# Einstein Chair Mathematics Seminar

The Einstein Chair Mathematics Seminar is concentrated on the relationship between algebraic topology and quantum field theory.

This is like the opposite of JFK's famous saying "Ask not what your country can do for you, but rather ask what you can do for your country".

Namely beyond the noble desire of some mathematicians to clarify the foundations of quantum theory by the definitions and methodology of mathematics specifically to illuminate theoretical physics, there is a slightly different opportunity-mining theoretical physics as it is for the sake of mathematics.

Passing over in silence the commonly held conclusion that quantum field theory is physically successful in that it already contains a viable procedure for making predictions that are verifiable in high energy experiments, one can observe the success of the algorithms associated to the action principle, quantum style, in mathematics itself. There are rigorous and famous discussions in mathematics that are separated in mathematics but are actually unified in the minds of theoretical physicists by these algorithms of quantum field theory.

There are also unsolved areas of mathematics that suffer the same technical difficulties as those in quantum field theory; but while the former field outside of applications seems to languish in the math journals the latter field seems to flourish in the theoretical physics journals.

In the first example one can mention the celebrated invariants of differential topology (Donaldson and Vaughn Jones), of symplectic topology and algebraic geometry {Gromov and Witten), and of complex structures (Kodaira Spencer Griffiths...Kontsevich).

While for the second example one can mention intractable nonlinear PDEs like those appearing in 3D fluid dynamics. One may add in the second example the remark that important examples of tractable nonlinear PDEs, the integrable systems or hierarchies, seem to have a deep connection with quantum field theory and conversely they seem to have a deep role to play there.

There may be several veins of precious math material to mine from this opportunity and several methods.

I am particularly interested in the method of algebraic topology which associates linear objects (homology groups) to nonlinear objects with points ( manifolds...) just like quantum theory associates linear spaces of states to classical systems with points. The main character in algebraic topology is the nilpotent operator or boundary operator while in quantum field theory an important role is played by the nilpotent operators called Q and "delta" which encode whatever symmetry is present in the action of the particular theory and measure the obstruction to invariantly assign meaning to the integral over all paths.

In algebraic topology there is a powerful idea, due first to Stasheff but going beyond his famous and elegant concept of an infinitely homotopy associative algebra, which allows one to live with slightly false algebraic identities in a new world where they become effectively true. In quantum field theory the necessity to regularize or cutoff which sometimes destroys, but only slightly, identities expressing various symmetries and structures may provide an opportunity to use this powerful idea from algebraic topology.

Finally algebraic and geometric topology has always directed it efforts towards understanding in an algebraic way geometric objects like manifolds which are the classical models of spacetime, while quantum field theory often begins its specification of a particular theory with the classical action defined on the classical fields spread over spacetime and then proceeds to its algebraic algorithms.

All these connections suggest that one way to enter the mining business in the above sense is to define relevant algebraic structures with a nilpotent operator and formulate mathematically the intuitively clear physical idea of an "effective theory" as a kind of push forward of the entire algebraic structure in the new world or sense created by the idea underlying Stasheff's famous example of an A-infinity algebra.

The format of the seminar is generous regarding time and allows a robust exchange of information between the expositor and the other participants- who usually ask a lot of questions.

The first talk in the seminar is usually from 2- 3:15pm. After tea, the second lecture is from four until the discussion ends. The seminar takes place in Room 6417 on the 6th floor of the CUNY Graduate Center.

Videos of previous Einstein Chair seminars

DateDate: February 3, 2014

Speaker: Alistair Hamilton, Texas Tech University

Title: Compactifications and classes in the moduli space of curves

Abstract: In this talk I will describe a method of producing classes in compactifications of moduli spaces of curves using a framework that appears in quantum field theory under the heading of the Batalin-Vilkovisky formalism. These ideas go back to Kontsevich and his work on noncommutative geometry and Witten's conjecture. Part of the talk will be aimed at explaining some of the relevant background material, such as the orbi-cell decomposition of moduli space and results from the Batalin-Vilkovisky formalism. One of the goals of the talk will be to describe an analogue, dealing with compactifications of the moduli space, of a result due to Kontsevich describing the cohomology of the moduli space in terms of noncommutative geometry. The types of structures producing classes in such compactifications will be related to A-infinity structures, and the role of deformation theory in producing such structures will be explained.

Time permitting, a second construction of classes will be described. The result of pairing the homology and cohomology classes produced by these constructions may be expressed as a functional integral over a finite-dimensional space of fields. Computing such functional integrals can detect the nontriviality of these classes and some examples of this will be described.

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Date: October 28, 2013

Speaker: Prof. Ana Rechtman, Université de Strasbourg

Title: A non-amenable Folner foliation

Abstract: In this talk we will discuss the relation between amenable foliations and foliations with Folner leaves. Both notions are motivated by the corresponding ones on finitely generated groups, and in this context they are equivalent. In contrast, for foliations there are examples of non-amenable foliations with all its leaves Folner.

We will begin by discussing the definitions and previously know results.

We will construct an example of a non-amenable foliation with Folner leaves, using a foliated plug motivated by the work of Wilson on the construction of non-singular vector field on 3-manifolds with a finite a set of periodic orbits.

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Date: October 21, 2013

Speaker: Daniel Pomerleano, Univ of Tokyo

Title: A user friendly introduction to mirror symmetry

Abstract: From a purely mathematical point of view, mirror symmetry begins with an observation: there is a mysterious symmetry of [p,q] Betti numbers on pairs of six dimensional, compact, algebraic varieties with trivial canonical bundle. This coincidence extends into a dictionary relating holomorphic objects like holomorphic subvarieties and sheaves to symplectic geometry constructions of chain complexes associated to pairs of special sub manifolds of one half the total dimension. The generators are points of intersection between the two submanifolds and the differential is constructed from so called J-holomorphic Riemann surfaces whose boundary lies on these submanifolds and define paths between these points of intersection.

All of these observations demand a geometric explanation. The conjecture of Strominger, Yau and Zaslow is one of the most promising approaches to understanding mirror symmetry and the construction of mirror pairs. From this point of view, mirror pairs involves 2n-manifolds where open dense sets are fibred by n-dimensional torii. These orbits usually develop singularities in the closure of the open dense set--- a typical example is provided by the momentum map of a symplectic manifold equipped with a Hamiltonian torus action of maximal dimension. The conjecture postulates, roughly speaking, that mirror symmetry can be implemented by dualization of torus fibrations. I will give a gentle introduction to this conjecture, focusing on the relevant geometry and topology. Then, I will describe an example, based on joint work with Kazushi Ueda and Kwokwai Chan.

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Date: October 7 & 14, 2013

Speaker: Joana Cirici, Freie Universitat Berlin

Title: Cofibrant models of diagrams: applications to rational homotopy

Abstract: Bousfield and Guggenheim reformulated the homotopy theory of differential graded algebras in terms of Quillen model structures, obtaining a description of the homotopy category in terms of homotopy classes of morphisms between minimal objects. Following this line, it would be desirable to establish an analogous formulation for the category of mixed Hodge diagrams: this category appears naturally in the study of the rational homotopy of complex algebraic varieties and involves a mixture of differential graded algebras with (several) filtrations, defined over the fields of rational and complex numbers.

