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\author{Gabriel C. Drummond-Cole}
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\title{Deformation Theory Workshop}
\begin{document}\maketitle
\section{Soibelman}
There are several motivations for this work.  Some is purely geometric.  Everyone knows the Gromov Witten invariants.  Fix a genus, map into a Calabi Yau, it's a counting problem.  Then there are also the lesser known Donaldson Thomas invariants.  Locally free sheaves of rank one on three dimensional Calabi Yau manifolds over $\cC$.  The number of paremeters is more or less the same; this is the case when we count with fixed Chern character.  The number of parameters in both cases is 1 plus the dimension af the second homology.  The conjecture was that you can sort of identify them.  The best known work was by \wh, where they first stated rigorously the counting problem for the right hand side, and then explained in what sense these should be equal.  This is proven for a wide class of three dimensional Calabi Yaus.  The left hand side belongs to string theory while the right hand side belongs to gauge theory.  You work with vector bundles with flat in antiholomorphic direction connections.  There should be a general categorical framework in which such equalities can be understood.  In Gromov Witten, it's more traditional so something has been done, by Costello, Kontsevich, myself and others, to understand this as a problem about algebras over a prop.  Although the formulas do not exist, hopefully it will be done at some point.  For Donaldson Thomas, it was widely open.  In geometry there are some examples, a general understanding but it was beyond methods of $A_\infty$ categories.  Physicists speak obut the counting of $BPS$ states, but there is no mathematical definition.  What we have done is suggest a mathematical framework to treat this.  

In order to be more precise, let me recall the definition of a Calabi-Yau category and then connect to the geometry.

The definition can be given for an arbitrary field of characteristic zero over $k$.

A Calabi Yau category of dimension $d$ is a $k$-linear $A_\infty$ category (triangulated) endowed with a pairing of degree $d: Hom(E,F)\otimes Hom(F,E)\to k[-d]$. so that $W_N(a_1,\ldots, a_N)=(m_{n-1}(a_1,\ldots a_{N-1}),a_N)$ is cyclic invariant, where $m$ is the higher \wh.

Now the potential $W_E(\alpha)=\sum_{n\ge 2} \frac{W_n(\alpha,\ldots,\alpha)}{n},\alpha\in Hom(E,E)[1]$.  Here I can in fact take $Ext^1(E,E)$ [by some magic I didn't understand].

An example: let's say $X$ is a complex three dimensional Calabi Yau manifold, compact.  We can work with a $dg$ model for the category of derived sheafs.  This means for $\cC$ that I take $(\Omega^{0,*}(X),\bar{\partial})-dg$ modules, which are projective as graded modules.

The pairing $(f,g)=\int_X Tz(fg)\wedge \Omega^{3,0}_X$ $E$ is a bundle over $X$, and now the potential is $$W_E(\alpha)=CS_{hol}(\alpha)=\frac{(\bar{\partial}\alpha,\alpha)}{2}+\frac{(\alpha\wedge\alpha,\alpha)}{3}$$

So we want to count critical points, which turn out to be holomorphic vector bundles.  We can fix Chern classes to specify the topological data.  This is an ill-defined problem because we need some kind of stability condition.  \wh did this with 4 dimensional Yang mills for two dimensional surfaces.  Even if you know what in this general categorical framework what are the stable or semistable objects, then there is the question of what to count.  Now you need to \wh discrete parameters, so you have a $K$-theory, so you can fix a class $\gamma\in K_o(\Cc)$.  Then you can look at semistable objects with fixed $\gamma$.  Then $n_\gamma=\# \Cc^{ss}_\gamma$ so $Ze(t)=\sum_\gamma n_\gamma t^\gamma$.  The number can be infinite, but if you have a Chern character, you can work with finite \wh lattices.  What to do if it's infinite?  It still can have good structures.  You might be able to compute the Euler characteristic $\chi(\Cc^{ss}_\gamma)$.  or the framework of Donaldson Thomas invariants, there were two ideas of \wh.  One is that if you have a moduli space $M$ which is good, meaning that it has a perfect obstruction theory, the dg Lie algebra that controls deformations has nonzero homology only in dimensions one and two in duality with one another, then there exists an integer valued constructible function $B:M\to \zZ$ and the invariants $\int_M B = \sum_{n\in\zZ}n\chi B^{-1}(n)$

In a sense this is a direct application of all of the ideas of \wh, who at the beginning of the 80s suggested to use the Euler characteristic as a measure.

So if $M$ is compact, proper, then there is a virtual fundamental class and the corresponding integral is $\int_{[M]^{vir}}1$.  This is one of his ideas but the second one was even better.  In fact, even from this point we can see how motivic integration can be \wh.  Here we have something that can be decomposed into the disjoint union, then the fundamental classes are additive.  There are several other things like this.  We can integrate over Hodge polynomial instead of Euler characteristic, for example.  

The second thing is what kind of function $B$ can appear.  If you have a smooth variety and you have a function $f\in \Oo(X)$ then the local invariant assigned to $x\in X$ is $(-1)^{\dim X}$ is the following.  It is $(-1)^{\dim X}(1-\chi(MF_x(f))$.  In order words we want $M$ to sit inside $X$ as the set of critical points of a function, or we need to extend this theory to allow us to work with less good schemes, because there are some restrictions, so the moduli space must be good, and this is difficult to prove.  We can see part of this story on the left blackboard because we have the function which is a potential.  The set of critical points is a set of objects.  Here there is a parallel, especially since \wh \wh, so there are some ingredients which can be used in order to combine this with three dimensional Calabi Yau categories.  For those who know something about motivic integration, the basic class of spaces are constructible sets, not just schemes, so we need to see some constructibility, so the categories should not just be Calabi Yau categories, they should have some additional structure.  The structure that we impose on the categories will be that they are ind-constructible.  The notion is quite general.  It is a kind of abstract nonsense, I don't want to give a lecture, but at least I can highlight some basic properties.  A constructible set is something in the boolean algebra of closed schemes.  So we have an $A_\infty$ category.  We want $Ob(\Cc)$ to be an inductive limit of constructible sets.  Since we can decompose into complements, it's a countable disjoint union of constructible sets (e.g., closed schemes).  If you have in mind that these are objects, then every object is acted on by its group of automorphisms, so $X_i$ is acted on by an affine algebraic group.

In fact what we are going to have is an ind-constructible set, the thing for morphisms, it's like an $A_\infty$ category but a little bit nonlinear.  For every $n$ we define the constructible ind-constructible bundle of morphisms of degree $n$. So we can restrict to $X_i\times X_j$, a constructible piece, and there we have the action of the group $G_i\times G_j$, so the bundle should be invariant under this action.  Also for technical reasons, we need that it be trivial if $n$ is very small.  I am not being very precise so you can disregard this condition.  What are the higher compositions?  Usually they are from tensor product of morphisms to morphisms, so we can take morphisms $Hom^{\ell_1}\otimes\cdots\otimes Hom^{\ell_n}$, well,
$$
p_{12}^* Hom^{\ell_1}\otimes\cdots\otimes p_{n,n+1}^*Hom^{\ell_n}\to p_{1,n+1}^*Hom^{\sum \ell_i +2 -n}
$$
These are pullbacks with respect to the corresponding factors.
Then we can take the $\bar{k}$ axioms of an $A_\infty$ category, unital, triangulated structure.

