Alexei Bondal

T-structures in commutative and non-commutative geometry.

We shall discuss some problems in commutative and non-commutative algebraic geometry which naturally appeal to constructing some t-structures in derived categories. We shall also discuss an approach to proving existence of such t-structures.

Tom Bridgeland

Cluster mutations and Donaldson-Thomas invariants'

I will try to explain how recent (not yet published) work of Kontsevich and Soibelman gives an interpretation of cluster mutations as a change-of-variable formula for generating functions of Donaldson-Thomas invariants.

Joe Chuang

Feynman diagrams and minimal models for operadic algebras (joint with A. Lazarev)

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 $A_\infty$-algebras. Further, 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.

Giovanni Felder

Holomorphic differential operators and Riemann-Roch-Hirzebruch formula

The Lefschetz number of a holomorphic differential operator acting on sections of a vector bundle on a complex compact manifold is the alternating sum of traces of the action on sheaf cohomology. Recent developments in deformation quantization led to new local formulae for the Lefschetz numbers, generalizing the Riemann-Roch-Hirzebruch formula (the case of the identity operator). The talk is based on joint works with Feigin and Shoikhet, with Engeli and with Tang.

Mikael Vejdemo-Johansson

On the computation of A-infinity algebras and Ext-algebras

For a ring R, the Ext algebra Ext_R^*(k,k) carries rich information about the ring and its module category. The algebra Ext_R^*(k,k) is a finitely presented k-algebra for most nice enough rings. Computation of this ring is done by constructing a projective resolution P of k and either constructing the complex Hom(P_n,k) or equivalently constructing the complex Hom(P,P). By diligent choice of computational route, the computation can be framed as essentially computing the homology of the differential graded algebra Hom(P,P).

Being the homology of a dg-algebra, Ext_R^*(k,k) has an induced A-infinity structure. This structure, has been shown by Keller and by Lu-Palmieri-Wu-Zhang, can be used to reconstruct R from Ext_R^{\leq 2}(k,k).

In this talk, we shall discuss the computation of Ext_R^*(k,k) and methods for computing an A-infinity structure on the Ext algebra. Examples will be drawn from group cohomology, where the computation of the Ext algebra has conditions from Benson and Carlson for recognizing whether a partial computation has the entire structure.

Domenico Fiorenza

Generalized periods of Kaehler manifolds

Let X be a compact Kaehler manifold, and let

H*(X) be its cohomology ring. We exhibit a natural L-infinity-morphism between the differential graded Lie algebra of holomorphic polyvector fields on Xand the mapping cone of the inclusion of differential graded Lie algebras

0àEnd(H(X)). The induced morphism of deformation functors maps the generalized deformations of X to the linear automorphisms of H*(X).

The restriction of this map to classical deformations is an infinitesimal lifting of Griffiths' periods map; moreover, when X is a Calabi-Yau manifold, one recovers Barannikov-Kontsevich's notion of generalized periods, Bogomolov-Tian-Todorov's lemma and Yukawa couplings. Joint work with Marco Manetti.

Victor Ginzburg

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

The hypersurface in ${\mathbbC}^3$ with an isolated quasi-homogeneous elliptic singularity of type $\tilde{E}_r,r =6, 7, 8$  has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r provides a semiuniversal Poisson deformation of that Poisson structure.

We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface.

To this end, we first deform the polynomial algebra ${\mathbb C}[x_1,x_2,x_3]$ to a noncommutative algebra with generators $x_1,x_2,x_3$ and the following 3 relations labelled by cyclic parmutations (i, j, k) of (1, 2, 3):

\[x_ix_j - tx_jx_i =\Phi_k(x_k), \Phi_k\in {\mathbb C}[x_k].\]

This gives a family of Calabi-Yau algebras ${\mathfrak A}^t(\Phi)$ parametrized by a complex number t \in ${\mathbb C}×$  and a triple $\Phi=(\Phi_1,\Phi_2,\Phi_3)$ of polynomials of specifically chosen degrees.

Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form ${\mathfrak A}^t(\Phi)<<\Psi>>$  where $<<\Psi> >\subset ${\mathfrak A}^t(\Phi )$ stands for the ideal generated by a central element $\Psi$  which generates the center of the algebra ${\mathfrak A}^t(\Phi)$  if $\Phi$ is generic enough.

Alastair Hamilton

Noncommutative geometry and compactifications of the moduli space of curves,

There is a theorem, due to Kontsevich, which expresses the homology of the moduli space of curves as the homology of a certain infinite-dimensional Lie algebra. In this talk, I will explain how this theorem can be extended to a compactification of the moduli space introduced by Kontsevich in his study of Witten's conjectures, by making use of a Lie bialgebra structure considered by many authors. If time permits, I will outline how this theorem can be applied to constructing classes in the moduli space.