The axioms for Quillen’s model categories are very powerful and they provide, not only a precise description of the maps in the homotopy category, but also higher homotopical structures. As a counterpart, there exist interesting categories from the homotopical point of view, which do not satisfy all the axioms. This is the case of diagram categories involving filtrations, where more specific techniques have to be introduced.

In this talk, I will describe a weaker axiomatic than the one provided by Quillen's model structures, but sufficient to study the homotopy category of certain diagram categories in terms of level-wise cofibrant (or minimal) objects, and to extend the classical theory of derived additive functors, to non-additive settings. The main example of application is the category of mixed Hodge diagrams, but the theory may be applied to other contexts, such as diagrams of topological spaces, or diagrams involving complexes over abelian categories.

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Date: September 30, 2013

Speaker: Ralph Kaufmann, Purdue University

Title: Feynman categories: Universal operations and Hopf algebras.

Abstract: After giving a brief definition of Feynman categories. We will discuss how the approach of Feynman categories to operadic type objects naturally leads to universal operations via colimits. Examples are pre--Lie and admissible Lie structures, Gerstenhaber brackets, etc..

We will then present how a co-operad with a mutliplication gives rise to a Hopf algebra structure. This construction unifies Connes-Kreimer Hopf algebras for renormalization, the one of Goncharov for multizetas and the one of Baues in his double co-bar construction.

It is also naturally embedded into the even larger context of Hopf algebras resulting from Feynman categories.

The first part is joint work with B. Ward and the second with I.Galvez-Carrillo and A.Tonks.

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**Date:**May 6, 2013

**Speaker:**Ana Rechtman, IRMA, Universite de Strasbourg

**Title:**"The minimal set of Kuperberg's plug"

**Abstract**: In 1993 K. Kuperberg constructed examples of smooth and real analytic

flows without periodic orbits on any closed 3-manifold. These examples continue to be the only known examples with such properties and are constructed using plugs.

After reviewing K. Kuperberg’s construction, I will present part of a study of the minimal set of Kuperberg’s plug.

**Title:**"Existence of periodic orbits of geodesible vector fields on 3-manifolds"

**Abstract:**TBAA non-singular vector eld on a closed manifold is geodesible if there is a Riemannian metric making its orbits geodesics. We will study the existence of periodic orbits for such vector elds on closed 3-manifolds.

On 3-manifolds, K. Kuperberg constructed examples of vector elds without periodic orbits. On the other hand, C. H. Taubes proved that the Reeb vector eld of a contact form has periodic orbits. Reeb vector elds are geodesible, and also suspensions are geodesible. If we assume that the ambient manifold is either dieomorphic to the three sphere or has non trivial second homotopy group, we will prove the existence of a periodic orbit for volume preserving geodesible vector fields.

Volume preserving geodesible vector elds form a subset of the vector elds satisfying the time-independent Euler equations, we will explain how the above result extends to the second family of vector elds.

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**Date:**March 4, 2013

**Speaker:**Joana Cirici, Freie Universitat Berlin

**Title:**E1-Formality of Complex Algebraic Varieties

**Abstract:**I will first recall the construction of the filtration on the cohomology of complex algebraic varieties, called the weight filtration and defined by P. Deligne. I will then introduce the notion of rational homotopy type and, extending the Formality Theorem for the rational homotopy type of compact Kähler manifolds, I will show that every complex algebraic variety (possibly open and/or singular) is E1-formal, which means that its rational homotopy type is entirely determined by the first term of the spectral sequence associated with the multiplicative weight filtration.

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Date: February 25, 2013

Speaker: Joana Cirici, Freie Universitat Berlin

Speaker: Dr. Alastair Hamilton Title: Compactifications and classes in the moduli space of curves. Abstract: In this talk I will describe a method of producing classes in compactifications of moduli spaces of curves using a framework that appears in quantum field theory under the heading of the Batalin-Vilkovisky formalism. These ideas go back to Kontsevich and his work on noncommutative geometry and Witten's conjecture. Part of the talk will be aimed at explaining some of the relevant background material, such as the orbi-cell decomposition of moduli space and results from the Batalin-Vilkovisky formalism. One of the goals of the talk will be to describe an analogue, dealing with compactifications of the moduli space, of a result due to Kontsevich describing the cohomology of the moduli space in terms of noncommutative geometry. The types of structures producing classes in such compactifications will be related to A-infinity structures, and the role of deformation theory in producing such structures will be explained. Time permitting, a second construction of classes will be described. The result of pairing the homology and cohomology classes produced by these constructions may be expressed as a functional integral over a finite-dimensional space of fields. Computing such functional integrals can detect the nontriviality of these classes and some examples of this will be described.

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Date: February 11, 2013

**Speaker:**Dan Pomerleano, Kavli IPMU, University of Tokyo

**Title:**Symplectic homology of affine varieties and string topology

**Abstract:**We will examine the question of which cotangent bundles have nice symplectic compactifications as projective varieties. We will explain the motivations from deformation theory and mirror symmetry behind conjectures concerning the symplectic homology of symplectic manifolds which have such compactifications. Finally, we will describe what we are able to prove thus far.

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**Date:**December 10, 2012

**Speaker:**Prof. Bruno Vallette, Universite' NICE

**Title:**Homotopy theory of algebraic structures

**Abstract:**The goal of the talk will be to explain how one can do homotopy theory for algebraic structures, i.e. differential graded algebras over an operad, using the Kiszul duality theory for operads. This theory gives cofibrant resolutions (minimal models) for operads and thus define good notions of homotopy algebras with infinity-morphisms. I will use this approach to prove two main theorems: the homotopy transfer theorem and the fact that infinity-quasi-isomorphism are 'invertible'.

Finally, I will endow the category of coalgebras over the Koszul dual cooperad with a model category structure equivalent to the model category of algebras over the original operad. This provides us with a model category structure (without equalizers) on homotopy algebra with infinity-morphisms. In this case, a notion of homotopy relation for infinity-morphisms is given by the Lawrence-Sullivan dg Lie algabra model for the interval.

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**Date:**December 3, 2012

**Speaker:**Prof. Bruno Vallette, Universite' NICE

**Title:**Homotopy trivialization of the circle action

**Abstract:**One can study the homotopy theory of Batalin-Vikovisky algebras using two models of the associated operad: the Koszul model and the minimal model. This allows us to recover and generalize a well-known result of Barannikov-Kontsevich-Manin: the underlying homology groups of a dg BV-algebra satisfying the d/d-bar lemma carry a Frobenius manifold structure. This operadic approach provides us with higher structure which allows us to recover the homotopy type of the original BV-algebra.

We introduced a weaker but optimal condition which ensures the homotopy trivialization of the A-operator (circle action) called "Hodge-to-de Rham degeneration data", which permits us to apply the aforementioned theorem in Poisson geometry and in Lie algebra cohomology. For instance, we can endow the de Rham cohomology of Poisson manifolds (and more generally Jacobi manifolds) with a natural homotopy Frobenius manifold structure (extending the wedge product and allowing one to reconstruct the homotopy type of the de Rham original algebra).

