an evidence for the conjecture, I will prove that every 3 manifolds can be embedded in a topological way and such an embedding can be made a smooth one after a single stabilization. As a corollary of the proof, I will prove that integer/rational cobordism group is generated by Stein fillable 3 manifolds. And if time permits, I will give some idea on how one can try to obstruct smooth embeddings of 3 manifolds in symplectic 4 manifolds.

Online link: https://gatech.zoom.us/j/93504092832?pwd=V29FVVFlcEtwNWhkTnUyMnFqbVYyUT09

Many proposed interplanetary space missions, including Europa Lander and Dragonfly, involve trajectory design in environments where multiple large bodies exert gravitational influence on the spacecraft, such as the Jovian and Saturnian systems as well as cislu- nar space. In these contexts, an analysis based on the mathematical theory of dynamical systems provides both better insight as well as new tools to use for the mission design compared to classic two-body Keplerian methods. Indeed, a rich variety of dynamical phenomena manifest themselves in such systems, including libration point dynamics, stable and unstable mean-motion resonances, and chaos. To understand the previously mentioned dynamical behaviors, invariant manifolds such as periodic orbits, quasi-periodic invariant tori, and stable/unstable manifolds are the major objects whose interactions govern the local and global dynamics of relevant celestial systems.

This work is focused on the development of numerical methodologies for computing such invariant manifolds and investigating their interactions. After a study of persistence of mean-motion resonances in the planar circular restricted 3-body problem (PCRTBP), techniques for computing the stable/unstable manifolds attached to resonant periodic orbits and heteroclinics corresponding to resonance transitions are presented. Next, I will focus on the development of accurate and efficient parameterization methods for numerical calculation of whiskered quasi-periodic tori and their attached stable/unstable manifolds, for periodically-forced PCRTBP models. As part of this, a method for Levi- Civita regularization of such periodically-forced systems is introduced. Finally, I present methods for combining the previously mentioned parameterizations with knowl- edge of the objects’ internal dynamics, collision detection algorithms, and GPU computing to very rapidly compute propellant-free heteroclinic connecting trajectories between them, even in higher dimensional models. Such heteroclinics are key to the generation of chaos and large scale transport in astrodynamical systems.

In this talk, we focus on designing computational methods supported by theoretical properties for optimal motion planning and optimal transport (OT). 

Over the past decades, motion planning has attracted large amount of attention in robotics applications. Given certain
configurations in the environment, the objective is to find trajectories which move the robot from one position to the other while satisfying given constraints. We introduce a new method to produce smooth and collision-free trajectories for motion planning task. The proposed model leads to short and smooth trajectories with advantages in numerical computation. We design an efficient algorithm which can be generalized to robotics applications with multiple robots.

The idea of optimal transport naturally arises from many application scenarios and provides powerful tools for comparing probability measures in various types. However, obtaining the optimal plan is generally a computationally-expensive task, sometimes even intractable. We start with the entropy transport problem as a relaxed version of original optimal transport problem with soft marginals, and propose an efficient algorithm to obtain the sample approximation for the optimal plan. We also study an inverse problem of OT and present the computational methods for learning the cost function from the given optimal transport plan. 
 

Many real life processes that we would like to model have a self-exciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time.  Examples of processes that have this property are earthquakes, crime in a neighborhood, or emails within a company.  In 1971, Alan Hawkes first used what is now known as the Hawkes process to model such processes.  Since then much work has been done on estimating the parameters of a Hawkes process given a data set and creating variants of the process for different applications.

In this talk, we propose a new variant of a Hawkes process, called a self-limiting Hawkes process, that takes into account the effect of police activity on the underlying crime rate and an algorithm for estimating its parameters given a crime data set.  We show that the self-limiting Hawkes process fits real crime data just as well, if not better, than the standard Hawkes model.  We also show that the self-limiting Hawkes process fits real financial data at least as well as the standard Hawkes model.