Okay, so examples of such categories.  If you have $\Dd^{\Cc}(X)$, then you have $Perf(X)$, and we can consider $Perf(A-mod)^{proj}$.  You can \wh and then consider nilpotent extensions.  Secondly we should take care about direct summands.  This can be dealt with in the language of \wh.  This is by solving polynomial equations of increasing size.  So we can get that the objects are a union of schemes.  The morphisms are then a long story.  Now let me continue.  If you have ind-constructible structures.  

I should briefly mention pieces of the theory of motivic functions, and actually may be returning to the beginning of my lecture when I get to stability.  If you fix \wh then it will be required to be a constructible set.  Then the question will be how to count something using the constructible pieces.  But what kind of functions we consider, so suppose we have a constructible set $X$ over $k$, it's a reminder, so what is the ring of motivic functions?  It's one of the simplest rings we use.  It's generated by symbols $\pi:S\to X$, so it should be thought of as a constructible family.  There is one condition, that if we decompose $S$ into a disjoint union, this is additive.  If you have a constructible morphism of two such things $f:X\to Y$ then you have pullback and pushforward $f_*:Mot(X)\to Mot(Y)$ and $f^*:Mot(Y)\to Mot(X)$.

Okay, so then the direct product gives the product structure and it is indeed a ring.  This ring contains, well, let's consider a special case when $X$ is a spectrum of the field.  Then we have a module of the affine line, and typically we are going to add the inverse of the motive of the affine line, so we extend this by adding $[\lL^{-1}]$, so certainly this is a module over \wh of a point.

All right, and there is a variety of other rings of motivic functions, for example those that take care of constructible sets acted on by groups $Mot^G(X)$.  In this case we consider $G$-equivariant families and all decompositions are required to be equivariant.

All right, and then since I do not have much time, let me tell you how this machinery can be applied to produce some algebraic object related to an arbitrary ind-constructible category.  This object is a motivic Hall algebra, this is where my invariants will lead, later when I define them.  This is kind of an independent thing.  This is the algebra of an ind-constructible $A_\infty$ category.  It's a generalization of the derived Hall algebra.  Here we will work with an arbitrary field.  We can have nonzero or zero characteristic.  I can consider motivic functions $[Y_1\to Ob\ \Cc]$.  All right.  And obviously they generate an Abelian group formally.  How to multiply such things?  $$[\pi_1:Y_1\to Ob\ \Cc][\pi_2:Y_2\to Ob\ \Cc]=\sum_n [W_n\to Ob\ \Cc]\lL^{-n}$$ where $W_n=\{(y_1,y_2,\alpha),y_i\in Y_i,\alpha\in Ext^1(\pi_2(y_2),\pi_1(y_1)$, and the condition is $\sum_{i\le 0} (-1)^i dim Ext^i(\pi_2(y_2),\pi_1(y_1))=n$
We can consider constructible sets which are just pairs of points, and then we get a slightly modified formula, $[E_1][E_2]=q^{-\sum (-1)^i \dim Ext^i(E_2,E_1)\sum_{\alpha\in Ext^1(E_2,E_1)}[E_\alpha]$
The theorem si that $H(\Cc)$ is associative.  The proof uses deformation theory so I think my time is up so maybe I stop here today and next time I will discuss the invariants and maybe some applications to some counting problems and maybe some applications to \wh algebras.
\section{Zenalian}

So, thank you for having me.  I would like to talk a little bit about the deformation complex and its algebraic properties and discuss recent work with Tradler and Costello.  The simplest setting one can imagine is the following.  Suppose you have a one dimensional complex vector space.  How can I deform this keeping fixed the underlying real structure.  You are starting with a two dimensional real vector space, and you're asking yourself, what's the structure of all $\cC$-linear structures.  You're considering maps $J:V\to V$ so that $J^2=-\id$.  This is a set, but it has more structure.  It's a metric space.  You can define the distance between $J_1$ and $J_2$ in the following way.  You can take $J_1$, and in its complex structure you can get a Hermitian metric $\sqrt{z\bar{z}}$, and now look at the circle $e^{i\theta}$ from the second structure.  Then you can look at the ratio of the elipse's radii.  Take the log, and that turns out to be a metric space.  You look at geodesics and they're like, well, you get the Poincar\'e disc with the hyperbolic metric.  So this structure is a Riemann surface with an infinitessimal metric.  It's a hyperbolic disk.  We're defining a functor from the category of two dimensional oriented real vector spaces to hyperbolic disks, simply connected Rimann surfaces, compelete, with constant $-1$ curvature.  Look at a point, look at its tangent space.  What is this space of infinitessimal deformations?  It's $\varphi:V\to V$ which commute with $J$.  It wants to be an algebra, but it can't quite be.  This, of course, has consequences.  If you take a $2$ dimensional smooth surface, there is this bundle of hyperbolic disks sitting on top of any smooth surface.  A section of this is a varying family of $J$s.  This is in effect saying that there's a one to one correspondence between sections and complex structures.  This gives you a visual picture of what a deformation of a complex structure is.  Infinitessimally you will get a vector field along a curve.  There is a canonical integration along geodesics.  Then these are maps $\varphi: TM\to TM$ that anticommute with $J$.  You can write these things and they are called Beltrami differentials, $f(z,\bar{z})=\frac{\partial}{\partial z}\otimes d\bar{z}$.

You look at the derivative of a vector field and there is a part that doesn't distort circles and a part that does, $\bar{\partial}V$.  So really what you're doing is taking $H^1(M,TM)$.  We're interested in problems that have this nice deformation property.  This doesn't hold in higher dimensions, but you can deal with Calabi Yau manifolds.  For reasons that are defendible, you should consider all of the homology, with values in the extended sheaf $H^*(M,\wedge^\cdot TM)$.  You should always work at the level of complexes.  The complex is $\Gamma(M,\wedge \bar{T}^*M\otimes \wedge TM)$.  The right hand piece is holomorphic so you have a $\bar{\partial}$ resolution.  Similarly, you can consider a complex where you do the same but with $T^*M$ on the right.  This will have a $\partial$ resolution.  In the context of a Calabi Yau manifold, you get an isomorphism between these two complexes, which can be tensored with the identity, and the differentials don't match.  So this second $\partial$ moves up as $\Delta$.  Kontsevich \wh, the first page, this goes to say that this is a differential BV algebra.  It's a square zero differential, not a derivation.  You look at the discrepancy, it gives you a binary operation, and after a shift it gives you a Lie algebra.  This is the structure of a deformation complex, this differential graded $BV$-algebra.  That's a satisfactory answer, but when you look at the induced action, on the homology you have $\Delta$, but unfortunately it vanishes on homology.  So you have to do more work.  Baranikov Kontsevich do this and you get what is called a Frobenius manifold structure in the neighborhood of zero.  An equivalent way of describing this is, saying that this complex is a $BV$ algebra is saying that it's an algebra over the homology of the framed little disks.  Since this is a complex, it's a differential operator.  A more proper understanding of why this is zero is that this is the action of $H_*(\bar{M}_{n,1})$.  This map takes some doing to make.

