## Seminars and Colloquia by Series

### Fast Algorithm for Invariant Circle and their Stable Manifolds: Rigorous Results and Efficient Implementations

Series
Dissertation Defense
Time
Friday, July 9, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE: Bluejeans: https://gatech.bluejeans.com/6489493135
Speaker
Yian YaoGeorgia Tech

In this dissertation, we present, analyze, and implement a quadratically convergent algorithm to compute the invariant circle and the foliation by stable manifolds for 2-dimensional maps. The 2-dimensional maps we are considering are mainly motivated by oscillators subject to periodic perturbation.

The algorithm is based on solving an invariance equation using a quasi-Newton method, and the algorithm works irrespective of whether the dynamics on the invariant circle conjugates to a rotation or is phase-locked, and thus we expect only finite regularity on the invariant circle but analytic on the stable manifolds.

More specifically, the dissertation is divided into the following two parts.

In the theoretical part, we derive our quasi-Newton algorithm and prove that starting from an initial guess that satisfies the invariance equation very approximately, the algorithm converges quadratically to a true solution which is close to the initial guess. The proof of the convergence is based on an abstract Nash-Moser Implicit Function Theorem specially tailored for this problem.

In the numerical part, we discuss some implementation details regarding our algorithm and implemented it on the dissipative standard map. We follow different continuation paths along the perturbation and drift parameter and explore the "bundle merging" scenario when the hyperbolicity of the map losses due to the increase of the perturbation. For non-resonant eigenvalues, we also generalize the algorithm to 3-dimension and implemented it on the 3-D Fattened Arnold Family.

### Physical Billiards and Open Dynamical Systems

Series
Dissertation Defense
Time
Thursday, July 8, 2021 - 10:00 for 2 hours
Location
https://bluejeans.com/675272964/8610
Speaker
Hassan AttarchiGeorgia Institute of Technology

This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a hole" of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map.

In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions.

In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas.

Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.

### Multiscale Problems in Mechanics: Spin Dynamics, Structure-Preserving Integration, and Data-Driven Methods

Series
Dissertation Defense
Time
Wednesday, July 7, 2021 - 10:30 for 2 hours
Location
BlueJeans: https://gatech.bluejeans.com/8515708345
Speaker
Renyi ChenGeorgia Tech

This thesis focuses on analyzing the physics and designing multiscale methods for nonlinear dynamics in mechanical systems, such as those in astronomy. The planetary systems (e.g. the Solar System) are of great interest as rich dynamics of different scales contribute to many interesting physics. Outside the Solar System, a bursting number of exoplanets have been discovered in recent years, raising interest in understanding the effects of the spin dynamics to the habitability. In part I of this thesis, we investigate the spin dynamics of circumbinary exoplanets, which are planets that orbit around stellar binaries. We found that habitable zone planets around the stellar binaries in near coplanar orbits may hold higher potential for stable climate compared to their single star analogues. And in terms of methodology, secular theory of the slow dominating dynamics is calculated via averaging. Beyond analyzing the dynamics mathematically, to simulate the spin-orbit dynamics for long term accurately, symplectic Lie-group (multiscale) integrators are designed to simulate systems consisting of gravitationally interacting rigid bodies in part II of the thesis. Schematically, slow and fast scales are tailored to compose efficient algorithms. And the integrators are tested via our package GRIT. For the systems that are almost impossible to simulate (e.g. the Solar System with the asteroid belt), how can we understand the dynamics from the observations? In part III, we consider the learning and prediction of nonlinear time series purely from observations of symplectic maps. We represent the symplectic map by a generating function, which we approximate by a neural network (hence the name GFNN). And we will prove, under reasonable assumptions, the global prediction error grows at most linearly with long prediction time as the prediction map is symplectic.

### Polyhedral and tropical geometry in nonlinear algebra

Series
Dissertation Defense
Time
Wednesday, June 30, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Cvetelina HillGeorgia Tech

This dissertation consists of various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. We give a broad introduction and discuss connections to applied algebraic geometry, polyhedral, and tropical geometry. The individual topics discussed are as follows:

• Interaction between tropical and classical convexity, with a focus on the tropical convex hull of convex sets and polyhedral complexes. Amongst other results, we characterize tropically convex sets in any dimension, and give a combinatorial description for the dimension of the tropical convex hull of an ordinary affine space.
• The steady-state degree and mixed volume of a chemical reaction network. We present three case studies of infinite families of networks. For each family, we give a formula for the steady-state degree and mixed volume of the corresponding polynomial system.
• Methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs.

Thesis may be viewed here.

### Applications of monodromy in solving polynomial systems

Series
Dissertation Defense
Time
Wednesday, June 16, 2021 - 12:00 for 1.5 hours (actually 80 minutes)
Location
ONLINE
Speaker
Timothy DuffGA Tech

Final doctoral examination and defense of dissertation of Timothy Duff, June 16, 2021

Date: June 16, 2021, 12:00pm EST

Title: Applications of monodromy in solving polynomial systems

Advisor: Dr. Anton Leykin, School of Mathematics, Georgia Institute of Technology

Committee:

Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
Dr. Gregory Blekherman, School of Mathematics, Georgia Institute of Technology
Dr. Richard Peng, School of Computer Science, Georgia Institute of Technology
Dr. Rekha Thomas, Department of Mathematics, University of Washington
Dr. Josephine Yu, School of Mathematics, Georgia Institute of Technology
Reader: Dr. Rekha Thomas, Department of Mathematics, University of Washington
---------------------------------------------------------------------------------------------------------
The thesis is available here:

Summary:

Polynomial systems of equations that occur in applications frequently have a special structure. Part of that structure can be captured by an associated Galois/monodromy group. This makes numerical homotopy continuation methods that exploit this monodromy action an attractive choice for solving these systems; by contrast, other symbolic-numeric techniques do not generally see this structure. Naturally, there are trade-offs when monodromy is chosen over other methods. Nevertheless, there is a growing literature demonstrating that the trade can be worthwhile in practice.

