# A conference honoring

Murray Gerstenhaber's 90th

&

Jim Stasheff's 80th birthday

# Conference Venue

Room 4C8, Department of Mathematics, University of Pennsylvania. The department is located in DRL (David Rittenhouse Labs), 209 S. 33rd Street, Philadelphia, PA 19104

## Organizers:

Vincent Coll, Lehigh University

Anthony Giaquinto, Loyola University Chicago

Tom Lada, North Carolina State University

Ping Xu, Pennsylvania State University

# Participants

Alberto Cattaneo, University of Zurich

Vincent Coll, Lehigh University

Vasily Dolgushev, Temple University

Yael Fregier, Artois University

Kenji Fukaya, Stony Brook University

Anthony Giaquinto, Loyola University Chicago

Miodrag Iovanov, University of Iowa

Martin Karel, Rutgers University of Camden

Yvette Kosmann-Schwarzbach, Ecole Polytechnique

Takashi Kimura, Boston University

Irena Kogan, North Carolina State University

Fujio Kubo, Hiroshima University

Tom Lada, North Carolina State University

Sylvain Lavau, University of Porto

Martin Markl, Mathematical Institute of the Czech Academy

Radmila Sazdonovic, North Carolina State University

Malka Schaps, Bar-Ilan University

Ekaterina Shemyakova,   University of Toledo

Mihai Staic, Bowling Green State University

Daniel Sternheimer, Rikkyo University and Universite de Bourgogne

Mathieu Stienon, Pennsylvania State University

Boris Tsygan, Northwestern University

Ron Umble, Millersville University

Alexander Voronov, University of Minnesota

Theodore Voronov, University of Manchester

Ping Xu, Pennsylvania State University

Marco Zambom, KU Leuven

# Accomodations

Participants are requested to book their own hotels.  Many  hotel options are available and can be searched for with standard travel websites.

## Monday March 5

9:00 – 10:00    Arrival and continental breakfast

10:00 – 10:10  Introductions and general announcements

10:10 – 10:40  Yvette Kosmann-Schwarzbach

10:50 – 11:20  Martin Markl

11:30 – 12:00  Vasily Dolgushev

12:00 – 1:30  -- Lunch

2:10 – 2:40  Ron Umble

3:30 – 4:00 -- Tea and discussion

## Tuesday March 6

9:30 – 10:00     Alberto Cattaneo

10:10 – 10:40   Kenji Fukaya

10:40 – 11:00  -- Tea

11:00 – 11:30  Alexander Voronov

11:40 – 12:10 Yael Fregier

12:10 – 1:40  -- Lunch

1:40 – 2:10 Sylvain Lavau

2:20 – 2:50  Mathieu Stienon

2:50 – 3:30 -- Tea and discussion

## Wednesday March 7

10:00  - 10:30   Ted Voronov

10:30 – 10:50   -- Tea

10:50 - 11:20 Ekaterina Shemyakova

11:30 - 12:00 Daniel Sternheimer

12:00 – 1:30  Lunch

1:30 – 2:00  Fujio Kubo

2:10 – 2:40  Malka Schaps

2:40 – 3:15 -- Tea and discussion

3:15 – 3:45 -- Mihai Staic

3:45 – 5:00 -- Break

5:30 – 7:00 Conference Dinner.  This will be held at Han Dynasty in University City - 3711 Market Street.  (details TBA)

## Thursday March 8

9:30 – 10:00    Miodrag Iovanov

10:10 – 10:40   Takashi Kimura

10:40 – 11:00  -- Tea

11:00 – 11:30 Boris Tsygan

11:40 – 12:10 Closing Remarks

12:10 –-  End of Conference

## Abstracts and Links to Slides

Alberto Cattaneo (Slides)

Title: BFV and Poisson

Abstract:  After recalling the notion of differential graded symplectic manifolds, we focus on the two particular cases of BFV structures, which are used to give a cohomological resolution of symplectic reduction, and of Poisson structures up to homotopy. We discuss how these two structures are actually related in the case of a field theory on manifolds with boundary and how quantization should be understood.