This rich homotopy theory for BV-algebras also gives another interpretation of Givental group action on Cohomological Field Theories (aka Frobenius manifolds) in term of gauge group action. This provides us with a nice relationship between the intersection theory of moduli spaces of curves and the algebraic homotopy theory.

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**Date:**November 26, 2012

**Speaker:**Nikos Apostolakis, Bronx CC

**Title:**On (achiral) Lefschetz fibrations over the disk

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**Series of talks:**

**Speaker**: Tatyana Khodorovskiy, Hunter College

**October 15, 2012: First talk:**

In this talk, we will review the history and major results of the topology of 4-dimensional manifolds, as well as discuss the differences with manifolds of dimension other than 4. We will give an outline of the methods used to study 4-dimensional manifolds, and discuss some important invariants that one can associate to a 4-dimensional manifold.

**October 22, 2012: Second talk:**

Kirby Calculus is a beautiful theory which describes a pictorial method of depicting and manipulating (smooth) 4-dimensional manifolds. This method is very useful for proving positive existence results in smooth 4-dimensional topology. We will draw lots of pictures and give many examples. We will also discuss how these manipulations of 4-dimensional manifolds affect the 3-dimensional manifolds that bound them.

**October 29, 2012: Third talk:**

Complex surfaces (complex dimension 2 or real dimension 4) are a rich source of examples of 4-dimensional manifolds, and are better understood than general 4-dimensional manifolds. We will give many examples and discuss their classification in relation to the geography of 4-dimensional manifolds.

Date: April 18, 2012

Speaker: Anil Hirani, University of Illinois at Urbana-Champaign

Title: Algebraic Topology for Computations

Abstract: Algebraic topology was once used only within mathematics and physics. Over the last decade or so it has made the transition to high dimensional data analysis, sensor networks, computer graphics, computational geometry, and numerical analysis. I will give two computational examples where algebraic topology brings unification and clarity. But more importantly, in one of the examples it also brings remarkable and unexpected computational efficiency. This has opened up a problem once considered hopelessly intractable by the computational geometry and graphics communities. The other example is computation of harmonic cochains, a fundamental step in vector finite element computations. The first example is the optimal homologous chain problem (OHCP) -- given a chain (usually a cycle), find the smallest one in its homology class. A few years ago it was shown that in mod 2 homology this problem is NP-hard even to approximate. We showed that for integer homology, and posing OHCP as 1-norm minimization, it can be solved in polynomial time for a large class of simplicial complexes. Our main theorem is that the boundary matrix is totally unimodular if and only if the complex is relatively torsion-free. For such complexes linear programming can be used to solve OHCP as an integer program. This is the first appearance of torsion in computational problems. It is also the first topological interpretation of total unimodularity. A geometric interpretation has been known since the early days of linear programming. We've also introduced variants of OHCP and solved an open problem in computational knot theory. We get the second example by replacing 1-norm by 2-norm and homology by cohomology. By using isomorphism between harmonic cochains and cohomology we solve the cohomologous harmonic cochain problem by minimization and get the best known numerical algorithm for it.

**Date**: February 29, 2012

**Speaker**: Bhargav Bhatt, Univ of Michigan

**Title**: Comparison theorems in p-adic Hodge theory

**Abstract**:A fundamental theorem in classical Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison was the subject of a series of conjectures made by Fontaine in the early '80s. In the last three decades, these conjectures have been proven by various mathematicians, and have had an enormous influence on arithmetic algebraic geometry. In my talk, I will first discuss Fontaine's conjectures, and why one might care about them. Then I will talk about some work in progress that leads to a simple conceptual proof of these conjectures based on general principles in derived algebraic geometry, and some classical geometry with curve fibrations.

**Date**: February 22, 2012

**Speaker**: Tushar Das, University of North Texas

**Title**: Infinite-dimensional models of hyperbolic space and related analogues of dynamics and discrete groups

**Abstract**:We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite-dimensional separable Hilbert space. We generalize most results of negative curvature and Gromov-hyperbolic settings to get to their geometric core and have greater scope for applications. Many of the essential ideas are already present when working in Hilbert space, although one must be careful with boundaries and non-geodesic scenarios. There are many examples that explain what is fundamentally different from the classical finite-dimensional setting. For starters, in infinite dimensions properly discontinuous actions are no longer strongly discrete (finitely many orbit points in arbitrary balls) and though a Poincare-type summation over the orbits being finite implies strong discreteness always, the reverse fails in infinite-dimensions. The existence of fixed points of isometries and their structure will be discussed - here one discovers interesting parabolic behaviour that's absent in finite dimensions. We characterize groups whose limit sets are compact and convex-cobounded in terms of radial points in the limit set. Schottky groups whose limit sets are Cantor sets provide a variety of interesting phenomena where extensions of the classical thermodynamic formalism (a la Bowen) prove strong results about the geometry and dynamical properties of their limit sets. We prove a generalization of the Bishop-Jones theorem, equating the Hausdorff dimension of the radial limit set with the Poincare exponent. Time permitting, we sketch the proof of the Ahlfors-Thurston theorem and develop Patterson-Sullivan theory for divergence type groups. Here there are examples of convergence type groups that do not admit a conformal measure. To end, we discuss a few problems/applications. Almost everything will be developed from scratch with an attempt to present the underlying geometric ideas behind the proofs – graduate students are very welcome.

**Date**: February 1, 2012

**Speaker**: Daniel Pormeleano, University of Berkeley

**Title**: Curved Topology and Tangential Fukaya Categories

**Abstract**:In this talk, we will look at non-commutative versions of Landau-Ginzburg models. More precisely, given a simply connected manifold M such that it’s cochain algebra, C^*(M), is a pure Sullivan dga, we will consider curved deformations of the algebra of chains on it’s loop space C_* (\Omega(M)) and consider when the category of curved modules over these algebras becomes fully dualizable. For simple manifolds, like products of spheres, we are able to give an explicit criterion, like the Jacobian criterion, for when the resulting category of curved modules is smooth, proper and CY and thus gives rise to a TQFT. We give Floer theoretic interpretations of these theories for projective spaces and their products, which involve defining a Fukaya category which counts holomorphic disks with prescribed tangencies to a divisor.

**Date**: November 30, 2011

**Speaker**: Justin Young, Indiana University

**Title**: Brace Bar-Cobar Duality and the E_2 cochain algebra

**Abstract**: After providing motivation for studying the E_2 algebra (roughly, d.g. algebra plus homotopy commutative) structure on the cochain complex of a space, we will consider the category of dg E_2 algebras. We will show that the classical bar-cobar duality between dg algebras and dg coalgebras can be enhanced to a duality between dg E_2 algebras and dg Hopf algebras. Then, we will discuss application of this duality to finding a commutative/Lie model for a space.

**Date**: November 9, 2011,

**Speaker**: 1:45pm - 2:45pm: Prof. Alessandra Iozzi, ETH Zurich

**Speaker**: 2:50pm - 3:50pm: Prof. Marc Burger, ETH Zurich

**Title**: Weakly Maximal and Casual Representations of Surface Groups, I and II

**Abstract**:We define the notion of a weakly maximal representation of a surface group and relate it to the previously studied notions of a maximal representation and a tight homomorphism. While these are defined in terms of bounded cohomology, we describe a particular kind of weakly maximal representation arising from a purely geometrical construction, namely the causal representations. We then present a structure theorem for weakly maximal representations, which leads to a new characterization of Teichmüller space.

