Studying Fluid Flows with Persistent Homology - Rachel Levanger

Explore the application of persistent homology in studying fluid flows through this comprehensive lecture. Delve into topics such as 2D Kolmogorov Flow, sublevel set persistent homology, and bottleneck distance in persistent homology. Examine the challenges of dealing with multiple time scales and learn about the Algebraic Stability Theorem for Generalized Interleavings. Investigate 2D Rayleigh-Bénard Convection, focusing on spiral defect chaos, Lyapunov vectors, and topological defects. Discover how persistent homology can be used to classify canonical patterns and identify critical point pairings. Explore the concept of persistent homology defects and the challenges of finding correspondences in persistence diagrams. Compare linear interpolation with actual flow in fluid dynamics. Extend the discussion to 3D Navier-Stokes equations and fully-developed turbulence, utilizing diffusion maps and landscape metrics. Uncover a new relationship in fluid dynamics and examine the Tsurushima Mechanism for HCCI. Conclude by learning how to differentiate chemical species using speed profiles and reaction profiles.

Intro
DYNAMICS
PATTERNS
2D Kolmogorov Flow
Sublevel Set Persistent Homology
Bottleneck Distance in per
Point Cloud Persistent Homology
Dealing with Multiple Time Scales
Algebraic Stability Theorem for Generalized Interleavings
2D Rayleigh-Bénard Convection : Spiral Defect Chaos
Lyapunov Vectors and SDC
Topological Defects + Canonical Pattern Classification
Critical Point Pairings and Persistent Homology
Persistent Homology Defects
Finding a Correspondence...
Tangled Vines in a Vineyard of Persistence Diagrams
Linear Interpolation vs. Actual Flow
BEYOND PERSISTENCE MODULES
3D Navier-Stokes: Fully-developed Turbulence
Diffusion Maps
Landscape Metric on per
Diffusion Map Embedding of 512 Vorticity Fields
RESULT:Uncovered a New Relationship
Tsurushima Mechanism for HCCI
Landscape l'Speed Profiles
Differentiating Chemical Species by Speed Profile
Reaction Profile: R1