165 research outputs found
Topological Hochschild cohomology and generalized Morita equivalence
We explore two constructions in homotopy category with algebraic precursors
in the theory of noncommutative rings and homological algebra, namely the
Hochschild cohomology of ring spectra and Morita theory. The present paper
provides an extension of the algebraic theory to include the case when is
not necessarily a progenerator. Our approach is complementary to recent work of
Dwyer and Greenlees and of Schwede and Shipley. A central notion of
noncommutative ring theory related to Morita equivalence is that of central
separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild
cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of
A. We introduce a notion of topological Azumaya algebra and show that in the
case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a
commutative ring this notion specializes to classical Azumaya algebras. A
canonical example of a topological Azumaya R-algebra is the endomorphism
R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod
2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over
the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the
determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2.
As far as we know this is the first calculation of THH(A,A) for a
noncommutative S-algebra A.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-29.abs.htm
Edge Label Inference in Generalized Stochastic Block Models: from Spectral Theory to Impossibility Results
The classical setting of community detection consists of networks exhibiting
a clustered structure. To more accurately model real systems we consider a
class of networks (i) whose edges may carry labels and (ii) which may lack a
clustered structure. Specifically we assume that nodes possess latent
attributes drawn from a general compact space and edges between two nodes are
randomly generated and labeled according to some unknown distribution as a
function of their latent attributes. Our goal is then to infer the edge label
distributions from a partially observed network. We propose a computationally
efficient spectral algorithm and show it allows for asymptotically correct
inference when the average node degree could be as low as logarithmic in the
total number of nodes. Conversely, if the average node degree is below a
specific constant threshold, we show that no algorithm can achieve better
inference than guessing without using the observations. As a byproduct of our
analysis, we show that our model provides a general procedure to construct
random graph models with a spectrum asymptotic to a pre-specified eigenvalue
distribution such as a power-law distribution.Comment: 17 page
BGWM as Second Constituent of Complex Matrix Model
Earlier we explained that partition functions of various matrix models can be
constructed from that of the cubic Kontsevich model, which, therefore, becomes
a basic elementary building block in "M-theory" of matrix models. However, the
less topical complex matrix model appeared to be an exception: its
decomposition involved not only the Kontsevich tau-function but also another
constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition
function. The BGW tau-function can be represented either as a generating
function of all unitary-matrix integrals or as a Kontsevich-Penner model with
potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page
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