**Date**: November 2, 2011

**Speaker**:Prof. Jae Suk Park, Yonsei University

**Title**: Homotopical Probability Space

**Date**: October 26, 2011

**Speaker**: Prof. Maria Hempel, ETH Zurich

**Title**: Rigidity of Surfaces:The Polyhedral case

**Date**: October 19, 2011

**Speaker**: Prof. Robert Guralnick, University of Southern California

**Title**: Maps from the Generic Riemann Surface

**Abstract**: In 1936, Zariski proved that for g greater than 6, there is no holomorphic map from the generic Riemann surface of genus g to the Riemann sphere so that the monodromy group of the branched covering representing the map is solvable. It was well known that the generic Riemann surface of genus 6 has a degree 4 map to the sphere and so its monodromy group is solvable. We generalize this by showing that an indecomposable map from the generic Riemann surface of genus g at least 4 of degree n to the sphere (that is one which is not a composition of branched coverings) has monodromy group an alternating or symmetric group of degree n at least 2g or greater than g/2 respectively. We will also discuss the case of rational maps on the Riemann sphere as well as analogs in positive characteristic.

**Date**: May 18, 2011

**Speaker**: Prof. Gregory Ginot, Université Pierre et Marie Curie

**Title**: Beilinson-Drinfeld algebras, Quantization and Todd genus

**Abstract**: Beilinson-Drinfeld algebras are a structure describing the observables of a quantization of a classical field theory. Following, Costello, Grady-Gwilliam, we will explain how the Todd genus is encoded in the existence of a quasi-isomorphism between two Beilinson-Drinfeld algebras, which, as spaces, are given by Hochchild homology of the Weyl algebra and the de Rham forms. This quasi-isomorphism follows from a general theorem on quantization of observables applied to a real Chern-Simons field theory that we will describe.

**Date**: May 11, 2011

**Speaker**: Prof. Gregory Ginot, Université Pierre et Marie Curie

**Title**: Factorization algebras and invariants of framed manifolds

**Abstract**: We will recall one definition of topological chiral homology, which is an invariant associated to framed manifolds of dimension n and E_n-algebras introduced by Lurie. We will explain how this invariant can be computed as a factorization algebra homology and can be extended to higher Hochschild homology when the algebra is actually commutative. We will also explain a few elementary consequences of these equivalences. This is joint work with Tradler and Zeinalian.

**Date**: May 4, 2011

**Speaker**: Prof. Gregory Ginot, Université Pierre et Marie Curie

**Title**: Factorization algebras and examples

**Abstract**: We will introduce the definitions of Factorization Algebras, which are a kind of "multiplicative" cosheaf studied by Costello-Gwilliam. Factorization algebras are closely related to several classical objects of study in algebraic topology such as mapping spaces, E_n-algebras, Hochschild homology and arise as well when studying the observables of (quantum) field theories. We will explain more precisely the aforementioned relationships and will give some elementary properties, as well as examples, of factorization algebras.

**Date**: March 23, 2011

**Speaker**: Prof. Ezra Getzler, Northwestern University

**Title**: A filtration of open/closed topological field theory

**Abstract**: we prove a higher analogue of the presentation of modular functors due to Moore and Seiberg, and extend it to open/closed field theory.

**Date**: March 2, 2011

**Speaker**: Prof. Albert Fathi, École Normale Supérieure de Lyon

**Title**: Aubry-Mather Theory, Lax-Oleinik semi-group, and viscosity solutions of the Hamilton-Jacobi Equation

**Abstract**: For Lagrangian systems Aubry-Mather theory is about what remains when KAM theorem cannot be applied. KAM theorem provides invariant tori which are Lagrangian, hence they are smooth solutions of the Hamilton-Jacobi theory. As was discovered 15 years ago, when there are no smooth solutions weak solutions come into play and Aubry-Mather theory can be recovered from them: this is weak KAM theory. The lecture will give an introduction to the subject; in the second part we will provide details, and some more recent developments.

**Date**: December 8, 2010

**Speaker**: Prof.Daniel Meyer, Helsinki University

**Title**: Invariant Peano curves of Expanding Thurston maps

**Abstract**: Let f be a self mapping of the two sphere which is a branched covering. We assume that the orbits of the critical points are finite and that the map is expanding in the sense that the components of the iterated preimage of a certain simply connected domain have diameters tending to zero. We show a sufficiently high iterate F of f is the quotient of the expanding self mapping g of the circle with the same degree as F. This means there an intertwining mapping C from the circle onto the two sphere (i.e. a Peano curve) so that FC=Cg. This generalizes a result of Milnor and corresponds to a result by Cannon-Thurston in the Kleinian group case. Furthermore F can be obtained by a geometric construction called "mating". This construction due to Douady-Hubbard is a way to geometrically combine two polynomials to form a rational map.

**Date**: November 17, 2010

**Speaker**: **CANCELLED** 3pm: Prof. Yakov Sinai, Princeton University

**Title**: Renormalization group technique and burgers system

**Abstract**: The talk will be about the construction of solutions of the Burgers systems which develop singularities in finite time. It will consist of five parts: 1. General Outline of RGM 2. Burgers system, a survey. 3. The derivation of the equation for the fixed point. 4. The linearized spectrum. 5. Main result of blow ups in complex-valued solutions of BS.

**Speaker**: 3:30-4:40pm: Prof. Yevsey Nisnevich, Courant Institute

**Title**: Adeles, Grothendieck topologies & G-buncles for semisimple groups G

**Abstract**: Conclusion of mini-course (SIMONS LECTURES/EINSTEIN CHAIR LECTURES/ARITHMETIC GEOMETRY TEAM LECTURES)

**Date**: November 10, 2010

**Speaker**: 1:30-3:15pm: Prof. Ralph Kaufmann, Purdue University

**Title**: Algebraic structures from odd operads and real blow ups

**Abstract**: Some classical algebraic structures like Gerstenhaber's bracket on the Hochschild complex have an operadic origin. We discuss generalizations of these operations coming from different operadic type settings. This includes BV operators and master equations. The natural setting for these considerations is the theory of (twisted) generalized operads. We also relate these constructions to the topological level by using moduli spaces and real blow-ups. This is joint work with Ben Ward and Javier Zuniga.

**Speaker**: 3:30-4:40pm: Prof. Yevsey Nisnevich, Courant Institute

**Title**: Adeles, Grothendieck topologies & G-buncles for semisimple groups G

**Abstract**: Continuation of mini-course (SIMONS LECTURES/EINSTEIN CHAIR LECTURES/ARITHMETIC GEOMETRY TEAM LECTURES)

**Date**: November 3, 2010

**Speaker**: 1:30-3:15pm: Joseph Hirsh, GSUC

**Title**: Relative Categories, Infinity Categories, and Simplicial Localization

**Abstract**: We will describe a passage, called simplicial localization, from categories Cwith collections of "weak equivalences" W to (infty,1) categories. We will explain how this passage can be seen as an equivalence between two versions of the theory of homotopy theories. Time permitting, we will introduce another (equivalent) version of the simplicial localization which is more useful for computations.

