Spectral numbers in Floer theories

The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ``nondegenerate spectrality'' axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.Comment: The main algebraic lemma (the former Theorem 1.6) has been given a more general statement and a shorter proof. This allows us to generalize the main results to cover Floer theories with integral or local coefficients. Submitted version. 13 page

Comment on Ryder's SINBAD Neurosemantics: Is Teleofunction Isomorphism the Way to Understand Representations?

The merit of the SINBAD model is to provide an explicit mechanism showing how the cortex may come to develop detectors responding to correlated properties and therefore corresponding to the sources of these correlations. Here I argue that, contrary to the article, SINBAD neurosemantics does not need to rely on teleofunctions to solve the problem of misrepresentation. A number of difficulties for the teleofunction theories of content are reviewed and an alternative theory based on categorization performance and statistical relations is argued to provide a better account and to come closer to the practice in neuroscience and to powerful intuitions on swampkinds and on broad/narrow content