Student Analysis Seminar

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Quantum theory, uncertainty principles, and contractive estimates (Part I)

Speaker: Omar Abdelghani, CIMS

Location: Warren Weaver Hall 517

Date: Tuesday, November 18, 2025, 11 a.m.

Synopsis:

It is known that in classical electrodynamics, Coulomb systems are unstable and tend to collapse, so atoms cannot exist in a classical universe.
We will show that this is not so in quantum theory:  Nonrelativistic N-body Coulomb systems are stable and cannot collapse. We will discuss why the the traditional appeal to Heisenberg's uncertainty principle does not imply stability, how Sobolev inequalities are stronger uncertainty principles that recover stability (as well as well-posedness) in 3 dimensions, and how these inequalities can be equivalently encoded in certain semigroup inequalities. If time permits, we will also sketch how these semigroup formulations of Sobolev inequalities are useful for proving similar stability results in the infinite dimensional setting of two dimensional quantum field theory.