1,207 research outputs found
Abstract Hodge decomposition and minimal models for cyclic algebras
We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy
Feynman diagrams and minimal models for operadic algebras
We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A∞-algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras
Cocommutative coalgebras: homotopy theory and Koszul duality
We extend a construction of Hinich to obtain a closed model category
structure on all differential graded cocommutative coalgebras over an
algebraically closed field of characteristic zero. We further show that the
Koszul duality between commutative and Lie algebras extends to a Quillen
equivalence between cocommutative coalgebras and formal coproducts of curved
Lie algebras.Comment: 38 page
Maass waveforms and low-lying zeros
The Katz-Sarnak Density Conjecture states that the behavior of zeros of a
family of -functions near the central point (as the conductors tend to zero)
agrees with the behavior of eigenvalues near 1 of a classical compact group (as
the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo
and Sarnak proved that the behavior of zeros near the central point of
holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal
matrices for suitably restricted test functions . We prove similar
results for families of cuspidal Maass forms, the other natural family of -functions. For suitable weight functions on the space of
Maass forms, the limiting behavior agrees with the expected orthogonal group.
We prove this for \Supp(\widehat{\phi})\subseteq (-3/2, 3/2) when the level
tends to infinity through the square-free numbers; if the level is fixed
the support decreases to being contained in , though we still uniquely
specify the symmetry type by computing the 2-level density.Comment: Version 2.1, 33 pages, expanded introduction on low-lying zeros and
the Katz-Sarnak density conjecture, fixed some typo
Combinatorics and formal geometry of the master equation
We give a general treatment of the master equation in homotopy algebras and
describe the operads and formal differential geometric objects governing the
corresponding algebraic structures. We show that the notion of Maurer-Cartan
twisting is encoded in certain automorphisms of these universal objects
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Cocommutative coalgebras: homotopy theory and Koszul duality
We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commutative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras
Koszul-Morita duality
We construct a generalization of Koszul duality in the sense of Keller-Lefèvre for not necessarily augmented algebras. This duality is closely related to classical Morita duality and specializes to it in certain cases
Distribution of Missing Sums in Sumsets
For any finite set of integers X, define its sumset X+X to be {x+y: x, y in
X}. In a recent paper, Martin and O'Bryant investigated the distribution of
|A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They
also conjectured the existence of a limiting distribution for |A+A| and showed
that the expectation of |A+A| is 2n - 11 + O((3/4)^{n/2}). Zhao proved that the
limits m(k) := lim_{n --> oo} Prob(2n-1-|A+A|=k) exist, and that sum_{k >= 0}
m(k)=1.
We continue this program and give exponentially decaying upper and lower
bounds on m(k), and sharp bounds on m(k) for small k. Surprisingly, the
distribution is at least bimodal; sumsets have an unexpected bias against
missing exactly 7 sums. The proof of the latter is by reduction to questions on
the distribution of related random variables, with large scale numerical
computations a key ingredient in the analysis. We also derive an explicit
formula for the variance of |A+A| in terms of Fibonacci numbers, finding
Var(|A+A|) is approximately 35.9658. New difficulties arise in the form of weak
dependence between events of the form {x in A+A}, {y in A+A}. We surmount these
obstructions by translating the problem to graph theory. This approach also
yields good bounds on the probability for A+A missing a consecutive block of
length k.Comment: To appear in Experimental Mathematics. Version 3. Larger computations
than before, conclusively proving the divot exists. 40 pages, 15 figure
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