Vasily Dolgushev (Slides)

Title:  What Drinfeld could have replied to Deligne

Abstract: My colleagues keep asking me if there is an explicit construction whose input is a Drinfeld associator and whose output is a formality quasi-isomorphism for the brace operad BR. In my talk, I will give a positive answer to this question by showing that there are no obstructions for constructing a formality quasi-isomorphism from the Cobar-Bar resolution of BR to the chain operad of parenthesized braids. In principle, Drinfeld could have sketched this construction as a possible response to Deligne's famous letter from 1993. Of course, this story would not be complete without Tamarkin's paper "Formality of chain operad of small squares" from 1998 and Fiedorowicz's recognition principle. Details of this construction will appear in a joint textbook (in preparation) with Boris Tsygan.

Yael Fregier

Title:  Derived brackets = hamiltonian flow

Abstract:  Schlessinger and Stasheff have discovered in the 80's $L_infty$-algebras. These algebras nowadays play an important rôle in many different areas, in particular via deformation theory. A very efficient tool to build such algebras is the derived bracket construction of T. Voronov. In this talk we will give a geometric interpretation of this construction and if time permits some applications of this new point of view.

Johannes Huebschmann (Slides)

Title: A Lie coalgebra generalization of Fedosov quantization via homological perturbations relative to a skew 2-tensor over a Lie-Rinehart algebra

Abstract: See Slides

Miodrag Iovanov

Title:  Deformations of incidence algebras and applications to representations of finite dimensional algebras

Abstract:  We provide a unified approach, via deformations of incidence algebras, to several important types of representations with finiteness conditions, as well as the combinatorial algebras which produce them. We show that over finite dimensional algebras, representations with finitely many orbits, or finitely many invariant subspaces, or distributive coincide, and further coincide with thin modules in the acyclic case. Incidence algebras produce examples of such modules, and we show that algebras which are locally hereditary, and whose projective are distributive, or equivalently, which have finitely many ideals, are precisely the deformations of incidence algebras, and hence they are the finite dimensional algebra analogue of Prufer rings. New characterizations of incidence algebras are obtained, such as they are exactly algebras which have a faithful thin module. A main consequence is that any thin module comes from an incidence algebra: if V is either thin, or V is distributive and A is acyclic, then A/ann(V) is an incidence algebra and V can be presented as its defining representation. We classify thin/distributive modules, and respectively deformations, of incidence algebras in terms of first and second cohomology of the simplicial realization of the poset. A main consequence is obtaining a complete classification of thin modules over any finite dimensional algebra. Their moduli spaces are multilinear varieties, and we show that any multilinear variety can be obtained in this way. A few other applications, to Grothendieck rings of combinatorial algebras, to graphs and their incidence matrices, to linear algebra (torus actions on matrices), and to a positive answer to the no-gap conjecture" of Ringel and Bongartz, in the distributive case, are given. Other results in the literature are re-derived.

Title:  Stasheff’s  A_infinity  algebras and homotopy Gerstenhaber algebras

Abstract:  We’ll going to demonstrate that Stasheff’s minimal Ainfty algebra structure can be interpreted as a twisting cochain in the Hochschild cochain complex with respect to Gerstenhaber’s circle product. Furthermore, using so called higher brace operations, forming a homotopy Gerstenhaber algebra structure in Hochschild complex, it is possible to develop obstruction theory for degeneracy of Stasheff's  Ainfty  (and Cinfty) structures. The obstructions lay in Hochschild (or Harrison) cohomologies.

Takashi Kimura

Title:  Equivariant TQFTs and relations between representations

Abstract:  An equivariant topological quantum field theory is an algebra over an operad of decorated surfaces with an action of a finite group. We explain how, in many geometric realizations, they arise from relations between representations constructed from a group algebra, relations which can be expressed in terms of decorated trivalent trees.

Yvette Kosmann-Schwarzbach (Slides)

Title:  The Gerstenhaber bracket, Stasheff’s higher brackets and other brackets

Abstract: I shall consider two questions. How did the now classical brackets and graded brackets acquire their names? Which new brackets have appeared on the mathematical horizon since Murray Gerstenhaber’s and Jim Stasheff’s fundamental contributions?