**Speaker**: 3:30-4:40pm: Prof. Yevsey Nisnevich, Courant Institute

**Title**: Adeles, Grothendieck topologies & G-buncles for semisimple groups G

**Abstract**: Continuation of mini-course (SIMONS LECTURES/EINSTEIN CHAIR LECTURES/ARITHMETIC GEOMETRY TEAM LECTURES)

**Date**: October 27, 2010

**Speaker**: 1:30-3:15pm: Prof. Scott Wilson, Queens College

**Title**: Period extensions of holonomy to mapping spaces.

**Abstract**: Bismut showed that holonomy, defined on the loopspace of a manifold with bundle and connection, extends to a closed periodic form on the loopspace of the manifold. Moreover, the restriction of this class to constant loops gives the classical chern character of the bundle. I will begin this talk by giving a new presentation of this result using a variation on hochschild complexes and iterated integrals that is suitable for locally trivialized bundles. Next, I will introduce the notion of an abelian gerbe with connection. A similar variation of higher hochschild complexes and their iterated integrals will be applied to show that, for an abelian gerbe with connection over M, the 2-holonomy of the connection extends to a closed periodic form on the space M^T of maps of a torus into M. Finally, it's known (and I'll explain) how an abelian gerbe with connection on M induces a line bundle with connection on LM, and I'll describe how the two constructions from above, on LLM and M^T, are related. This is joint work with Thomas Tradler and Mahmoud Zeinalian.

**Speaker**:3:30-4:40pm: Prof. Yevsey Nisnevich, Courant Institute

**Title**:Adeles, Grothendieck topologies & G-buncles for semisimple groups G

**Abstract**: Continuation of mini-course (Please see below)

**SIMONS LECTURES/EINSTEIN CHAIR LECTURES/ARITHMETIC GEOMETRY TEAM LECTURES **

**Speaker**: Prof. Yevsey Nisnevich, Courant Institute

**Title**: Adeles, Grothendieck topologies & G-bundles for semisimple groups G

**Locations**: STONYBROOK MATHEMATICS AND CUNY GRADUATE CENTER

**{TWO PARTS EACH DAY AND INDEPENDENT AT EACH SITE}**

**STONYBROOK ** 2 & 3:30pm [Tues, Oct 12 & Fri, Oct 15: Room 4-125] [Mon, Oct 18 & Friday Oct 22: Room P-131]

**CUNY GRAD CENTER [ROOM 6417] WED 3:30 & 5pm October 13, 20**

**OPEN TO ALL- EXPERTS AND NON EXPERTS**

1) Preliminaries on schemes: Noetherian schemes, closed, non-closed and generic points, the local rings and the residue fields of points, the structure sheaf. The Krull dimension of schemes. The functor of points Y -> X(Y) of a scheme X with the values in another (variable) scheme Y and the Grothendieck-Yoneda embedding. Fiber product of schemes, fibers of a morphism of schemes f: X -> Y.

2) Some important classes of morphisms of schemes: unramified, flat, etale and smooth morphisms. Regular and singular points on a scheme.

3) Grothendieck topologies: the general definition and some functorial properties. The principle examples: the Zariski, Nisnevich, etale and faithfully flat quasi-compact topologies on the category of schemes and its subcategories (in the increasing order). The concepts of points and local rings of points for Grothendieck topologies. Sheaves on a Grothendieck topology and a fiber of a sheaf over a point on a topology. Henselian rings and their main properties, Henselization of a ring. The description of local rings and fibers of sheaves over points for the first 3 topologies listed above and appearance of Henselian rings for the Nisnevich and etale topologies.

4) Cohomology of sheaves of abelian groups on a Grothendieck topology. Some functorial properties. Cohomological dimensions of the principle topologies.

5) The principal homogeneous spaces for a sheaf of (possibly) non-abelian groups G on a topology \tau, locally trivial in this topology (= \tau G-torsors). Cohomology of non-abelian sheaves on a topology \tau and non-abelian exact sequences for a sheaf of subgroups H of G.

6) Group schemes - definition and the principle examples: additive G_a, multiplicative G_m, algebraic tori, full linear group GL_m over an arbitrary base scheme. Semisimple and reductive group schemes over a base scheme.

7) Adeles for schemes of dimension 1 with values in a group scheme G. Henselian adeles. The Nisnevich topology as a mean for a geometrization of adeles. Nisnevich cohomology and adelic class groups.

8) The principle vanishing theorems for the class groups and the Nisnevich and etale cohomology of semisimple group schemes over general Dedekind (= regular 1-dimensional) rings. Cohomological expressions for these invariants in the non-simply connected case. Finiteness theorems for Dedekind rings with finitely generated divisor class groups.

9) Applications of the vanishing theorems: proof of the conjecture of Grothendieck and Serre on the Zariski local triviality of rationally trivial etale G-torsors over regular base of dimension 1 and 2 and for Henselian local regular rings of an arbitrary dimension.

10) Further applications: extensions of the Uniformization Theorem for the moduli space of G-vector bundles over a smooth irreducible projective curve C/k onto arbitrary fields k and general semisimple group schemes G over C. (It was previously known for an algebraically closed field k and a constant semisimple group scheme G/k only and it was due mainly to Drinfeld-Simpson (1995).

**Date:** September 22, 2010

**Speaker:** 2-2:50pm-Prof. Edson de Faria, University of Sao Paulo

**Title:** Unbounded conformal distortion - David homeomorphisms via Carlson boxes

**Abstract:** We construct a family of examples of increasing homeomorphisms of the real line whose local quasi-symmetric distortion blows up almost everywhere, which nevertheless can be realized as the boundary values of David homeomorphisms of the upper half-plane. The construction of such David extensions uses Carlson boxes.

**Speaker:** 3-3:50pm-Prof. Jae-Suk Park, Research Institution, Seoul

**Title:** Algebraic Models of QFT and corresponding Homotopy Invariants

**Abstract:** This talk is about an attempt to understand quantum field theory mathematically. I shall explain the notion of a binary QFT, which shall be viewed as an algebraic model for quantum field theory with a binary product. First a binary QFT algebra is a certain algebraic structure defined over k[[h]] such that its classical limit (h=0) is a commutative differential graded algebra (CDGA) over k. Then a binary QFT is a binary QFT algebra with a QFT cycle, whose classical limit is a cycle of the CDGA. I shall define notions of quantized observables and their quantum expectation values and quantum correlation functions as certain homotopy invariants of this algebraic setup. An exact solution for all quantum correlation functions shall be presented under certain conditions. Such a theory is part of a natural family parametrized by a smooth-formal moduli space which has certain coordinates. These generalize those of flat or special coordinates in topological string theories. Time permitting I shall briefly explain a program to go from or quantize a CDGA with "cycle" to obtain a binary QFT.

