Variants of Lojasiewicz's Gradient Inequality
Explore the intricacies of Lojasiewicz's gradient inequality and its variants in this 45-minute lecture by Krzysztof Kurdyka from Université Savoie Mont Blanc, presented at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the celebrated inequality's significant consequence of uniformly bounding gradient trajectory lengths between two levels, leading to the existence of trajectory limits when approaching critical levels. Discover how this concept inspired René Thom's conjecture about trajectory tangents at the limit. Examine various versions and extensions of the inequality, including a variant applicable to maps with values in finite-dimensional vector spaces. Learn how this variant demonstrates the boundedness of submanifold volumes transverse to the kernels of map differentials, drawing parallels to the bounded length of gradient trajectories.