Dominic Joyce

Kuranishi (co)homology: a new tool in symplectic geometry

Moduli spaces of J-holomorphic curves in a symplectic manifold  are in general not smooth, but are topological spaces with the structure of a Kuranishi space, in the sense of Fukaya-Ono. To define symplectic Gromov-Witten invariants, Lagrangian Floer homology, contact homology, etc., it is necessary to define a "virtual cycle" for the moduli spaces, usually in singular homology. Roughly speaking, one takes a multivalued transverse perturbation of the moduli space, and triangulates it by singular simplices to get a virtual chain. For Gromov-Witten invariants, in which the moduli spaces are without boundary, this virtual cycle construction is fairly painless. In Lagrangian Floer homology, where the moduli spaces have boundary and corners, it is acutely painful. The issue of choosing virtual cycles for moduli spaces which are compatible at the boundary and corners with other choices of virtual cycles is one of the principal problems in the main reference [FOOO] on Lagrangian Floer homology, which is an absurdly complex long horrible mess.

In this talk I describe some new technology which gives a fresh, far cleaner approach to the virtual cycle problem for J-holomorphic curve moduli spaces.

I define homology and cohomology theories of orbifolds in which the (co)chains are Kuranishi spaces, with maps to the target domain, and some extra "gauge-fixing data". I prove these are isomorphic to singular homology, and compactly-supported homology. Then for Gromov-Witten theory, Lagrangian Floer homology etc., there is no need to form virtual cycles: we can just regard the moduli space itself as a (co)chain in the (co)homology theory.

Bernhard Keller

The periodicity conjecture via 2-Calabi-Yau categories

The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov, Kuniba-Nakanishi and Ravanini-Valleriani-Tateo. It asserts that a certain discrete dynamical system associated with a pair of Dynkin diagrams is periodic and that its period divides the double of the sum of the Coxeter numbers of the two diagrams. The conjecture was proved by Frenkel-Szenes and Gliozzi-Tateo for the pairs (A_n, A_1), by Fomin-Zelevinsky in the case where one of the diagrams is A_1 and by Volkov when both diagrams are of type A. We will sketch a proof of the general case which is based on Fomin-Zelevinsky's work on cluster algebras and on the theory relating cluster algebras to triangulated 2-Calabi-Yau categories. An important role is played by Amiot's construction of such categories from finite-dimensional algebras of global dimension two.

Alastair King

Brane tilings and Calabi-Yau algebras

String theory suggests how to construct non-commutative crepant resolutions of toric Gorenstein 3-fold singularities. I will try to explain some of the ideas involved.

Calin Lazaroiu

Framed  bicategories and defects in Landau-Ginzburg  models.

I show that defects in B-type Landau-Ginzburg models naturally form a framed bicategory with duality in the sense of May and Sigurdsson. I also discuss connections with the planar algebras of Jones.

Jean-Lois Loday

The A-infinity operad is an A-infinity coalgebra

We construct a geometric diagonal on the Stasheff polytope through a simple formula which enables us to show the assertion of the title.

Sergei Merkulov

Pro(p)file of Batalin-Vilkovisky quantization, homotopy transfer and simplicial BF theory

Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of a wheeled dg prop which, on the one hand, is isomorphic to the cobar construction of the prop of unimodular Lie 1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop of unimodular Poisson structures. These results allow us to apply properadic methods for computing formulae for a homotopy transfer of a unimodular Lie 1-bialgebra structure on an arbitrary complex to the associated quantum master function on its cohomology. We shall show that Losev-Mnev's simplicial BF theory for unimodular Lie algebras can be naturally extended to the case of unimodular Lie 1-bialgebras (and, eventually, to the case of unimodular Poisson structures). Using a finite-dimensional version of the Batalin-Vilkovisky quantization formalism it is rigorously proven that the Feynman integrals computing the effective action of this new BF theory describe precisely homotopy transfer formulae obtained within the wheeled properadic approach to the quantum master equation. Quantum corrections (which are present in our BF model to all orders of the Planck constant) correspond precisely to what are often called ``higher Massey products" in the homological algebra.

Gabriele Mondello

Natural triangulations of Riemann surfaces with boundary and Weil-Petersson Poisson structure

Natural cellularizations of the Teichmueller space via ribbon graphs have been invented using decorated hyperbolic surfaces of Jenkins-strebel differential. We show that these constructions are interpolated by infinitely grafted hyperbolic surfaces with boundary. We also discuss the behavior of the Weil-Peterson symplectic structure in the flat limit.

Alexander Odesskii

Algebraic structures connected with pairs of compatible associative algebras

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A, D, E-type.

