Invented by Andreas Floer in 1988 to solve Arnold's
conjecture, (symplectic) Floer Homology is a machinery to relate the
existence of periodic trajectories of an Hamiltonian flow on a
symplectic manifold M with the homology groups of M, analougously to
Morse Homology. Indeed this is done by developing an infinite
dimensional Morse theoretic framework adapted to a certain functional
(the action functional) on the loop space of M, whose critical points
are the periodic trajectories of the given hamiltonian flow.
Despite the topological nature of the results, the construction is
technically quite heavy, involving hard analysis and elliptic systems of
Together with Andrei Agrachev and Antonio Lerario we are developing a
method to construct such infinite dimensional homology invariants using
only soft and essentially finite dimensional tools. In my talk I will
present our approach.
The main feature consists in approximating the loop space with finite
dimensional submanifolds of increasing dimension, we do this with the
language of control theory, and then interpret asymptotically the
information provided by classical Morse theory
- PDE CDT Lunchtime Seminar
There is a beautiful mathematical theory of how independent agents tend to synchronise their behaviour when weakly coupled. Examples include how audiences spontaneously rhythmically applause and how nearby pendulum clocks tend to move in sync. Another famous example is that of the London Millennium Bridge. On the day it opened, the bridge underwent unwanted lateral vibrations that are widely believed to be due to pedestrians synchronising their footsteps.
In this talk Alan will explain how this theory is in fact naive and there is a simpler mathematical theory that is more consistent with the facts and which explains how other bridges have behaved including Bristol's Clifton Suspension Bridge. He will also reflect on the nature of mathematical modelling and the interplay between mathematics, engineering and the real world.
Alan Champneys is a Professor of Applied Non-linear Mathematics at the University of Bristol.
Please email firstname.lastname@example.org to register.
The Oxford Mathematics Public Lectures are generously supported by XTX Markets.
- Oxford Mathematics Public Lectures
Pick's theorem is a century-old theorem in complex analysis about interpolation with bounded analytic functions. The Kadison-Singer problem was a question about states on $C^*$-algebras originating in the work of Dirac on the mathematical description of quantum mechanics. It was solved by Marcus, Spielman and Srivastava a few years ago.
I will talk about Pick's theorem, the Kadison-Singer problem and how the two can be brought together to solve interpolation problems with infinitely many nodes. This talk is based on joint work with Alexandru Aleman, John McCarthy and Stefan Richter.
- Functional Analysis Seminar
Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set admitting a countable cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an exactness result about the completion functor in the category of ultrametric locally convex vector spaces, and in particular we prove that a strict morphism between these spaces has closed image if its kernel is Fréchet.
- Algebra Seminar