When you look at this picture, passing to the compactification of moduli space has to do with triviality of a circle action.  One can draw pictures and see that passing to the Deligne Mumford compactification you need for these circle actions to be trivial.  Okay.  

You can sort of see where the circle action comes into play.  That's look at the moduli space of three points on the Riemann sphere.  The moduli space is a single point.  The next interesting thing is four points.  Three of these are fixed and the moduli space is the placement of the fourth point.  Before compactification, it is the sphere minus three points, which is not compact.  You must compactify somehow.  SO you can look at the geodesic separating two points from two other points, and you see what happens when this approaches zero.  If you look at it in the plane, you can consider an annulus, and the modulus of the annulus goes to infinity.  There's a geodesic neck that shrinks as this goes to zero.  What happens when you compare hyperbolic metrics, the lengths in the smaller domain are bigger.  If this geodesic shrinks to zero, the one in the plane shrinks to zero.  So the picture at infinity are two curves kissing at a point.  It doesn't matter if they're like this or that, there are no moduli for the kissing.  You could not go all the way to the Deligne Mumford compactification.  You could put a tangent vector, that's the real compactification.  The precise statements of this are in the latest paper of Kontsevich et al., and they describe how when the circle action is trivial you can pass to the compactification depending on how you trivialize that circle action.  

Now, this was sort of, the story in genus zero, but one would like to understand more structure on this deformation complex, so what will we do?  First of all, you can't continue using geometry, you must use algebra.  At least I know no way of doing it.  You can push the algebra away in one aspect.  So for example, you can ask, is this an algebra over the genus zero moduli space, or the higher moduli space.  There are two answers I know.  Many people have worked on the algebraic structure on the Hochschild complex.  One way is that you use this Kontsevich idea that deformations of the complex structure is the bounded derived category of coherent sheaves.  You have $\Gamma(M,\wedge^\cdot \bar{T}^*M\otimes \wedge TM)$.  So you start by saying this is Hochschild of the category of bounded coherent sheaves $CH_.(\quad D^b(M))$.  Then you can use Costello's machinery to get an action of the chains TCFC.  This will give you an action of the uncompactified moduli space.  You still need to talk about the triviality of the circle action.  In a moment I will talk about another example related to string topology, not to deformation theory.  I think the methods of $A_\infty$ categories are not the best way of doing this.  The other approach is to start with this bounded derived category of coherent sheaves, and this category is controlled by one object, a single algebra $A$ whose homology is this deformation complex.  Start from $D^b(M)$ and then pass to the strong generator $E$ and then look at $End(E)$ and resolve it, $\Omega^{\cdot,*}(M,End(E))$.  Then you get this thing, and can do this other sort of way of doing things which is more analytic, using heat kernels and so on.  This is joint with Tradler and Costello.  This, well, I'm not going to give a precise description because it's involved.  The input is a Hermitian Calabi Yau elliptic space.  It's a space controlling an algebra.  Then using some construction you get a TCFT.  Once again it's open, you get de Rham currents on the open moduli space acting on this algebra.  The case in which it doesn't hold, the much simpler thing to study, is not to take this algebra, but instead take the de Rham complex.  You can think of this in two ways.  You can push the structure to the homology and get an $A_\infty$ algebra and get \wh, but you look at what happens when you want to pass to the Deligne Mumford compactification.  The circle action over here is highly nontrivial.  You start to saturate and get equivariant loops, you look at things and see that you can't get to the compactified.  You don't know what field theory describes this guy.  There are proposals and conjectures, one by Sullivan, but it's not fully understood.  I want to say something in defense of an analytic approach.  Let me try to address that a little bit.

There's this hope when string topology comes about, since it's local and uses intersection theory, the invariants is something that is not a homotopy invariant.  After a while people started working on this, and it's not true, it's a homotopy invariant.  Now you start to look, well, although the original constructions are given by local duality, you could use global duality.  There's still the hope that when you understand circle actions, well, small loops you have to fill them in.  You need a texture, a piece of space to contract over.  There's a feeling that the proper understanding of the Hochschild of this would give rise to structures that are not homotopy invariant.  There are good arguments for and against it.

Let me talk about the distinction between local and global Poincar\'e duality.  If you have a space $X$, you can ask yourself whether the homology and cohomology are related in any way.  Some spaces have Poincar\'e duality.  You have a top chain, you have $H_n(X)=\zZ$, and you think of $[X]\cap:H^.X\to H_.X$, and if this is an isomorphism you call your space a Poincar\'e duality space.  Manifolds have this property but many other spaces have this property.  The distinction is quite well understood in the language of $L$ theory.  You can give a nice description of local Poincar\'e duality at the cohomology level.

What is local Poincar\'e duality?  You look at a space $X$, and now you can look at $X\injects X\times X$ as the diagonal.  Now you can look at this triple, the pair $X\times X-\Delta\subset X\times X$ and you can look at the long exact sequence of cohomology:
$$
H^.(X\times X,X\times X-\Delta)\to H^.(X\times X)
$$
What is this thing we are talking about, capping with the fundamental class?  It's a map from cohomology to homology.  If you think about this as an element, it lives in $H_.(X\times X)$.  This is something that every Poincar\'e duality space has.  When you look at the inverse you get a map from $H_.\to H^.$, and interpret that as an element of $H^.(X\times X)$, and that's the Thom class of the diagonal.  Now you ask yourself if it's in the image of this map or not.  If it is, then you have local Poincar\'e duality.  A Poincar\'e duality space is a homology manifold if and only if the Poincar\'e duality is local.  You can talk about duality globally, these don't have invariants.  If you demand that the duality be local, suddenly these are homology manifolds, they have characteristic classes.  If you start to intersect cycles as in string topology, if you do them locally, you may get invariants of manifolds and not just of the homotopy type.

Okay, so now, what does this have to do with cyclic $A_\infty$ structure or Calabi Yau structure.  Either this definition is not appropriate for string topology, or the way that we have been assigning an $A_\infty$ Calabi Yau category to a manifold is not sensitive enough.  But the only ways that I know where you can go from a space to a Calabi Yau category are homotopy invariant.  One way is to consider the minimal $C_\infty$ structure and then take a unique lift into a $C_\infty$ cyclic structure, or any other way I know are homotopy invariants.  There may be a more sensitive way that is not, but I don't know.  Now let me talk a little bit about this main sort of theorem.  If there is time I can talk about homotopy triviality of \wh.

All right.  So this definition is due to Kontsevich.  A Hermitian Calabi Yau elliptic space is, well, let's define it in steps.  An elliptic space is, you need to have a manifold $M$ and a bundle of $\zZ_2$-graded algebra over it.  You look at the sections $\Aa$ of this bundle $E\to M$ and now you ask for an operator $Q$ of odd degree on $\Aa$ which is a differential, $Q^2=0$, and an elliptic operator.  You say this space is Calabi-Yau if there is a trace which factors through densities.  You want it to be Hermitian, so that you have a star operator that goes from $\Aa$ to $\Aa$ and you want when you define $\langle a,b\rangle$ as $tr(a*b)$ you want this to be a Hermitian metric.