**Date:** September 1, 2010

**Speaker:** Prof. Qian Yin, University of Michigan

**Title:** Expanding Thurston Maps and Cell Decompositions

Abstract: Thurston maps are branched covering maps over the 2-sphere with finite post-critical sets. In this talk, we are going to define expanding Thurston maps and give natural cell-decompositions of the 2-sphere induced by the dynamics of an expanding Thurston map following Bonk and Meyer. These cell decompositions help us understand the structure of expanding Thurston maps. As time permits, we will see how these cell decompositions give us a Gromov hyperbolic space, and describe some very nice properties of it.

**Date:** May 19, 2010

**Speaker: **Prof. John Baez, University of California Riverside

**Title:** Electrical Circuits

Abstract: While category theory has many sophisticated applications to theoretical physics — especially quantum fields and strings — it also has interesting applications to a seemingly more pedestrian topic: electrical circuits. The pictorial resemblance between circuit diagrams and Feynman diagrams is an obvious clue, but what is the underlying mathematics? This question quickly leads us to an interesting combination of category theory, symplectic geometry, complex analysis and graph theory. Moreover, electrical circuits are just one example of 'open systems': physical systems that interact with their environment. While textbooks on classical mechanics usually focus on closed systems, open systems are more important in engineering, and their mathematics is arguably deeper and more interesting.

**Date:** April 28, 2010

**Speaker:** Prof. H.T. Yau, Harvard University

**Title:** Lattice gases, large deviations, and the incompressible Navier-Stokes equations

Abstract: We study the incompressible limit for a class of random particle systems on the cubic lattice in three space. For starting probability distributions corresponding to arbitrary macroscopic finite energy initial data the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore explicit exponential rates for the convergence (large deviations) are obtained.

**Date:** April 21, 2010

**Speaker:** Prof. Gregory Ginot, Université Pierre et Marie Curie

**Title:** "Hochschild chain complex over spaces and topological chiral homology."

Abstract: This is a joint work (in progress) with T. Tradler and M. Zeinalian. We will review the notion of Hochschild chain complex over spaces, following Pirashivili, which, roughly speaking, is a (homotopy) bifunctor from the category of commutative algebras and the category of spaces to the category of DG-commutative algebras. We will explain how this theory can be interpreted as a special kind of (unoriented) extended (\infty,n)-dimensional topological field theories and how it is related to topological chiral homology in the sense of Lurie.

**Date: **March 10, 2010

**Speaker:** David Chateur, Universite' de Lille

**Title:** "On Bivariant chains of PL-manifolds"

**Abstract:** Let M be a closed oriented PL-manifold, in this talk we will present a bivariant chain theory for M. This bivariant chain complex is naturally quasi-isomorphic to the PL-chains of M and its singular cochain complex via a nice chain model of the Poincaré duality map. We will apply this construction to show that Mc-Clure's partial commutative intersection product is equivalent to the cup product.

**Date:** February 24, 2010

**Speaker:** Alexander Shnirelman, Concordia University

**Title:** "Long-time behavior of 2-dimensional flows of ideal incompressible fluid"

**Abstract: **Consider the motion of ideal incompressible fluid in a bounded 2-d domain. It is described by the Euler equations which, in spite of their deceptive simplicity, are hard to investigate. For the initial velocity field smooth enough, the Euler equations have a unique solution for all time, and it's natural to ask what is its long-time asymptotics. The physical experiments and computer simulations show a nontrivial, counterintuitive picture of a huge attractor in the space of incompressible velocity fields, consisting of stationary, periodic, quasiperiodic and, possibly, chaotic solutions. This picture appears to contradict the conservative nature of the Euler equations; this is similar to contradiction between the microscopical reversibility of the molecular motion and macroscopical irreversibility of thermodynamical processes. I am going to demonstrate the results of computer simulation and physical experiments on the fluid motion, and discuss connections of this problem with analysis, dynamical systems and even topology.

**Date:** February 10, 2010

**Speaker:** Ruth Lawrence, Hebrew University

**Title:** "On quantum knot and 3-manifold invariants"

**Abstract:** Given a semi-simple Lie algebra and a choice of representation on each component, there is defined an invariant of links in the 3-sphere as a polynomial in a parameter q known as the colored Jones polynomial. Using a surgery presentation of 3-manifolds, this can be extended to invariants of compact oriented 3-manifolds (and more generally of links embedded in such 3-manifolds) dependent on a root of unity q, namely the Witten-Reshetikhin-Turaev (WRT) invariant. The Ohtsuki invariant of rational homology spheres is a formal power series-valued invariant which may be constructed out of congruence properties of WRT invariants, and should be considered as an asymptotic expansion of the WRT invariant, in a suitable sense. Collectively these invariants are known as quantum invariants, as distinct from classical topological invariants obtainable with classical techniques of homology and homotopy. This will be a survey talk in which we define and discuss properties and structure of these invariants, ranging across integrality, congruence, almost modularity, holomorphicity and asymptotic structure. No prior knowledge of quantum invariants will be assumed.

**Date:** February 3, 2010

**Speaker:** Borya Shoiket, University of Luxembourg

**Title:** "What is the categorical generalization to bialgebras of the monoidal category of bimodules over an algebra."

**Abstract:** Let B be an associative algebra with a coassociative coalgebra structure which is compatible in the sense that the comultiplication is a map of algebras. In other words B is an associative and coassociative bialgebra,or bialgebra for short. Such an algebraic structure has a deformation theory which is controlled by a "deformation complex" D(B), in particular its second cohomology. D(B) is called the Gerstenhaber-Schack complex for its creators Gerstenhaber and Schack at the University of Pennsylvania. It is conjectured that D(B) has a homotopy commutative product and a homotopy Lie bracket of degree -2 and that these are compatible by a homotopy derivation property. So far, no explicit construction of any of these pieces of the structure is known. We will present a construction of a structure on the cohomology of D(B) of the sort that would result if the conjecture were true. To do this we make the additional assumption that B is a hopf algebra. This means there is an anti-automorphism S intertwining multiplication and comultiplication in a diagram you can see on wikipedia. An analogous contruction in the case of the complex controlling deformations of an associative algebra A (the Gerstenhaber- Hochshild complex) is due to S. Schwede and uses the monoidal structure on the category of A-bimodules. In particular, Schwede gives a conceptual construction of the Gerstenhaber bracket on the corresponding Gerstenhaber-Hochschild cohomology. In the case of bialgebras what replaces the category of bimodules is the category of tetramodules. This category admits two different monoidal structures. These two structures are compatible in a rather non-trivial way. Tetramodules over B have a 2-monoidal category structure. We will prove the following general theorem: let Q be an n-monoidal abelian category (with some mild assumptions), and let e be the unit object in Q. Then Ext/Q (e,e) has the structure that would result if the complex defining the Ext had a commutative product and compatible lie bracket of degree -n and all this up to homotopy.

**Date: **November 18, 2009

**Speaker:** Zhenghan Wang, Senior Researcher, Microsoft Corporation

**Title:** "Pictures of curve relations on surfaces can define 2+1 topological quantum field theories"

**Abstract:** Curves in surfaces are deceptively elementary and simple objects. By considering linear combinations of simple closed curves in surfaces and relations among them, we arrive at beautiful (2+1)-TQFTs, the so-called diagram TQFTs. Diagram TQFTs with specific idempotents due to Jones and Wenzlas relations can be considered as quantum generalizations of linearized mod two- homology. Using picture relations among trivalent graphs instead of simple closed curves, we can construct all of the (2+1)-TQFTs related to the constructions of Drinfeld. Such TQFTs are not only interesting in mathematics, but they are also likely to play a role in condensed matter physics, even possibly in the construction of quantum computers. If time permits, I will discuss possible applications to quantum computing.

