The
notion of operad was introduced to encode algebraic operations. This
theory arose in the 70's at the University of Chicago where it was
created in order to recognize iterated loop spaces in algebraic
topology. Since the early 90's, operads have encountered a renaissance.
Still then, they have been successfully used in many fields of
mathematics like algebraic topology, differential geometry, universal
algebra, algebraic combinatorics, category theory, mathematical physics
and computer science.
Lectures:
Operads and Koszul duality theory [Loday-Vallette]
The purpose of these lectures is to introduce the notion of operad, to
describe the Koszul duality theory and to explicit the applications
to homotopy algebras.
1)
Operad [Loday]
S-module, Schur functor, Operads,
non-symmetric operads, equivalent definitions of operads, End_A, free
operad,
cooperad. Mention of the variations (cyclic operad, properad,
prop).
A)
Twisting morphism [Vallette]
Differential graded algebra and
dg coalgebra, twisting morphisms, bar and cobar construction for
(co)algebras,
twisted tensor product, comparison lemmas, twisting morphisms fundamental theorem.
2)
Operadic twising morphism [Loday]
Differential graded operads and
dg cooperads, operadic twisting morphisms, bar and cobar construction
for (co)operads,
twisted tensor product, comparison lemmas, twisting morphism fundamental theorem.
B)
Koszul duality for associative algebras [Vallette]
Quadratic data, Koszul dual,
Koszul algebras and Koszul criterion, PBW bases, inhomogeneous Koszul
duality theory.
3)
Koszul duality for operads [Loday]
Quadratic
data, Koszul dual, Koszul operads and Koszul criterion, mention of
inhomogeneous Koszul duality theory,
A-infini algebras and Stasheff
polytope.
C)
Homotopy algebras [Vallette]
Three equivalent definitions of
homotopy algebras (algebra over the Koszul resolution, twisting
morphism, square-zero coderivation),
A_\infty-algebras and
L_\infty-algebras, \infty-morphisms of homotopy algebras, transfer
theorem, Massey products,
bar construction of homotopy algebras, rectification.
References:
J.-L. Loday and B. Vallette, Algebraic operads, book in preparation.
M. Markl, S. Shnider and J. Stasheff, Operads in algebra, topology and physics. Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, 2002.
S. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39-60.
V. Ginzburg, M.M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1995), 203-272.
E. Getzler, J.D.S. Jones, Operads, homotopy algebra and iterated
integrals for double loop spaces, preprint, arXiv:hep-th/9403055 (1994).
M. Kontsevich and Y. Soibleman, Deformations of algebras over operads and the Deligne conjecture.
Conférence Moshé Flato 1999, Vol. I (Dijon),
255--307,
Diagram rewriting [Lafont]
Keywords: Word problem and word rewriting, diagrams, sequential and parallel composition
Terminology: basic case, classical case, linear case, and quantum case
Examples: finite permutations, finite maps, linear boolean maps, and linear boolean permutations
Main tools: canonical form, termination, confluence, and critical peaks
References:
Y. Lafont, Towards an Algebraic Theory
of Boolean Circuits, Journal of Pure and Applied Algebra 184 (2-3), p.
257-310, Elsevier (2003)
http://iml.univ-mrs.fr/~lafont/pub/circuits.pdf
A. Burroni, Higher dimensional word problem with application to
equational logic, Theoretical Computer Science 115, 43-62,
Elsevier (1993)
http://people.math.jussieu.fr/~burroni/mapage/highwordpb.pdf
Y. Lafont, Algebra and geometry of rewriting, Applied Categorical Structures 15, p. 415-437, Springer-Verlag (2007)
http://iml.univ-mrs.fr/editions/preprint2006/files/lafont_agr.pdf
Y. Lafont & Pierre Rannou, Diagram rewriting for orthogonal
matrices: a study of critical peaks, Rewriting Techniques and
Applications, 19th International Conference, Hagenberg, Austria, July
15-17, LNCS 5117, Springer-Verlag (2008)
http://iml.univ-mrs.fr/~lafont/pub/orthogonal.pdf Homotopy theory of operads [Moerdijk]
1) Quillen model structures in the theory of operads
I will give the definition
of Quillen model structure, prove the existence of such - under
suitable conditions- on operads, on algebras over an operad, and on
modules over such an algebra.
2.) The Boardman-Vogt resolution and the homotopy coherent nerve
I
will describe a particular construction of cofibrant resolutions
of operads, due to Boardman and Vogt. The homotopy coherent nerve of a
topological category is adjoint to a special case of this construction.