Given this space you can talk about Hochschild homology, Hochschild chains $CH_.(A,A)$, and that's a complex, and you can talk about $Hom$ of this, multilinear maps of this to many copies, it's the direct sum of $CH^{\otimes n}\to CH^{\otimes m}$.  There's a model of the open moduli space given by ribbon graphs.  If you take appropriate de Rham currents you can make this into a nice PROP, and there is a nice map of PROPS from $\Omega_.(M)$ to the Homs of Hochschild of $A$ at the chain level.

And, so, the relationship between this and string topology is that when you look at the degree zero homology, it matches with the known operations defined topologically.  As far as whether this is homotopy invariant or not, I don't know.  But at least this is not, it's not doomed to fail in terms of giving invariants that are not homotopy invariant.  The data are not clearly homotopy invariant.  If you start with a Calabi Yau approach, this data is doomed to be homotopy invariant.  I would be very interested to hear about a construction of a Calabi Yau category which expresses local Poincar\'e duality, not global.

If you have a circle action, you can identify completely what it means for it to be trivial.  This is encoded in a bundle over $\cC\pP^\infty$ with fiber $X$, this base being the classifying space of $S^1$.  Triviality means that $HX\to H^{eq}X$ is injective, and I think that by looking at an exact sequence, it's not.
\section{Terilla, quantizing deformation theory}
This will be a workshop talk.  So I wanted to discuss the idea of quantizing deformation theory, or to refer to higher genus deformation theory.  So I'll repeat some main points that have already been discussed.  The first thing I want to say is that deformation theory, as a theory itself, is the theory of $L_\infty$ algebras.  In order to justify this, what I should really say, to justify this I should qualify it, say deformation theory over commutative spaces.  How am I going to justify this?  What makes deformation theory, deforming complex structures, associative algebras, vector bundles with connections, quantum field theories, algebras over a properad, and what unifies all of these ideas is that when you deform an object $X$, you do it over a base space $B$.  What this means is that you consider deformations of $X$ as fibered over this base space.  For complex manifolds, $B$ has a base point and there's a fixed isomorphism of $X$ with a fiber over a base point.  The picture of associative algebras, you pick a parameter ring, so you deform it over that commutative ring.

So a particular deformation problem leads to a functor $F:\Aa\to Set$ from parameter rings to sets.  It's a functor because if you have a fibration you can pull maps back.  Okay, now there's one particularly nice functor from parameter rings to sets, and that's $Def_L$ for an $L_\infty$ algebra $L$, which I will think of as $(V,D)$, where $V$ is a graded vector space over a ground field of characteristic zero, and $D$ is a collection of multilinear maps on $V$ which satisfy some conditions.  So $Def_L$ is the Maurer-Cartan functor.  What is $Def_L$ of a parameter ring?  They are $\gamma\in V\otimes R:D(e^\gamma)=0$.  The example to have in mind is that the $L_\infty$ algebra is a differential graded Lie algebra, and then $D(e^\gamma)$ is $d\gamma+\frac{1}{2}[\gamma,\gamma]$.  In the notation that Soibelman was using, you want to think of $\frac{1}{2}$ as $\frac{1}{2!}$.  A couple of points that I'm thinking of to say.  You should probably think of $R$ as some kind of local ring, and then you may want this to converge so you may want $R$ to be Artinian.  This is a particularly nice example.  So there's this concept of a deformation problem, and I will call it $F$, thinking of it already as a functor from parameter rings to sets.  This is a definition, I guess.

\begin{defi}
A deformation problem $F$ is controlled by an $L_\infty$ algebra means that $F$ is naturally equivalent to $Def_L$.
\end{defi}
For example, deformations of complex structure on a complex manifold $X$ is controlled by a particular differential graded Lie algebra, the Dalbeaux resolution of the sheaf of polyvector fields. For associative algebras, this is the Hochschild complex.  You give a deformation problem to a capable deformation theorist and they should be able to work with it and figure out an $L_\infty$ algebra for you.

[what connection do you expect between the algebra for one structure and for another?]

You expect a new $L_\infty$ structure with a different differential.  When the deformations exist, you have a nice space in which you can discuss this, it's like the $L_\infty$ algebra is zero somehow.  Well, I don't know?

So the first words I said is this is a workshop talk, this is an example of how I meant it.  So basically if you deform the complex structure, it's like deforming the differential of the $L_\infty$ algebra.  There's a beautiful theorem of Manetti's from the 90s that says that every deformation functor $F$ is naturally equivalent to $Def_L$ for some $L_\infty$ algebra $L$.  You had a collection of examples, but now it is completely clear that you can find the $L_\infty$ algebra controlling a deformation functor.  So the two things you need to do is define a deformation functor, and then find the $L_\infty$ algebra.  SO you start by finding the tangent space of the functor, $F(k[\epsilon]/\epsilon^2)$, and this turns out to have a vector space structure, and then that's where you find the $L_\infty$ structure.  Then there's another theorem, that $\sigma:L\to L'$ a morphism of $L_\infty$ algebras gives a transformation of the functors, and this is a natural equivalence if and only if $\sigma$ is a quasiisomorphism.

That's my justification for the statement.  Now what about extending deformation theory?  One obvious way is to expand the parameter rings available.  I've already implicitly assumed you've made an extension, when I quoted Manetti, because you're dealing with graded differential parameter rings, so like $\zZ$-graded spaces.  So you can pass from ungraded to graded parameter rings.  In the notation, Merkulov has written about this.  In his notation you say something like $\Aa\to \Aa^\zZ$, Artin algebras to $\zZ$-graded Artin algebras.  This has a great impact.  For associative algebras, you're looking for elements in $(Hoch(A)\otimes R)^1$.  If $R$ is ungraded, these are Hochschild $1$ cochains.  But if $R$ is graded, this controls the theory of deforming $A$ in the category of $A_\infty$ algebras.  For complex structures, passing from ungraded to graded parameter rings, this is complex structures to what is geometrically called extended complex structures.  There are various descriptions of what this means in terms of generalized periods, a categorical description, and so on.

So changing parameter rings has a profound effect.  You could pass from commutative to not neccesarily commutative rings, which I'll call noncommutative rings.  This will have an interesting effect.  I don't know exactly how it works in these two examples.  A problem of Joseph Hirsch, a student in New York is to extend this, give the right axioms for Manetti's theorem, get something that will not be an $L_\infty$ algebra.  Soibelman said that if you have a functor that commutes with projective limits, you'll get a tensor algebra.  That will be a tensor algebra on the homology which you can think of with $0$ $L_\infty$ structure.

I want to describe another kind of passage to extending the parameter rings, so I want to pass to higher genus.  In order to do this in a way that isolates and highlights those higher genus features, I will stick to the commutative world.  Okay, so let me do some linear algebra first.

Okay.  So the first thing is, I want to say that there is an operator $\hat{\circ}_0$.  The reason I'm using this for my first object is I'm going to slowly remove these decorations.  This is a map $Hom(V^n,V^m)\otimes Hom(V^i,V^j)\to Hom(V^{n+i},V^{m+j})$, and this is kind of the only one we can imagine, that is, $f\otimes g\mapsto f\hat{\circ}_0 g$.  This is just $f\otimes g$, so $f\hat{\circ}_0 g(v_1\ldots, v_{n+i})= f(v_1,\ldots, v_n)\otimse g(v_{n+1},\ldots, v_{n+i})$.  Pictorially, you put these two corollas next to one another and then fuse the vertices or something.  It loks disconnected but you want to think of it as connected.

