Thread by @robertghrist

a technical thread for a technical paper:
"Cellular sheaves of lattices & the Tarski Laplacian"
with @hansmriess
arxiv.org/abs/2007.04099
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here's the setup: cellular sheaves are data structures over a cell complex (such as a social or neural network) that attach algebraic data to nodes, edges, etc.
they are wonderful mathematical entities.
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a sheaf of vector spaces has vector spaces atop cells, with dimensions varying from cell-to-cell.
these are all glued together by a network of linear transformations, respecting the base structure (cf., the adjacency matrix of the network)
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ok. if you have a sheaf of *inner product* spaces (think vector spaces w/bases), there is a beautiful *Hodge Laplacian* for the sheaf which generalizes the graph Laplacian.
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in previous work with @_jakobhansen , this Hodge Laplacian was shown to have nice spectral properties and is useful for computing the sheaf cohomology.
you can set up a "heat equation" and harmonic-flow your way to cohomology.
that's useful in, e.g., opinion dynamics.
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that's the recent past.
the present challenge: redo this, but for sheaves whose data types are not vector spaces, but *lattices*...
(think: lattices of subsets of a set under intersection/union, or Boolean algebras, e.g.)
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there's a problem: there is no natural sheaf cohomology (the appropriate category of lattices and lattice morphisms is not abelian).
you have a global section functor (H^0), but no higher sheaf cohomology, alas.
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here's the idea:
we define a Laplacian -- an endomorphism on 0-cochains (data distributions over vertices).
we argue that it "works" like a Laplcian should.
then we try to do harmonic flow.
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it's well-known that a discrete-time harmonic flow for signal processing on a graph uses the graph Laplacian L to diffuse real-valued signals via (I-L), where I is the identity.
the fixed point set of this map is the harmonic distributions (locally constant).
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main theorem: for lattice-valued sheaves over a cell complex, the fixed point set of (I-L), where L is this new *Tarski Laplacian* is the global sections of the sheaf.
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the name *Tarski Laplacian* is in homage to the Tarski fixed point theorem for lattices, which implies that for complete stalks, the global sections forms a quasi-complete nonempty sublattice.
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once you have this, it's possible to run consensus algorithms on lattice sheaves, compute global sections algorithmically, & more.
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but the Tarski Laplacian generalizes easily to higher dimensional cochains...
and the fixed point set of (I-L) is a complete quasi-sublattice of the higher cochain space.
so...
let's call this the *Tarski cohomology*
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this is cool: it's kind of like exploiting the Hodge theorem to get at sheaf cohomology. the latter doesn't seem to want to exist, but the former does...
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but: it's not an honest sheaf cohomology theory, alas.
in fact, there are a few alternatives one could propose.
the paper explores that a bit, with comparisons.
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(if you've made it this far, congratulations & apologies are perhaps in order)
(perhaps not in that order though)
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"WHY?!?" you ask.
well...
there are reasons and then there are hopes.
i'd rather not say too much yet.
but any time you can take the tools of homological data analysis and extend them to more general data types, that's a good sign of things to come.
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this is part of the Ph.D. thesis work of Hans Riess @hansmriess in @ESEatPenn Electrical/Systems Engineering at the University of Pennsylvania @Penn
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