References:
C. Berger, I. Moerdijk, Axiomatic homotopy theory for operads, Comm. Math. Helv. 78 (2003).
C. Berger, I. Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories, Topology 45 (2006).
J.M. Boardman, R.M. Vogt, Homotopy invariant algebraic structures on topological spaces, Springer LNM 347 (1973).
M. Hovey, Model Categories, AMS (1999).
V. Hinich, Homological algebra of homotopy algebras, Comm. Alg. 25 (1997).
Cochain models of topological spaces [Fresse]
The purpose of these lectures is to explain applications of operads
to the definition of cochain models
in algebraic topology.
Abstract :
pdf file
References:
[BF] C. Berger, B. Fresse, Combinatorial operad actions on cochains,
Math. Proc. Cambridge Philos. Soc. 137 (2004), pp. 135-174.
[FHT] Y. Félix, S. Halperin, J.-C. Thomas, Rational homotopy theory,
Graduate Texts in Mathematics 205, Springer-Verlag, 2001.
[F1] B. Fresse, Homology of partition posets and Koszul duality of operads,
in "Homotopy theory and its applications (Evanston, 2002)",
Contemp. Math. 346 (2004), pp. 115-215.
[F2] B. Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer-Verlag, 2009.
[F3] B. Fresse, The bar complex of an E-infinity algebra, preprint (2008).
[F4] B. Fresse, The iterated bar complex of E-infinity algebras and homology theories,
preprint (2008).
[F5] B. Fresse, Koszul duality of En-operads, in preparation.
[HS] V. Hinich, Schechtman, On homotopy limit of homotopy algebras,
in "K-theory, arithmetic and geometry (Moscow, 1984-1986)", Lecture Notes in Math. 1289, Springer-Verlag (1987), 240-264.
[K] M. Kontsevich, Operads and motives in deformation quantization,
in "Moshé Flato (1937-1998)",
Lett. Math. Phys. 48 (1999), 35-72.
[M] M. Mandell, E¥ algebras and p-adic homotopy theory,
Topology 40 (2001), pp. 43-94.
[R] A. Robinson, Gamma homology, Lie representations and E-infinity multiplications,
Invent. Math. 152 (2003), 331-348.
References [BF] and [F1-5] are available from the web page: http://math.univ-lille1.fr/~fresse/Articles.html
Slides : Part I,
Part II
Little cube operads and applications to algebraic topology [Lambrechts]
They main theme of these lectures
will the little cube operads, which are easy examples of
topological operads. They are ones of the first operads ever
introduced. They are used to recognize n-fold loop spaces.
References:
1) Boardman, J. M.; Vogt, R. M. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York, 1973.
2) May, J. P. The geometry of iterated loop spaces. Lectures Notes in Mathematics, Vol. 271. Springer-Verlag, Berlin-New York, 1972. PDF
Configuration spaces and Leibniz operad [Merkulov]
We give a detailed intruduction into
geometric and operadic descriptions of
some compactified configuration spaces.
In the first lecture we explain Fulton-MacPherson compactification
of a configuration space of n points in the Euclidean N-space and its
relation
to the minimal reolution of the operad of Leibniz algebras, and to Poisson
geometry.
In the second lecture we explain a new compactification of a configuration
space of n points
in the hyperbolic N-space, and its relation to the dg operad of strongly
homotopy automorphisms of Leibniz-infinity algebras,
and to exotic automorphisms of Lie bialgebra structures and of
Poisson structures.
References:
1) J.L. Loday, "Dialgebras" 7-46, LNM 1763 (2001)
2) M. Kontsevich, "Deformation Quantization of Poisson Manifolds", Lett. Math. Phys. vol.66, 157–216, (2003).
3) S.A. Merkulov "Exotic automorphisms of the Schouten algebra of polyvector fields", arXiv:0809.2385 (2008)
Koszul duality in algebraic topology [Sinha]
We survey the topology which led to the original bar and cobar
constructions,
for both associative algebras and coalgebras and for Lie algebras and
commutative coalgebras. These constructions are often viewed as part of
the
larger theory of Koszul duality of operads, so this survey is meant to
offer an
historical perspective on the most prominent cases of that theory. We
also
explain recent work which shows that Hopf/linking invariants for
homotopy are
at the heart of the duality between commutative algebras and Lie
coalgebras.
References:
1) D.P. Sinha
"Koszul duality in algebraic topology: an historical perspective",
arXiv:1001.2032
Prerequisite:
Elements of homological algebra, notions of category
and functor (adjoint functors), notions on the representations of the
symmetric groups.
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Last update: January 23rd,
2009