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Quantitative estimates for the Hopf lift in fractional Sobolev spaces
- Prelegent(ci)
- Adam Grzela
- Afiliacja
- University of Warsaw
- Język referatu
- angielski
- Termin
- 27 maja 2026 12:30
- Pokój
- p. 4060
- Seminarium
- Seminarium Zakładu Równań i Analizy
The classical lifting problem asks: given a ``good map'' $\Pi: E \to N$ (such as a Riemannian covering, fiber bundle, etc.) and a map $u : M \to N$, can one construct a so-called lift --- that is, a map $U : M \to E$ such that $u = \Pi \circ U$ --- or show an obstruction to doing so? In the continuous setting, this is a purely topological problem, but new questions arise when considering Sobolev mappings. In particular, given $u$ in a fixed Sobolev class, one would like to establish Sobolev regularity for the lift, preferably with direct estimates in terms of the original mapping.
In this talk, we briefly discuss the classical results, focusing primarily on the special case of the Hopf bundle $H: \mathbb{S}^3 \to \mathbb{S}^2$. We then present a new result concerning Hopf lifts of maps in fractional-order Sobolev spaces, along with some interesting consequences for obtaining quantitative estimates of the Hopf invariant.
Based on joint work with Katarzyna Mazowiecka, Armin Schikorra and Jean Van Schaftingen