Fujio Kubo

Title: An elementary performance of the algebraic deformation theory composed by Gerstenhaber

Abstract: I will discuss about the classification and deformations of associative algebras of low dimension. These algebras are with or without an identity and ones over the field of real numbers. To illustrate the variety of the structure constants, I will ,in my own way, get deeply involved in Gerstenhaber's algebraic deformation theory of 1964 and Gabriel's work on the geometric classification of 1975. I will also touch on the finite-dimensional non-commutative Poisson algebras and the process to find the appropriate controlling cohomology of algebraic systems, for example, Lie triple systems.

Kenji Fukaya

Title: Representability of A_infinity functors.

Abstract: I will explain how the idea A infinity version of Yoneda lemma is useful to show Floer homology can be defined for certain Lagrangian submanifold. One important idea using category theory is use representability of functors to obtain certain object in the category. This is an example of application of such idea in A infinity category.

Sylvain Lavau (Slides)

Title:  Foliations and BRST formalism

Abstract:  Every foliation F on a smooth manifold M admitting a resolution induces a Lie infinity-algebroid structure on this resolution. If the foliation F is finitely generated, then the generators of F can be seen as a set of constraints defined on the phase space T*M. In that case the corresponding resolution of F can be seen as a sub-resolution of the Koszul-Tate resolution appearing in the BFV-BRST procedure. Then we show that the homological vector field Q encoding the universal Lie infinity-algebroid of F is the restriction of the BRST differential s to the zero section of T*M.

Martin Markl (Slides)

Title: Distributive laws between the Three Graces (with Murray Bremner)
Motto: All algebras are equal, but some algebras are more equal than others.

Abstract:  Experience teaches us that the most common classes of algebras are the Three Graces - associative, commutative associative, and Lie - together with others that combine these in a specific way. The most prominent example of a combined structure are Poisson algebras which are combinations of Lie and commutative associative algebras by means of a quadratic homogeneous distributive law.    Our aim was to investigate whether the commonly known combinations of the Three Graces are the only possible ones via such a distributive law. In my talk I will present results of our on-going work including some "exotic" examples.   Our research was facilitated by advances in computer-assisted mathematics, and in particular the computer algebra system Maple worksheets written by the first author expressly for this project extended hand calculations of the second author dating from some 20 years ago.

John McCleary (Song - Some Love to Sing)

Title:  On factorization and chromatic graph homology  (with Vladimir Baranovsky)

Abstract: Factorization homology, introduced by Ayala, Francis, and Tanaka, generalizes Hochschild homology. Chromatic homology, a comultiplication-free Khovanov-type theory for graphs constructed by Helme-Guizon and Rong, approximates Hochschild homology when applied to a circle. We complete this picture by relating factorization and chromatic homology.

Malka Schaps (Slides)

Title:  Structure and deformations of group blocks

Abstract:  There are two approaches to deformations of finite dimensional algebras, an approach deforming structure constants as in Gerstenhaber's original 1964 paper, and an approach using the quiver and relations in which it is the relations which are deformed, where one must check flatness. Gerstenhaber's work on the Donald-Flanigan problem about deformations of group blocks generally used this second approach, and in the process, determined a lot of information about the structure of blocks of the symmetric group, some other reflection groups, and blocks of elementary abelian defect group.

Ekaterina Shemyakova (Slides)

Title: Obstructions to factoriations of differential operators on the algebra of densities on the line

Abstract:  Algera of densities was introduced in 2004 by H.Khudaverdian and Th.Voronov in connection with Batalin-Vilkovisky geometry. It is a commutative algebra with unity and invariant scalar product naturally associated with every manifold (and containing the algebra of functions). It gives a convenient framework to consider differential operators acting on densities of different weights simultaneously. We shall show that factorization of differential operators acting on densities on the line is different from what we know for the
classical case, where factorizations always exist and their structure is known due to Frobenius theorem. We explicitly describe the obstruction to factorization of generalized Sturm-Liouville operator in terms of a solution of the corresponding classical Sturm-Liouville equation.