For me, the symmetric algebra of $V$, $SV$, is a quotient of the tensor algebra $TV$, and so I have $\pi:TV\to SV$ and I also have a symmetrization map $SV\to TV$, and this is also true for $S^nV$ and $T^nV$.  So this allows me to transfer $\hat{\circ}_0$ to $\circ_0$ which will be from $Hom(SV,SV)\otimes Hom(SV,SV)\to Hom(SV,SV)$.  For simplicity, let me call $Hom(SV,SV)$ by $P$.  I can now define $\circ_1:P\otimes P\to P$.  How will I do this?  By transfering a product on the tensor algebra $\hat{\circ_1}: Hom(T^n V,T^m V)\otimes Hom^{T^iV,T^jV)\to Hom(T^{n+i-1}V,T^{m+j-1}V)$.  I am doing a partial composition.  It's a single gluing.  This is a noncommutative product, the top is different from the bottom.  Here's a fact which you can check.  $\circ_1$ with no hat is on the symmetric algebra.  So the fact is that it's a Hochschild cocycle of $(P,\circ_0)$.  It's pretty easy to see; I think I will skip.  It's a fun thing, you just draw these diagrams and see everything cancels.

The Hochschild differential of $\circ_1$ will act on three things.  It will be $(f\circ_0 g)\circ_1 h$ plus the other terms and you get zero.  In this setting I don't know how many of you already did it in your seat, but now it's natural to quantize the algebra $P$ with the $\circ_0$ product, quantize it in the $\circ_1$ direction.  It's commutative and associative.  So you quantize this to obtain a star product $P[[\hbar]]\otimes P[[\hbar]]\to P[[\hbar]]$ which has the form $f\star g=f\circ_0 g+\hbar f\circ_1 g+\hbar^2 f\circ_2 g+\cdots$ and so on.

Now before I elaborate on that, I want to say that $\circ_1$ is interesting.  This $\circ_1$ product encodes the $L_\infty$ concept.  That's like a title; the proposition is that a degree one map $D$ in $Hom(SV,V)$ which of course sits inside $Hom(SV,SV)$ defines an $L_\infty$ algebra structure on $V$ if and only if $D\circ_1 D=0$.  I don't want to lift this as a coderivation, I just want to use it literally.  If you see $\circ_k$ in this star product, it has a simple description in terms of multiple gluings.  $\hat{\circ}_k$ is just gluing the last $k$ outputs of $f$ to the first $k$ inputs of $g$.  This space $P$ is kind of like $S(V\oplus V^*)$, and you can--
[Bruno asks a question]
Yes, but I want to think of the star product.  The quantization of deformation theory is the passage from pairs $(V,D)$ where $D$ is in $Hom(SV,V)$.  Let me make a comment, if I'm talking about $Hom(SV,V)$, then $\circ_1$ is the same as $\star$ on odd elements.  So I want to pass from $(V,D)$ to $(V,H)$ where $H\in Hom(SV,SV)[[\hbar]]$ and $H\star H$ is zero.

Here's another theorem
\begin{thm}(Drummond-Cole, Tradler, Terilla)
\\
An element $H$ of degree one in $Hom(SV,SV)[[\hbar]]$ satisfying $H\star H=0$ is the same as giving $V$ the structure of an algebra over the properad $cobar(coFrob)$.  \end{thm}Okay.  So let me just write down something about this.

You can write down $H$ as a sum over genera as $\sum \sigma_g \hbar^g$ and you can decompose $\sigma_g$ as $(\sigma_g)_i^j$ where this is a map $S^i V\to S^j V$.  It's clear that this describes a multialgebraic structure on $V$.  What structure is it?  It's a theory of properads, Bruno Vallette's theory, which is a theory of multigenus many inputs many outputs theory.  There's a very simple coproperad called coFrobenius, and all we've done is said that this collection of relations is precisely the same as cobar(coFrobenius).

[can we understand genus without knowing about properads?]

Yes.  Let me make an analogy.  $L_\infty$ is the same as saying cobar(cocommutative).  This is essentially in the original paper by Ginzburg and Kapranov.  So this brings me back to the original theme of parameter rings.  So the commutative operad is something that has a one dimensional space of operations $Com(n)$.  If you have a commutative algebra, a representation of the commutative operad, there's a unique map from $W^{\otimes n}\to W$.  If I have any number of inputs, it doesn't matter how I multiply them.  I can multiply them in any order.  This is Com.  Now because you have a one dimensional operation, Com plays the role of the unit when you tensor operads.  If $L=(V,D)$ is an $L_\infty$ algebra, or any operad, and $A=(W,\cdot)$ is a commutative associative algebra, then $V\otimes W$ with $(D\otimes \cdot)$ is again an $L_\infty$ algebra.  You were looking to solutions of the Maurer Cartan equation in $V$ tensor a parameter ring, and it was essential that you end in the same category of $L_\infty$ algebras.

A Frobenius algebra $F$ has two operations $\cdot$ and $\Delta$, which are commutative and cocommutative, associative and coassociative, and have a compatibility.  The advantage is that for every $n$ inputs and $m$ outputs, and $g$ genus, this is one dimensional.  These relations assemble so that any way of taking four to two with one genus it's the same.  So I want to spend at least sixty seconds talking about motivation, although I've been pretending that the talk is self-motivated.  Let me make a quick statement which I will call a program.  It is to develop a theory of functors of Frobenius algebras governed or controlled by algebras over cobar(coFrob).  What i didn't tell you is that if you have an algebra over any properad and tensor with a Frobenius algebra, you get an algebra over that properad back.