**Date:** October 21, 2009

**Speaker:** Joan Millès, Université Nice

**Title:** "André-Quillen cohomology theory of an algebra over an operad"

**Abstract:** Following the ideas of Quillen and by means of model category structures, Hinich, Goerss and Hopkins have developped a cohomology theory for (simplicial) algebras over a (simplicial) operad. Thank to Koszul duality theory of operads, we describe the cotangent complex to make these theories explicit in the differential graded setting. We recover the known theories as Hochschild cohomology theory for associative algebras and Chevalley-Eilenberg cohomology theory for Lie algebras and we define the new case of homotopy algebras. We study the general properties of such cohomology theories and we give an effective criterium to determine whether a cohomology theory is an Ext-functor. We show that it is always the case for homotopy algebras.

**Date: **April 22, 2009

**Speaker:** Gregory Ginot, Université Pierre et Marie Curie

**Title:** "Hochschild (co)homology over spaces and String Topology"

**Abstract:** We will explain how one can define Hochschild (co)chain complex associated in a functorial way to any space X, CDG algebra A and A-module M. We will explain the relationships between these Hochschild (co)homology theories and string (or Brane or Surface) topology.

**Date:** March 11, 2009

**Speaker:** Max Lipyaskiy, Columbia University

**Title: **"Geometric cycles in classical topology and Floer theory"

**Abstract:** I will introduce a new approach to Floer theory based on mappings of Hilbert manifolds into a target space. After describing the general framework for the theory, I will discuss the relationship of the Floer theory of the cotangent bundle of a manifold to the classical homology its loopspace.

**Date:** December 3, 2008

**Speaker:** Victor Turchin, Kansas State University

**Title:** "Hodge decomposition in the homology of long knots" (Joint work in progress with G. Arone and P. Lambrechts)

**Abstract:** We will describe a natural splitting in the rational homology and homotopy of the spaces of long knots Emb(R^1,R^N). This decomposition arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting will be presented. Based on this generating function one can show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. There are two more motivations to study this decomposition. First, it is related to the study of the homology of higher dimensional knots Emb(R^k,R^N). Second, it is deeply related to the question whether Vassiliev invariants can distinguish knots from their inverses.

**Date:** November 19, 2008

**Speaker:** Prof. Carl-Friedrich Boedigheimer, Mathematical Institute University of Bonn

**Title:** "Models for the Moduli Spaces of String Theory"

**Abstract:** Let M = M(g,m,n) be the moduli space of surfaces of genus g with n incoming and m outgoing boundary curves. Theses moduli spaces have attracted much attention in recent years for their importance in string theory (either of physical or mathematical origin). We shall give a description of M as a finite cell complex. The cells are given by simultaneous conjugation classes of q-tuples of permutations in the p-th symmetric group, where p < 2h + 1, q < h + 1, and h = 2g - 2 + n + m, and the number of cycles of the permutation in the last component of the q-tuple (respectively, the permutation in the first component) is n (respectively, m). Furthermore, we shall describe the operad structure of all such moduli spaces in terms of these models.

**Date:** November 12, 2008

**Speaker:** Prof. Jean Louis Loday, CNRS et Université de Strasbourg

**Title:** First talk: "Combinatorial Hopf algebras".

**Abstract:** Many recent papers are devoted to some infinite dimensional Hopf algebras called collectively "combinatorial Hopf algebras". Among the examples we find the Faa di Bruno algebra, the Connes-Kreimer algebra and the Malvenuto-Reutenauer algebra. We give a precise definition of such an object and we provide a classification. We show that the notion of preLie algebra and of brace algebra play a key role.

**Second talk:**"Homotopy associative algebras and Stasheff polytope".

**Abstract:** We construct an A-infinity algebra structure on the tensor product of two A-infinity algebras by providing a simple formula for a geometric diagonal of the Stasheff polytope. This formula is based on the Tamari poset structure on the set of planar binary trees. As a result the operad A-infinity gets a binary cooperation. We show that similar formulas give higher cooperations so that the operad A-infinity gets a structure of A-infinity coalgebra for the Hadamard product.

**Date:** October 22, 2008

**Speaker:** Prof. Alastair Hamilton, Univ. of Connecticut

**Title:** "Noncommutative geometry, compactifications of the moduli space of curves and A-infinity algebras."

**Abstract:** There is a theorem, due to Kontsevich, which states that the homology of the moduli space of curves can be identified with the homology of a certain infinite dimensional Lie algebra. This Lie algebra is constructed as the noncommutative analogue of the Poisson algebra of hamiltonian vector fields on an affine symplectic manifold. There is a compactification of the moduli space of curves which was introduced by Kontsevich in his study and proof of Witten's conjectures. It is defined as a certain quotient of the well-known Deligne-Mumford compactification. In the first part of this talk I will describe how Kontsevich's Lie algebra can be deformed into a differential graded Lie algebra whose homology recovers precisely the homology of this compactification of the moduli space. This is achieved through the use of an additional structure on this Lie algebra -- a Lie cobracket -- which makes Kontsevich's Lie algebra into a Lie bialgebra. Such structures have been considered by various authors including Chas-Sullivan, Movshev, Fukaya and Ginzburg-Schedler. I will explain how the relationship between the moduli space and its compactification is reflected algebraically in this framework -- the deformation parameters contain the extra information at the boundary of the moduli space. In the second part of the talk I will explain how the definition of an A-infinity algebra and one of its important generalisations known as a cyclic A-infinity algebra can be subsumed in this framework of noncommutative geometry using the notion of Maurer-Cartan moduli space. I will explain a simple construction which produces classes in the homology of any differential graded Lie algebra by exponentiating elements in its associated Maurer-Cartan moduli space. This construction can be used to produce classes in the moduli space from cyclic A-infinity algebras, as was observed by Kontsevich. The corresponding algebraic structures producing classes in the compactification of this moduli space seem to sometimes go under the heading of `quantum A-infinity algebras'. There is a natural deformation theory which controls the process of building a quantum A-infinity algebra out of a cyclic A-infinty algebra. I will explain how the problem of building a quantum A-infinity algebra out of a cyclic A-infinity algebra corresponds to the problem of extending a class defined on the moduli space to its compactification. I will explain how these ideas apply to a simple but important example.

