This Week's Seminars and Colloquia

Generic and non-generic synchronization configurations in networks of coupled oscillators

Series
Algebra Seminar
Time
Monday, October 3, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Tianran ChenAuburn University at Montgomery

Networks of coupled oscillators are studied in biology, chemistry, physics, and engineering. The Kuramoto model is a simple dynamical system that models the nonlinear interaction among coupled oscillators. It has received widespread attention since it is simple enough to be analyzed rigorously yet complex enough to exhibit interesting emergent behaviors.

One such emergent behavior is the spontaneous synchronization of oscillators into special configurations. In the past decades, rigorous analysis of such synchronization configurations has been the focus of intensive studies.

In this talk, we explore the new insight to this problem provided by an algebraic and tropical approach.

Geography of surface bundles over surfaces

Series
Geometry Topology Seminar
Time
Monday, October 3, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
İnanç BaykurUMass Amherst / Harvard

An outstanding problem for surface bundles over surfaces, closely related to the symplectic geography problem in dimension four, is to determine for which fiber and base genera there are examples with non-zero signatures. I will report on our recent progress (joint with M. Korkmaz), which resolves the problem for all fiber and base genera except for 18 pairs at the time of writing.

The existence of Prandtl-Batchelor flows on disk and annulus

Series
PDE Seminar
Time
Tuesday, October 4, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Zhiwu LinGeorgia Tech

For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao. 

The complexity of list-5-coloring with forbidden induced substructures

Series
Graph Theory Seminar
Time
Tuesday, October 4, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Speaker
Yanjia LiGeorgia Tech

The list-$k$-coloring problem is to decide, given a graph $G$ and a list assignment $L$ of $G$ from $V(G)$ to subsets of $\{1,...,k\}$, whether $G$ has a coloring $f$ such that $f(v)$ in $L(v)$ for all $v$ in $V(G)$. The list-$k$-coloring problem is a generalization of the $k$-coloring problem. Thus for $k\geq 3$, both the $k$-coloring problem and the list-$k$-coloring problem are NP-Hard. This motivates studying the complexity of these problems restricted to graphs with a fixed forbidden induced subgraph $H$, which are called $H$-free graphs.

In this talk, I will present a polynomial-time algorithm to solve the list-5-coloring $H$-free graphs with $H$ being the union of $r$ copies of mutually disjoint 3-vertex paths. Together with known results, it gives a complete complexity dichotomy of the list-5-coloring problem restricted to $H$-free graphs. This is joint work with Sepehr Hajebi and Sophie Spirkl.

Latin squares in extremal and probabilistic combinatorics

Series
Research Horizons Seminar
Time
Wednesday, October 5, 2022 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tom KellyGeorgia Tech

An order-n Latin square is an n by n array of n symbols such that each row and column contains each symbol exactly once.  Latin squares were famously studied by Euler in the 1700s, and at present they are still a central object of study in modern extremal and probabilistic combinatorics.  In this talk, I will give some history about Latin squares, share some simple-to-state yet notoriously difficult open problems, and present some of my own research on Latin squares.

3-Manifolds up to 1957

Series
Geometry Topology Student Seminar
Time
Wednesday, October 5, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Weizhe ShenGeorgia Institute of Technology

A three-manifold is a space that locally looks like the Euclidean three-dimensional space. The study of three-manifolds has been at the heart of many beautiful constructions in low dimensional topology. This talk will provide a quick tour through some fundamental results about three-manifolds that were discovered between the late nineteenth century and the Fifties.

A stochastic approach for noise stability on the hypercube

Series
Stochastics Seminar
Time
Thursday, October 6, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/86578123009
Speaker
Dan MikulincerMIT

Recording: https://us02web.zoom.us/rec/share/cIdTfvS0tjar04MWv9ltWrVxAcmsUSFvDznprS...

We revisit the notion of noise stability in the hypercube and show how one can replace the usual heat semigroup with more general stochastic processes. We will then introduce a re-normalized Brownian motion, embedding the discrete hypercube into the Wiener space, and analyze the noise stability along its paths. Our approach leads to a new quantitative form of the 'Majority is Stablest' theorem from Boolean analysis and to progress on the 'most informative bit' conjecture of Kumar and Courtade.

Recovery of quantum information: quantum Markov chains and matrix product states

Series
Math Physics Seminar
Time
Thursday, October 6, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Brian KennedySchool of Physics, Georgia Tech

The mathematical theory of the recovery of quantum states stored in a quantum memory, is intimately related to the subadditivity property of the entropy function, and the class of states known as quantum Markov chains. In this talk we will introduce some of the basic ideas of this area of quantum information theory. We discuss a theorem regarding recovery of a widely studied class of quantum states, the matrix product states, and its implication for the mutual information stored over separated regions of a one dimensional quantum memory. This is joint work with Pavel Svetlichnyy and Shivan Mittal.

The three-body problem and low energy space missions

Series
Time
Friday, October 7, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Marian GideaYU

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

The three-body problem, on the dynamics of three masses under mutual gravity, serves as a model for the motion of a spacecraft relative to the Earth-Moon or Sun-Earth system. We describe the equations of motion for the three-body problem and the geometric objects that organize the dynamics: equilibriums points, periodic and quasi-periodic orbits, and their stable and unstable manifolds. As it turns out, trajectories that follow these manifolds require zero energy cost. We describe several methods to design low energy spacecraft trajectories from Earth to Moon, as well as maneuvers to change the inclination of the orbit of a satellite relative to the ecliptic. This is based on joint works with E. Belbruno, F. Topputo, A. Delshams, and P. Roldan.   
 

Sparse Cholesky factorization by greedy conditional selection

Series
ACO Student Seminar
Time
Friday, October 7, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Stephen HuanGeorgia Tech CS

Dense kernel matrices resulting from pairwise evaluations of a kernel function arise naturally in machine learning and statistics. Previous work in constructing sparse transport maps or sparse approximate inverse Cholesky factors of such matrices by minimizing Kullback-Leibler divergence recovers the Vecchia approximation for Gaussian processes. However, these methods often rely only on geometry to construct the sparsity pattern, ignoring the conditional effect of adding an entry. In this work, we construct the sparsity pattern by leveraging a greedy selection algorithm that maximizes mutual information with target points, conditional on all points previously selected. For selecting k points out of N, the naive time complexity is O(N k^4), but by maintaining a partial Cholesky factor we reduce this to O(N k^2). Furthermore, for multiple (m) targets we achieve a time complexity of O(N k^2 + N m^2 + m^3) which is maintained in the setting of aggregated Cholesky factorization where a selected point need not condition every target. We directly apply the selection algorithm to image classification and recovery of sparse Cholesky factors. By minimizing Kullback-Leibler divergence, we apply the algorithm to Cholesky factorization, Gaussian process regression, and preconditioning with the conjugate gradient, improving over k-nearest neighbors particularly in high dimensional, unusual, or otherwise messy geometries with non-isotropic kernel functions.

Smooth structures on open 4-manifolds III

Series
Geometry Topology Working Seminar
Time
Friday, October 7, 2022 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John EtnyreGeorgia Tech

One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.  

The Entropy Compression Method

Series
Graduate Student Colloquium
Time
Friday, October 7, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Abhishek DhawanGeorgia Tech Math

The Lovasz Local Lemma is a powerful tool to prove existence of combinatorial structures satisfying certain properties. In a constructive proof of the LLL, Moser and Tardos introduced a proof technique that is now referred to as the entropy compression method. In this talk I will describe the main idea of the method and apply it to a problem easily solved using the LLL. I will also describe recent applications of the idea to various graph coloring problems.