The symplectic field theory from Givental, Eliashberg, Fukaya, gives, conjecturally, a square zero element in a Weyl algebra on Reeb orbits of this type.  Quantum field theory in the Park approach, and certain differential $BV$ examples, like the type that we've heard about a lot.  You see this, I can't say much about the $BV$ example.  An old observation in quantum field theory is that infinitessimal deformations of quantum field theory correspond to physical observables.  The infinitessimal deformations give you the physical observables.  Does a deeper understanding of the deformation theory give you a better physical understanding of observables?  This gives a possible algebraic approach to quantum field theory, so that if you have an $\hbar$ enriched deformation theory, just studying the moduli space is not enough, this will tell you all correlation functions, and the step would be to understand this better and I'm considering this as a step in that direction.  Thank you.
\section{Merkulov}
Since it is a workshop, I can talk about things understood partially, so about exotic $L_\infty$ automorphisms of the Shouten Lie algebra.  The machinery applies to any Lie structure from de Rham theory on configuration spaces.  In particular, there are only two examples, Shoikhek the other one.  There will probably come more.  That was just implicit construction of another.  This Lie algebra codifies obstructions, but you want to quantize infinite dimensional structures.  This encodes obstructions to deformation quantization.  Since it was annuncd my work on $BV$, I will spend five minutes to formulate the main message.  In that paper there are three main points.  \begin{enumerate}
\item The category of quantum BV manifolds, odd symplectic manifold along with a density of a special form.  Any odd symplectic manifold you can get a Laplacian $\Delta$ which squares to zero.  So you can take $V\oplus V^*[1]$.  So in local coordinates you get $e^{\Gamma(x,\hbar)/\hbar}$ times a canonical half-density.  In any symplectic geometry, the category is a groupoid.  Things must be isomorphisms, you cannot compare symplectic things of different directions.  Physicists start with a density on a big manifold and integrate on a Lagrangian.  I want a category on which such integration is a morphism.  The main idea is that once you have an odd symplectic structure and a density, you can get a dgBV structure.  It's a morphism so that the pullback on functions is a morphism of dgBV algebras.  The main result is that, well, the boundary conditions for $\Gamma$, well, with them you can show that $\Gamma$ starts with quadratic terms and higher and that defines a differential.  So I have a $dg$ category.  The theorem is that in any dimension, any master function is isomorphic to a minimal and contractible one.  These are as good as $L_\infty$ algebras.  Similar phenomena go for Lie, and so on.  Many complicated structures come from something simple.  The second result is that the derived category of unimodular Lie^n bialgebras.  You have a Lie product and coproduct in degree $0$ and $1$ and then the adjoints of the product and coproduct are zero.  So representations of this guy are \wh.  What does this give me, I can use Kozsul duality and so on to get homotopy transfer formulae to see what is induced on homology.  So it looks like a Feynmann integral, and then I used an idea of Mnev, and constructed a QFT with a very simple Lagrangian.  I computed the Feynman integral of the Lagrangian over $\hbar$.  So I choose a good submanifold to do this over, and you get a homotopy transfer.  The theorem is the same as what happens at all order of $\hbar$.  This is a commercial for the paper.  Now I will come to this.

How many, well, look at Poisson structures on a vector space.  One can produce a one parameter family so that $\frac{d\alpha_t}{dt}$, let me draw this tetrahedron.  An edge that comes in is a derivative.  An output edge means the sum is contracted.  So this is something like $\frac{\partial \alpha^{ij}}{\partial x^a\partial x^b \partial x^c}\frac{\alpha^{a}}{}$[\wh]

Let $g$ be a Schouten algebra on a vector space.  He claims there is a Lie infinity morphism from this to itself.  There is no differential, so there are just maps $F_n$ which satisfy relations, and $F_1$ is always zero.  For any Maurer Cartan element, $F(\alpha)=\sum t^{n-1}F_n(\alpha,\ldots,\alpha)$ is again a Maurer Cartan element.  So he claims that there is a $L_\infty$ morphism that is $F(\alpha)=\alpha + \cdots$ where the next term is a tetrahedron.

So I claim I have computed all the other terms.  The general term will be some graph belonging to $n$ vertices, $2n-2$ edges, plus some operator associated to this graph.  Then there will be a correction term $c_\Gamma\Phi_\Gamma$.  The weight will be an integral, $c_\Gamma=\int_{\hat{C}_{n,0}}\widehat{e\in \Gamma}\omega_e$, which depends on a propagator $\omega_e$, a one form of $\hat{C}_{2,0}$.  So what is $\Phi_\Gamma$?  Given a graph with $n$ vertices, I want to get a map from the $n$ power of $g$ (polyvector fields) to $g$.  Decorate each vertex with a polyvector field.  Take their product.  Edges are ordered.  Each edge is determined by an ordered pair of integers, it's beginning and end.  So I differentiate over $x^i \psi_j$.

What is $c_\Gamma$?  So my claim is that a family of exotic $L_\infty$ automorphism is nonempty.  If $\omega(z_i,z_j)$ is a propagator, it will work.  But also $\omega^{op}$ will work.  The propagators are not interesting, but interesting things come from antipropagators.  What happens if I take the opposite of the full Kontsevich propagator?  The answer is that in fact, if we come back we can measure its action by vector fields.  Let us apply it to a polyvector field.  In fact, the wheeled graph will contribute.  There will be one through $n$ and then we will have things that go into one.  This is a graph $\omega_n$, and this is $\Phi_\omega$ applied to $\alpha$ $n$ times.  These are the Bernouli numbers (for $n$ even).

What happens if I take opposite to one half of the Kontsevich propagator?  This gives me the weight $\zeta(n)$ for all $n$.  I will not have time, but the first case when the Bernouli number appears.  This can be applied to any Maurer Cartan element in this algebra.  Let me take polyvector fields which have only $\alpha_0$ and $\alpha_1$ nonzero, and $\alpha_1$ is a linear Poisson structure, then the equation implies that $\alpha_0$ is an invariant polynomial.  If I plug in I get \wh deformation.  It generalizes \wh to arbitrary Maurer Cartan elements.

It's a generalized special \wh.  Right, now, maybe I should say the following.  The question is, we're interested in any automorphism.  We're more interested in those that are not homotopy trivial in the sense of homotopy theory.  I think these guys are homotopy trivial, and all of these terms here don't change homotopy type, so only the odd ones change.  Can I find another version where \wh disappear immediately.  If I work with the braid version of configuration spaces, only odd will contribute, and I will talk a little more later.

This is application of one simple formula.   What is the formula?  You take the configuration of ordered numbered points in the upper half plane and take the quotient with respect to the action, modulo the action of the group $\lambda z+a$.  It uses the Euclidean geometry to compactify.  This is the Kontsevich notation, the boundary will be of the following form.  If $k$ points come together, you are left with spaces like this, and you will be left with \wh where it comes to \wh.

There will appear new \wh that I haven't defined.  Let me remind you of this guy.

Let me begin with another definition.  $C_n$ is configurations of $n$ ordered point in $\cC$ modulo $\lambda z+a$ where $a$ is now allowed to be a complex number.  I'm going to say the proposition, this is a manifold with corners, so it has a face complex.  Then the proposition is that $\oplus C_.(\hat{C}_n)$ is a dg free operad which is a minimal resolution of the following structure ($Z_\infty$); a $Z$-algabra is a degree $-1$ binary multiplication which satisfies 

$(12)3 + (13)2 + 1(23)=0$  If we assume commutativity, it is Jacobi, but this is just Leibnitz maybe.  

Ret me embed $C_{n,0}$ into the following one.  Take $(S^1)^{n(n-1)}\times [0,\infty)^{n(n-1)}\times \rR\pP^{n(n-1)(n-2)}$.  I map $z_1,\ldots, z_n$ to $\prod_{i\ne j} \wh(z_i-z_j)\times \prod h(z_i,z_j)\times h(z_i,z_j)h(z_j,z_k)h(z_k,z_i)$  So $\hat{C_{n,0}}$ is just the closure of this.

I have codimension one boundary when they come together, $\boundary \hat{C}_{n,0} = C_{n-r+1,0}\times C_r$

[A bunch of pictures]

This compactification should have to do with $L_\infty$ morphisms.  Given two of these, if I denote the operations like this, a morphism is a number of components, it's a machine that has inputs in one color and outputs in the other.  What are the equations?  In graphical terms they look as follows.  You have this and you plug in as follows.