**Date:** September 17, 2008

**Speaker:** Prof. Vasiliy Dolgushev, UC Riverside

**Title:** "Proof of Swiss cheese conjecture"

**Dates:** September 10 - September 14, 2008

**Speakers: **Please see the schedule

**Title:** "FRG CUNY Workshop"

To view (please allow time to download talks): http://vvf.streamhoster.com/ViewVirtualFolder.aspx?vfid=5e8cab41-9c97-4f03-af50-3c1d117b86d9

Date: May 28, 2008

**Speaker:** Nathan Habegger, Université de Nantes

**Title:** "Vassiliev invariants and related invariants in 3 dimensions"

**Date:** May 21, 2008

**Speaker:** Michael Freedman, Microsoft

**Title:** "Positivity of the universal pairing in dimension=3 "

**Abstract:** This will be a mathematics talk explaining arxive:math/0802.3208. The topic is a positivity property of the sesquilinear pairing defined by gluing a superposition of three manifolds with a fixed boundary Y to a superposition with boundary (Y bar) - Y with reversed orientation. Proof uses all aspects of the theory of 3-manifolds from Dehn to Thurston to Witten to Perelman. One motivation for this study comes from condensed matter physics. I mention it bellow in the hope that it may attract a few physicists to the lecture (who would also be welcome to leave at any point they choose as the discussion will necessarily be in the language of topology.) Surface layer physics, particularly the two-dimensional electron gasses which generate the fractional quantum Hall effect (FQHE), are presently being investigated as a possible substrate for the construction of a quantum computer. The mathematical concept of a (2+1)-dimensional Unitary topological quantum field theory (UTQFT) provides the link between the lowest energy properties of surface layer physics and topology. Under this mapping developed by Witten, Read, Moore, and others, a bounded 3-manifolds map to quantum mechanical state on the bounding surface. A consequence of a celebrated result of Cumrum Vafa (Harvard) that for no single UTQFT is this mapping "injective," i.e., separates all three manifolds with fixed boundary. A crucial question is whether, taken together, a family of UTQFTs might successfully reflect all of 3-manifold topology (that is, be injective). This paper (arxiv:math/0802.3208) by Freedman, Walker, and Calegari shows that this is, at least, possible by producing a complexity function "c" on closed 3-manifolds with the same formal property for gluings: c(AB) or= max( c(AA), c(BB)) with equality holding only if A=B, as would a partition function of a UTQFT which had been (miraculously) liberated from the constraints of Vafa's theorem. This situation is in direct opposition to the state of affairs in 3+1 dimensions. There (arxiv:math/0503054) it was found that much of the interesting detail of 4-manifolds (Donaldson and Seiberg-Witten invariants) were not reflected in the structure of (3+1)-dimensional UTQFTs. This establishes a fundamental distinction between the quantum mechanics of two- and three-dimensional systems.

**Date:** May 14, 2008

**Speaker:** Tom Lada, North Carolina State University

**Title:** "Homotopy Algebras and Brace Algebras "

**Abstract:** We will review the concept of L-infinity algebras from several points of view, including the relationship with brace algebras. We will also discuss several types of actions of such algebras, such as L-infinity modules and OCHAS (open closed homotopy algebras). Several concrete examples will be exhibited.

**Date:** April 23, 2008

**Speaker: **Jae-Suk Park, Yonsei University, Seoul, Korea

**Title: **"Minimal model of QFT "

**Abstract:** This talk is about an effort to understand quantum field theory (QFT) mathematically by studying what we call commutative quantum algebras (CQA). We begin with a simple but important example of CQA; Fix a ground field k with char(k)=0, and let h be a formal parameter. Def) A BV quantum algebra is a triple (C[[h]], K, m), where 1. (C[[h]], m) is a graded commutative associative k[[h]]-algebra, where C[[h]] is free as a k[[h]]-module such that (C= C[[h]]/h C[[h]], m ) is a graded commutative associative k-algebra, 2. (C[[h]],K) is a cochain complex over k[[h]]. 3. The failure of K being a derivation of m is a derivation of m and is divisible by h. We denote Q as the restriction of K to C, which gives the classical complex (C, Q) over k. The notion of BV quantum algebra is derived from so called the BV quantization scheme. As for cultural background, the BV quantization scheme is supposed to associate a BV quantum algebra to a given "classical field theory". Then there is an art, mastered by physicists, of doing "Feynman path integrals" involving a "quantum master action functional" and a choice of "gauge fixing", on which the result of path integral is supposed to be independent.

**Date:** April 16, 2008

**Speaker: **Moira Chas, Stony Brook University

**Title:** "New results about Goldman's lie bracket for closed curves on a surface "

**Date: **January 23, 2008

**Speaker:** Daniel Sternheimer, Keio University, Japan

**Title:** "The deformation philosophy of quantization and noncommutative analogues of space-time structures"

**Abstract:** Deformations in physics and mathematics are part of a deformation philosophy. This philosophy was promoted in mathematical physics in joint work with Moshe Flato dating back to the 70's. One development, especially its realization on manifolds which I understand has been discussed in this seminar, is deformation quantization. This refers to deformations of commutative algebra structures into non commutative algebra structures. Another development, the deformations of algebras related to classical Lie groups leads to the so-called quantum groups with interesting connections again to topology and the topics of this seminar. One may also think of objects dual to noncommutative algebras, the so-called quantum spaces, as deformations of classical spaces, the objects dual to commutative algebras. Expressing usual geometry in terms of algebra that makes sense for noncommutative algebras leads to a rich field in mathematics called noncommutative geometry. Deforming the space-time of Einstein, Lorentz and Minkowski and its Lie group of symmetries leads to a fruitful object which together with its group of symmetries is referred as AdS or "anti de Sitter space". The study of AdS has significant physical consequences. One example is that massless particles in four dimensional space-time like photons become, in a way compatible with quantum electro dynamics, composites of massless particles in three dimensional space-time called singletons. This is part of a general correspondence between the four dimensional space time AdS theory where the geometry is related to string theory, and a three dimensional space time theory which is defined using non abelian connections and is invariant under conformal transformations. Thus the latter is a CFT, a conformally invariant quantum field theory. In physics this correspondence lead to many developments, and now there is a rich part of theoretical physics that is referred to by the name AdS/CFT correspondence. In the first part of the lecture before tea I will give an elementary introduction to these deformation ideas and survey some of these areas, always insisting on the conceptual aspects. In the second part after tea I shall attempt to develop further any points which the audience requests. We will describe an ongoing program in which anti de Sitter would be quantized in some regions related to black holes. We speculate that this could explain a universe in constant expansion and that higher mathematical structures might provide a unifying framework. Apparently higher mathematical structures such as L_infinity and A_infinity algebras are often discussed in this seminar. No prior specific knowledge will be assumed in the first part which will prepare somewhat for the second part.

**Date: **October 2, 2007

**Speaker:** Alain Connes (Collège de France)

**Title: **“Noncommutative geometry and physics”

**Date:** October 10, 2007

**Speaker: **Prof. Dirk Kreimer, IHES

**Title:** “Hochschild Cohomology in renormalizable Quantum Field Theory ”

**Abstract:** We review the structure of perturbative renormalization from the viewpoint of Hopf algebras in Feynman graphs. We first rederive Zimmermann's forest formula, and how it is used in quantum field theory (QFT). We try to emphasize four points:

- how to obtain renormalized amplitudes in QFT
- how do QFT Green functions compare to the polylogarithm
- how do coideals in the Hopf algebra connect to internal symmetries of the theory
- how to go beyond perturbation theory

**Date:** November 7, 2007

**Speaker:** Borya Shoikhet (IHES)

**Title:** "Koszul duality in deformation quantization and topological quantum field theory"