Homological Percolation: The Formation of Giant Cycles

Explore the fascinating world of homological percolation and the formation of giant cycles in this hour-long lecture by Omer Bobrowski. Delve into a higher-dimensional analogue of percolation theory, examining the emergence of "giant" cycles in random point clouds generated over manifolds. Learn about the phase transitions describing the birth-time of these giant cycles and their significance in differentiating between signal and noise in persistence diagrams. Discover an unexpected connection to the Euler characteristic curve and gain insights into topics such as persistent homology, continuous percolation, and Gaussian random fields. Understand the applications of these concepts in probability theory and statistical physics, and explore future directions in this cutting-edge research area.

Intro
PERSISTENT HOMOLOGY
THE MAXIMAL PERSISTENCE OF NOISE
TYPICAL BEHAVIOR
CROSSING PROBABILITIES
CONT. PERC. IN A BOX - SETUP
CONT. PERC. - GIANT COMPONENTS
"GIANT" K-CYCLES
HOMOLOGICAL PERCOLATION
MAIN RESULT
SIGNAL VS. NOISE
H, PERCOLATION
DUALITY - PROOF
H.- -PERCOLATION
BIG PICTURE?
EULER CHARACTERISTIC
EC & PERCOLATION?
SIMULATIONS
PERMUTAHEDRAL SITE-PERCOLATION
GAUSSIAN RANDOM FIELDS
SUMMARY & FUTURE WORK