[\wh]

Let me finish with some motivations and open questions.  For any Drinfel'd associator there is an $L_\infty$ morphism, but a $L_\infty$ quasiisomorphism is invariant, so there exists one back, not unique.  Given two, they are related by an element of the group, so we can use one formality map and then go back with another.  This a $L_\infty$ automorphism.  How much is left?  The conjecture is that $GT$ acts nontrivially as $L_\infty$ automorphism of the Lie algebra.

In fact, if the conjecture is true, we need just one explicit family and then we see how it acts here, if it's true.  The correct version is the braid version.  I just work with a \wh point.  In a sense it's much much better.  There's a reason it's better.  By analogy with what I have said, there are \wh spaces of $n$ unordered points in $\cC$ and $n$ unordered points in the upper half plane.  Then, remember my statement that we had $Lieb_\infty$.  The braid version is $L_\infty$.  So we have a new geometric version, that is precisely the $L_\infty$ operad, $C_.(\bar{B})$ is $L_\infty$.

All wheeled graphs $W_{2n}=0$.

\section{Bruno Vallette}
Thank you very much.  It's a great pleasure to be here.  My goal here is to give the general framework for deformations of algebraic structures.

If you start from a category of algebras, for me, the toy model is the category of associative algebras.  By algebras I also mean modules, bialgebras, coalgebras, and so on.  The first goal is to model the category of algebras with one algebraic object, which lives in the world of Operads, sometimes this is an operad or a colored operad or a properad or a prop or whatever, that is what Operad with a capital O means.  You would like to define the deformation theory of this.  So the first thing is to find a ``good'' resolution of this operad.  So if $Ass$ is your operad, then Stasheff's $A_\infty$ is a ``good'' resolution.  So what does this operad model?  It models your algebra up to homotopy.  So what you get is the category of homotopy algebras, here $A_\infty$ algebras.  So I would like to be able to define a theory of deformations.  So now I would like a Lie theoretical interpretation.  I will see as John said, it may be an $L_\infty$ algebra.  So from this we define a deformation theory.  As a biproduct with this you also get to prove the theorem of transfer of structure.  This is also called the homological perturbation lemma.  You get the Hochschild chain complex and then the transfer theorems.

So we begin with operations, a monoidal category, monoids there, and then a category of algebra.

So the first structure you encounter is the Steenrod operations, so that is a stable module over the Steenrod algebra.  These are operations with one input and one output.  I want to encode these into one object.  When I compose them, I take the tensor product.  So I work in $(Vect, \otimes)$, and a monoid is an associative algebra.  So an associative algebra is a type of algebra.  If you study singular homology, you have the cup product, which gives you the category of associative algebras.  Now I want to consider this category.  It's a space with operations, several inputs but one output.  Then if you study for example the Hochschild cochain complex, Gerstenhaber defined a Lie bracket, so you want to be able to use Lie algebras and Gerstenhaber algebras.  So you can also consider the category of $BV$ algebras, but I also want to take care about the action of the symmetric group.  So the category is that of $\sS$-modules along with $\circ$ the composition operation.  This category of Operads is that of operads.  The category of associative algebras sits inside the category of operads because Vect with $\otimes$ sits within $\sS$-modules with $\circ$

So you want next to think of Frobenius bialgebras, or in string topology, you have involutive (or not) Lie bialgebras, and also Hopf algebras, bialgebras, and so on.  These operations have several inputs and several outputs.  We would like to consider these in one way.  Algebras are ``modules'' over an operad.  So we have $\sS$-bimodules, with a composition product which has the composition of operations.  The category of monoids is that of properads.  This gives you a picture to encode your algebras.

Let me give two other directions.  If you have several spaces, you can add colors.  If you have traces, you can add wheels.  It is very likely that an algebraic category can be modeled with an Operad.

So now I want to consider $End_A, Hom(A^{\otimes n},A^{\otimes m})$.  If they are concentrated in $n=m=1$ then they form an associative algebra.  For $m=1$ it is an operad and for arbitrary $n$ and $m$ is a properad.

So a $\pP$-algebra structure on $A$ is a morphism of Operads $\pP\to End_A$.  Now I will go to step two and do homological algebra.
\begin{thm}[Merkulov, Vallette] The category of dg properads has a model category structure.
\end{thm}
The idea of a proof is that each time you have a properad you can forget the composition and get a dg $\sS$ bimodule.  This admits a left adjoint, the free properad.  Then you use Quillen to transfer this theorem.  I will not consider free things, I will consider quasifree properads.  That is one so that the underlying module is free but the derivation might not be free.  Why these objects?  If you remember what people used to do with homological algebra, on the level of modules, the good resolutions are projective resolutions.  What is a projective module?  It is a direct summand of a free module.  Then there is a model category structure on modules where the cofibrant resolutions are direct summands of free modules, retracts of quasifree modules.  This is true if things are concentrated in positive degree.  If the characteristic of the ground field is zero, then you can lift this and any properad has a quasifree resolution.  Let me denote it $P_\infty$.  There is no harm in using quasifree resolutions.  I know they exist and they have the properties I want.  

So what is the shape of such a resolution.  I have a free properad and $d$ is a derivation.  This is completely characterized by the image of the generators.  So then you have $F(X)$ the free properad, which is on graphs.  You take operations from $X$ and do them in any formal composition.  The composition is grafting.  This means any time I have an operation from $X$, it produces a sum which I cannot control, $\delta_1(x)+\delta_2(x)+\ldots$.  This is the same as some data $\delta_n:X\to F(X)^{(n)}$.  So far I have just said $d$ being a derivation is characterized by the image of the generators.  But we have $d^2=0$.  So this equation is equivalent to the following one.  You apply one $\delta_k$ and then you sum over all the possible ways of doing this and you get zero.  Now I want to translate and make specific that this is the structure I want on $X$.  Once again, this is nice, not explicit.  If you give me a category of algebra, then I can do the Operad.  But what about the resolution?  Do we have data to make $X$ explicit?  And $\delta$?  People studying syzygies, they have modules, they want resolutions.  They work in the category of modules over a polynomial ring.  Here we work in the category of properads.  If you have a properad which is quadratic, meaning that the relations are linear combinations of compositions of two operations, all of these are defined by generating operations which are compositions of two operations. in this case it is very likely that you can apply Koszul duality (Priddy, Ginzburg and Kapranov, Vallette).  This gives you an explicit construction of $X$ and $\delta$.  You get $X$ and then $P^\antishriek$, with $\delta_n=0, n\ne 2$ When you split into two and then two, this is somehow coassociative.  This really deserves the name coproperad.  It is really a comonoid in this monoidal category.

Not all of these fall into this pattern.  If you have quadratic plus linear relations, such as $BV$, where the Lie bracket measures the default of $\Delta$ from being a derivation of the product.  This relation is not quadratic.  This does not fall into this theory.  This is some filtered Koszul duality.  That's exactly what Priddy did in his first paper.  Why did he do that?  He wanted resolutions of the Steenrod algebra.  These two are quadratic and then have linear terms in this.  We extended this theory to properads.  What happens here?  We are not far from Kozsul duality theory.  Then I can consider the quadratic properad associated with this.  Then I still have $\Delta$ but you need to add one linear term, so we will have $\delta_1$ and $\delta_2$.  So the $\delta_2$ is coassociative and you have a coproperad, and you get a $dg$ coproperad.