Travis Schedler

Differential operators and Batalin-Vilkovisky structures in noncommutative geometry

Abstract: We will explain how to extend Grothendieck's formalism of differential operators to associative algebras.  The operators act not on the algebra itself, but on a certain "Fock space" that is a commutative algebra in a certain symmetric monoidal category closely related to wheeled PROPs.  As an application, one obtains differential operators on representation varieties of the algebra.  Likewise, one obtains a definition of Batalin-Vilkovisky structures (and thus the Calabi-Yau condition) for associative algebras which gives rise to the same type of structure on representation varieties.

Sergey Shadrin

BCOV theory via Givental group action on CohFT.

I am going to show how one can apply Givental's group action in order to give a simple closed formula for the canonical genus expansion of the Barannikov-Kontsevich solutions of WDVV equations.

Richard Szabo

D-branes, duality and noncommutative correspondences

We describe a new categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of $C^*$-algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. The construction exploits noncommutative versions of Poincare duality and orientation in a purely algebraic framework of separable C^*algebras.

Ulrike Tillmann

n-fold cobordism categories and their homotopy type

Abstract: In the axiomatic approach to CFT and TQFT of Segal and Atiyah cobordism categories and their higher dimensional analogues play an important role. In recent work with Galatius, Madsen and Weiss we identified the  homotopy type of  d-dimensional cobordism categories.

This result extends to cobordism n-categories in a natural way. Indeed, the homotopy type of the n-fold cobordism category is a (non-connected) deloop of the (n-1)-fold cobordism category.

Andrew Tonks

Homotopy Batalin-Vilkovisky algebras

Joint work with I. Galvez and B. Vallette

We extend Koszul duality theory to properads defined by quadratic and linear relations. The operad encoding BV algebras is shown to be Koszul in this sense. This allows us to give an explicit small quasi-free resolution for it, which defines the notion of homotopy BV-algebras. We then describe deformation theory for (homotopy) BV-algebras.

Victor Turchin

Higher dimensional long knots (joint work with G.Arone and P. Lambrechts)

It turns out that the rational homology of the space of embeddings R^k in R^N, N>2k+1, in some sense depends only on the parity of k and N. More precisely this homology admits a double splitting (by means of the Goodwillie-Weiss calculus machinery), and each bilayer is the same for k's and N's of the same parities. Amazingly this biperiodicity starts from k=1. The generating function of the Euler characteristics of the bilayers will be presented, which gives a strong lower bound on the homology dimensions of the above spaces. It will be also shown how this homology can be interpreted from the point of view of the deformation theory of operads

Ronald Umble

$A_\infty$-bialgebras and Matrahedra (joint with S. Saneblidze)

Let $m,n\in {\mathbb N}$. We introduce free matrad $H_\infty$ which is a bigraded module generated by singletons $\theta^n_m$ in each bidegree $m,n$ and define an $A_\infty$-bialgebra as an algebra over $H_\infty$. We construct the matrahedra $KK=KK_{nm}=KK_{mn}$ of which $KK_{1n}=KK_{n1}$ is the Stasheff associahedron $K_n$ and realize $H_\infty$ as cellular cochains of $KK$.

Sasha Voronov

The n-category of cobordisms and TQFTs

Since Atiyah and Segal described the notions of Topological Quantum Field Theory (TQFT) and Conformal Field Theory (CFT), mathematicians have been looking for a suitable n-category framework to describe cobordisms with corners and a more general TQFT involving such cobordisms as a functor from that higher category of cobordisms with corners to a higher category of vector spaces. The problem deals with such standard difficulties of higher category theory as weak vs. strict axioms, coherence, suitable diagrams, etc.: almost every higher category theorist has this problem in the back of her head, but nobody seems to have gotten through to a satisfactory solution. In the talk, I will describe the set up Mark Feshbach and I have found.

Theodore Voronov

Higher Poisson Brackets and Differential Forms

Abstract: We show how the relation between Poisson brackets and symplectic forms extends to the case of an inhomogeneous multivector field and/ or inhomogeneous differential form. We obtain a generalization of the notion of symplectic structure and a construction of  'Koszul type' brackets in this setting. (Work in progress with H. Khudaverdian.)

Michael Weiss

Topological invariance of rational Pontryagin classes via sheaf theory

Novikov proved in the 1960’s that the map in rational cohomology induced by a homeomorphism between differentiable manifolds M and N takes the Pontryagin classes of TN to the Pontryagin classes of TM. The methods used by Novikov became immensely influential, but never ceased to look miraculous. In this talk, another proof of Novikov’s theorem will be described (ongoing joint work with A Ranicki) which aims to be obvious, although perhaps neither short nor free of technicalities. It is based on sheaf theoretic ideas. In particular Verdier duality for sheaves is a major ingredient.