Mihai Doru Staic (Slides)

Title: Hochschild cohomology and generalizations

Abstract: We discuss two deformation theories that are controlled by the higher order Hochschild cohomology over the two-sphere and respectively the secondary Hochschild cohomology.  We also show how one can define a Gerstenhaber algebra structure on these two cohomology theories.

Daniel Sternheimer (Slides)

Title: The reasonable effectiveness of mathematical deformation theory in physics

Abstract:  New fundamental physical theories can, so far a posteriori, be seen as emerging from existing ones via some kind of deformation. That is the basis for Flato's deformation philosophy", of which the main paradigms are the physics revolutions from the beginning of the twentieth century, quantum mechanics (via deformation quantization) and special relativity. On the basis of these facts we explain how symmetries of hadrons (strongly interacting elementary particles) could emerge" by deforming in some sense (including quantization) the Anti de Sitter symmetry (AdS), itself a deformation of the Poincaré group of special relativity. The ultimate goal is to base on fundamental principles the dynamics of strong interactions, which originated over half a century ago from empirically guessed internal" symmetries.  We start with a rapid presentation of the physical (hadrons) and mathematical (deformation theory) contexts, including a possible explanation of photons as composites of AdS singletons and of leptons as similar composites.  Then we present a model generating" framework in which AdS would be deformed and quantized (possibly at root of unity and/or in manner not yet mathematically developed with  noncommutative parameters").  That would give (using deformations) a space-time origin to the internal" symmetries of elementary particles, on which their dynamics were based, and either question, or give a conceptually solid base to, the Standard Model, in line with Einstein's quotation: The important thing is not to stop questioning. Curiosity has its own reason for existing."}

Mathieu Stienon (Slides)

Title:  Formality theorem for differential graded manifolds

Abstract:  The Atiyah class of a dg manifold (M,Q) is the obstruction to the existence of an affine connection that is compatible with the homological vector field Q. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory.  Using Kontsevich's famous formality theorem, Liao, Xu and I established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold (M,Q), there exists an L_oo quasi-isomorphism of dglas from an appropriate space of polyvector fields endowed with the Schouten bracket [-,-] and the differential [Q,-] to an appropriate space of polydifferential operators endowed with the Gerstenhaber bracket [[-,-]] and the differential [[m+Q,-]], whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold (M,Q) on the space of polyvector fields with the Hochschild--Kostant--Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology.

Ron Umble (Slides)

Title:  The coherent framed join and biassociahedra

Abstract: We construct the coherent framed join of finite ordered sets, form the reduced coherent framed join as an appropriate quotient, and use the combinatorics of the reduced framed join to construct the biassociahedra KK_{n,m}. In the ranges 1<= m <= 3  and 1 <= n <= 3, which are sufficient for our applications, the cellular chain complex C*(KK) is canonically isomorphic to the free matrad H_{\infty}.

Alexander Voronov (Limerick)

Title: Quantum deformation theory

Abstract: Classical deformation theory, whose creators we celebrate at this conference, is based on the Classical Master Equation (CME), a.k.a. the Maurer-Cartan Equation: dS + 1/2 [S,S] = 0. Physicists have been using a quantized CME, called the Quantum Master Equation (QME), a.k.a. the Batalin-Vilkovisky (BV) Master Equation: dS + h \Delta S + 1/2 {S,S} = 0. The CME is defined in a differential graded (dg) Lie algebra, whereas the QME is defined in a space V[[h]] of formal power series or V((h)) of formal Laurent series with values in a dg BV algebra V. One can anticipate a generalization of classical deformation theory arising from the QME, quantum deformation theory. Quantum deformation functor and its representability will be discussed in the talk.

Theodore Voronov (Slides)

Title:  Thick morphisms of supermanifolds and homotopy algebras

Abstract:  I will speak about a new sort of morphisms of (super)manifolds, which are not maps, but contain smooth maps as a particular case. They induce non-linear pull-backs on functions. For homotopy Poisson structures they give L-infinity morphisms for the brackets.