What about more than two?  So if $P$ is quadratic, the coproduct is a morphism of algebras.  So what about Hopf algebras?  So what do you do?  You kill the higher things, and it's $(qP)^\antishriek$ and perturbate with higher $\delta_n$.  This is exactly the definition of a homotopy coproperad.

So next let us move on to $P_\infty$ algebra structures.  This is a morphism from $P_\infty\to End_A$.

[A little motivation... I was furiously debating something he wrote on the board.]

A promised a Lie interpretation of this.  The Lie theoretical interpretation, wher is the Lie algebra?  I would like to have a Lie algebra in which I could find Maurer-Cartan elements.  So I can look at properad maps $(F(X),End_A)$, these are characterized by the image of the generators, these are maps $Hom_\sS(X,End_A)$, and these satisfy some sort of relation, that should transfer to some subset of this space.  If You look at $Hom(X,End_A)$, this is a map from a (homotopy) coproperad to a properad.  Let us take the simplest example, you have maps from a coalgebra to an algebra.  This is the convolution properad.  It may have to be homotopy if you started with a homotopy properad.  Now you need to look at the space, this structure induces a convolution $L_\infty$ algebra.  It's a $dg$Lie algebra precisely when Kozsul.  The general Maurer Cartan equation exists here, you can consider the space of solutions.

So you have, you take a properad, you take a resolution, you get an $L_\infty$ algebra that controls deformations.  This is just a definition of homotopy algebra.  Now I would like to define the deformation theory of these guys.

The deformation theory starts from a vector space $A$, you want to check the structure, you make this algebra $g_A$ explicit, where you have your Maurer Cartan solution, which is your structure.  Now you twist it and consider the twisted $L_\infty$ structure.  You just change the differential.  This produces the tangent homology, and this has the right to be considered a chain complex.  This is a deformation complex.  The complex John had before, he considered $A=V$, he considered exactly this algebra, inside this algebra, $H\star H=0$ and these are exactly the solutions of Maurer Cartan.

Let me give you examples.  If you consider the deformations of Ass, you get the Hochschild complex.  For Lie, you get the Chevalley Eilenberg complex.  You recover all of the cases in the literature.  You do not have first the complex and then the Lie algebra, you get both at the same time.  If you take any resolution of the prop of bialgebras, you get the Gerstenhaber \wh complex which controls the deformation of Lie bialgebras.

I don't need $End(A)$ and consider $Q$ any operad.  For example if you look at $Ass$ and the Poisson operad, you get something that is related to something Vassiliev.  Then you need a cotangent complex a la Quillen.  I want to finish with two nice things quickly.  What is the transfer of structure?  That's the following problem.  You start with two chain complexes and you ask that they form a strong deformation retract.  If you have $\pi:W\to V$ and $\iota: V\to W$ with $\pi\iota=id$ and $h$ a homotopy from $\iota\pi$ to the identity.

Let me state the theorem.  There exists a $P_\infty$ algebra structure on $V$, this is stupid, such that this extends to a map of $P_\infty$ algebras.  So far I cannot claim I can prove this.  This is the same as the case for operads.  And what is the idea?  Consider $g_W$, which controls $Hom_\sS(X,End_W)$, and you have a Maurer Cartan element here.  You are looking for a Maurer Cartan element in $g_V$, and the only thing you need is a map $End_W\to End_V$ along which I push forward.  Why am I so careful?  I want a map to be not a Lie map but an $L_\infty$ morphism.  So you want a weak morphism, one of homotopy operads.

So you get the image of a Maurer Cartan element will be a Maurer Cartan element.  How do you construct this?  For operads, it is easy.  If you have an operad, you can compose independently the inner edges.  It's easy to construct.  For properads it's a real mess.  Two words about the case of BV algebras.  We treated them with the filtered $BV$ theory.  This gives a deformation complex and then there is a nice statement, I finish with this because I like it very much.  If you have $\alpha$ a $G_\infty$ structure on $A$, you can consider $H(g_A^\alpha)$, as a $G_\infty$ algebra.  Consider this for $n=-4,-6$, etc.  If these homology groups are zero, then this can be lifted to a homotopy BV algebra.  This shows a long story.  I'd like to give another proof of cyclic Deligne conjecture.  So there is not a lot of hope.  Then you could hope to lift the structure of $G_\infty$ algebra given by \wh.

Thank you very much.

[Post talk continuation]

What is a BV algebra?  It is an algebra generated by a binary commutative product, a Lie product which is degree one, and $\Delta$ of degree $1$.  This is subject to the Gerstenhaber, associative, Liebnitz, Jacobi, and the Lie bracket measures the derivation of $\Delta$ from deriving the product.  The only problem is the linear relation for the bracket.  I can kill this guy and get qBV.  I could have done something else.  I could have taken $qBV$ to be this, and take all the possible quadratic relations, the minimal set of generators and a maximal set of relations.  Then I want to kill the linear parts.

The first theorem is to see if this is Koszul.  It is.  You take the cobar construction of the Koszul dual, and you resolve qBV.

The space will be a choice of $\delta^n$, a choice of $m$ and then a Lie element of ary $i_j$ for $j$ up to $m$.  So now I want to deform my resolution.  I need to put in another differential $d_1$.  The difference between $BV$ and $qBV$ is that $d(m_{1,1}^1)$ is equal to something in qBV but in $BV$ you want it to also have the bracket.  So the lemma is that there exists a unique coderivation $d_1$ of $F^c(sV)$, which are characterized by the projection on the generators, by $F^c(V)\to V$, and then it exists on $qBV^\antishriek$ (you have to check).  Then $d_1^2=0$ and $d_1$ commutes with $d_2$ because $d_1$ is a derivation of the coproduct ($d_2$).  It follows from regular Koszul duality.

So $BV^\antishriek$ is defined as $qBV, d_1$.  What you have at the end by Poincar\'e Birkhoff Witt is that $BV$ and $qBV$ are the same as $\sS$-modules.  The graded guy associated to $BV$ is the same thing too.  The relations you wrote are not homogeneous.  But you get a filtration annd this is the graded associated.  So how do you lift a $BV$ to homotopy BV?  So this is $k[\delta]\circ Com^1\circ Lie^2$.  This is $k\oplus k\delta\oplus\cdots$  You can split this into $k\circ Com^1\circ Lie^2 \oplus (k\delta\oplus\cdots)\circ Com^1\circ Lie^2$, so this is $G^\antishriek+ \overline{k[\delta]}\circ G^\antishriek$.

I don't want to consider this.  I want to think of this as an Abelian extension, so forget that it's a coproperad and consider it as a comodule.  When I dualize, I get $Hom_\sS(\overline{k[\delta]}\circ G^\antishriek,End(A))$ is a dg Lie module over $Hom_\sS(G^\antishriek,End_A)$.  They live in $H^{-2}$ of this module.  Given a power of $\delta$, $H^{-2}Hom(\delta^{(n)}\circ G^\antishriek,End_A)$, and this is isomorphic to $H_{-2}s^nHom (G^\antishriek,End_A)$ which is $H_{2-2n}(Hom_\sS (G^\antishriek, End_A)$.  If this is $0$ for all $n$ then it is okay.

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