In this thesis, we consider a framework for efficient monodromy computation which rivals the state-of-the-art in homotopy continuation methods. We show how its implementation in the package MonodromySolver can be used to efficiently solve challenging systems of polynomial equations. Among many applications, we apply monodromy to computer vision---specifically, the study and classification of minimal problems used in RANSAC-based 3D reconstruction pipelines. As a byproduct of numerically computing their Galois/monodromy groups, we observe that several of these problems have a decomposition into algebraic subproblems. Although precise knowledge of such a decomposition is hard to obtain in general, we determine it in some novel cases.

### Algorithmic Approaches to Problems in Probabilistic Combinatorics

Series
Dissertation Defense
Time
Thursday, June 10, 2021 - 10:00 for
Location
ONLINE
Speaker
He GuoGeorgia Institute of Technology

The probabilistic method is one of the most powerful tools in combinatorics: it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method:
1. We study the structural properties of the triangle-free graphs generated by a semirandom variant of triangle-free process and obtain a packing extension of Kim’s famous R(3, t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse.
2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n).
3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory.
4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.

The Bluejeans link of the defense is https://gatech.bluejeans.com/233874892

### Persistence of Invariant Objects under Delay Perturbations

Series
Dissertation Defense
Time
Thursday, May 6, 2021 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE at https://bluejeans.com/137621769
Speaker
Jiaqi YangGeorgia Tech

We consider functional differential equations which come from adding delay-related perturbations to ODEs or evolutionary PDEs, which is a singular perturbation problem. We prove that for small enough perturbations, some invariant objects (e.g. periodic orbits, slow stable manifolds) of the unperturbed equations persist and depend on the parameters with high regularity. The results apply to state-dependent delay equations and equations which arise in electrodynamics. We formulate results in a posteriori format. The proof is constructive and leads to algorithms.

This is based on joint works with Joan Gimeno and Rafael de la Llave.

### On Scalable and Fast Langevin-Dynamics-Based Sampling Algorithms

Series
Dissertation Defense
Time
Friday, April 23, 2021 - 15:00 for 1.5 hours (actually 80 minutes)
Location
ONLINE
Speaker
Ruilin LiGeorgia Institute of Technology

Langevin dynamics-based sampling algorithms are arguably among the most widely-used Markov Chain Monte Carlo (MCMC) methods. Two main directions of the modern study of MCMC methods are (i) How to scale MCMC methods to big data applications, and (ii) Tight convergence analysis of MCMC algorithms, with explicit dependence on various characteristics of the target distribution, in a non-asymptotic manner.

This thesis continues the previous efforts in these two lines and consists of three parts. In the first part, we study stochastic gradient MCMC methods for large-scale applications. We propose a non-uniform subsampling of gradients scheme to approximately match the transition kernel of a base MCMC base with full gradient, aiming for better sample quality. The demonstration is based on underdamped Langevin dynamics.

In the second part, we consider an analog of Nesterov's accelerated algorithm in optimization for sampling. We derive a  dynamics termed Hessian-Free-High-Resolution (HFHR) dynamics, from a high-resolution ordinary differential equation description of Nesterov's accelerated algorithm. We then quantify the acceleration of HFHR over underdamped Langevin dynamics at both continuous dynamics level and discrete algorithm level.

In the third part, we study a broad family of bounded, contractive-SDE-based sampling algorithms via mean-square analysis. We show how to extend the applicability of classical mean-square analysis from finite time to infinite time. Iteration complexity in the 2-Wasserstein distance is also characterized and when applied to the Langevin Monte Carlo algorithm, we obtain an improved iteration complexity bound.

### Contact structures on hyperbolic L-spaces

Series
Dissertation Defense
Time
Friday, April 23, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Hyunki MinGeorgia Tech

Ever since Eliashberg distinguished overtwisted from tight contact structures in dimension 3, there has been an ongoing project to determine which closed, oriented 3–manifolds support a tight contact structure, and on those that do, whether we can classify them. This thesis studies tight contact structures on an infinite family of hyperbolic L-spaces, which come from surgeries on the Whitehead link. We also present partial results on symplectic fillability on those manifolds.

### Approximate Schauder Frames for Banach Sequence Spaces

Series
Dissertation Defense
Time
Friday, April 16, 2021 - 16:00 for 1.5 hours (actually 80 minutes)
Location
ONLINE
Speaker
Yam-Sung ChengGeorgia Institute of Technology

The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space $\ell_1$. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space $\ell_1^n$. We prove that for any $N \geq n$, FUNTFs of length $N$ exist for real $\ell_1^n$. To show the existence of FUNTFs, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements. However, a different method of frame constructions allows us to prove the existence of FUNTFs for real $\ell_1^n$ of lengths $2n-1$ and $2n-2$ that do not have repeated elements.

The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space $\ell_1$. A Schauder frame provides a reconstruction formula for elements in the space, but need not be associated with a frame inequality. Our main theorem on this topic establishes a set of conditions under which an $\ell_1$-type of frame inequality is applicable towards unconditional Schauder frames. A primary motivation for choosing this set of hypotheses involves appropriate modifications of the Rademacher system, a version of which we prove to be an unconditional Schauder frame that does not satisfy an $\ell_1$-type of frame inequality.