A list of speakers from previous seminars follows in reverse chronological
order. [Videotapes of all seminars are available for viewing upon
request.]
The Einstein Chair Mathematics Seminar is concentrated on the relationship
between algebraic topology and quantum field theory. 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 seminar begins around half past twelve in the Mathematics Commons Room with
lunch and informal discussion. This is followed by a regular lecture format
from two until tea at four. After tea, there is more discussion with the remaining
enthusiasts for a variable period.
The seminar takes place on the 4th and the floor of the
CUNY Graduate Center’s new home in the old B. Altman building (365 Fifth Avenue)
located diagonally across from the Empire State Building.
Date: May/June, 2007
Speaker: Prof. Prof. Nathalie Wahl, University of Copenhagen
Date: May 2, 2007
**Joint two hour talk with Prof. Fred Gardiner and Prof. Jun Hu**
Title: “Associahedra in Teichmueller theory”
Time: 2-3pm
Speaker: Prof. Fred Gardiner, GSUC/Brooklyn College
Abstract: We explore four ways the Stasheff associahedra of A- infinity algebras is related to the Teichmueller theory of finitely many moving points on the boundary of the conformal disc
- the finite earthquake theorem,
- the infinitesimal finite earthquake theorem,
- Teichmueller's least conformal distortion theorem,
- the cylindrical quadratic different.
Time: 3-4pm
Speaker: Prof. Jun Hu, GSUC/ Brooklyn College
Abstract: We focus on how associahedra are related to the spaces of finite earthquake maps, measures and the geometric structure for a compactification of the deformation space of n distinct cyclically marked points on the boundary of the conformal disc.
Date: April 25, 2007
Time: 3:30pm-
Speaker: Prof. Janko Latschev,Ludwig-Maximilians-University
Title: “Symplectic field theory and string topology”
Date: April 18, 2007
Time: 10:45am-12:45pm
Speaker: Prof. Janko Latschev, Humboldt University
Title: "Symplectic field theory and string topology"
Abstract: In this talk I will report on the status of joint work in progress with Kai Cieliebak on the relation of the two theories in the title.
SFT was introduced by Eliashberg, Givental and Hofer and studies holomorphic curves in symplectic manifolds with cylindrical ends. String topology was introduced by Chas and Sullivan and studies new algebraic structures found on the chains on the loop space of a manifold. The link between the two theories is given by the study of holomorphic curves with Lagrangian boundary conditions. The relevance of string topology to this problem was first noticed by Fukaya.
Depending on audience interest and background, I will review the relevant parts of both theories, and discuss how they are related. Time allowing, I will also discuss some technical aspects of the program and give a conjectural picture of the ingredients of a full relative symplectic field theory.
Date: March 28, 2007
Speaker: Prof. Leonid Chekhov (Steklov Mathematical Institute)
Room: 6417
Title: “Quantum Teichmuller spaces”
Abstract: We describe the quantization of teichmuller spaces of Riemann surfaces with holes in terms of shear coordinates and introduce the algebra of geodesic functions satisfying quantum skein relations.
For particular choices of the geodesic basis (and for surfaces of higher genus), these bases are related to problems of monodromy preserving deformations of matrix linear differential equations.
We extend the description to surfaces with marked points on boundaries, construct the corresponding classical and quantum shear coordinates and geodesic functions and describe the algebras obtained. In the list of these algebras there are A(n) algebras corresponding to the disc with n marked points on the boundary and D(n) algebras corresponding to annulus with n marked points on the boundary.
Special All day conference on Quantum Theory, Manifolds and Moduli spaces
Date: March 21, 2007
Time: 2-4pm
Speaker: Prof. Kevin Costello, Northwestern University
Title: Quantising Chern-Simons theory
Abstract: I'll talk about how one can construct perturbative
Chern-Simons invariants in greater generality than the constructions of
Kontsevich and Axelrod-Singer. The construction relies on the renormalisation
techniques I described in a recent talk at CUNY. The result is a certain
algebraic structure on the cohomology of an oriented manifold.
Time: 5-6:30pm
Speaker: Prof. Pavel Mnev, Petersburg Department of Steklov Institute of Mathematics
Title: Mini-Course ”On simplicial BF theory”
Abstract: I will discuss the construction of a "simplicial BF
theory", the field theory with finite-dimensional space of fields,
associated to a triangulated manifold, that is in a sense equivalent to
topological BF theory on the manifold (with infinite-dimensional space of
fields). This is done in framework of simplicial program - program of
constructing discrete topological field theories. I will also discuss the relation
of these constructions to homotopy algebra.
Date: March 14, 2007
Time: 1:30-2:45pm
Speaker: Prof. Bruno Vallette, Université Nice Sophia-Antipolis
Title: Deformation theory of representations of prop(erad)s
Abstract: To any morphism of prop(erad)s, we define a chain complex, à la Quillen, which mesures its deformations. In this way, we give a conceptual explanation of well known cohomology theories (Hochschild, Harrison, Chevalley-Eilenberg, Gerstenhaber-Schack). We will show that this chain complex has a rich algebraic structure, generalizing some operations (braces) involved in the proof of Deligne's conjecture. As a corollary, we get a Lie bracket (up to homotopy) whose Maurer-Cartan elements will be studied. For instance, structures of (bial)gebra up to homotopy over a prop(erad) will be interpreted in this way. [Joint work with Sergei Merkulov]
Time: 3-4:15pm
Speaker: Prof. Scott Wilson, School of UMN Twin Cities
Title: The Algebra of Chains
Abstract: One way to understand an algebraic on the chains of a space or a manifold is to relax the condition on where the algebraic structure is defined. In particular, much is gained by reducing or enlarging either the domain or range of the operations. I'll describe such "partially defined algebraic structures" abstractly, and give examples related to the classical diagonal map and intersection product. Together we'll see how, for a manifold, these two fit together to give a (partial) chain-level open Forbenius algebra. This is recent joint work with G. Friedman and J. McClure, building off work of Goresky-MacPherson, McClure and ideas of Sullivan on open Frobenius algebras.
Time: 5-6:15pm
Speaker: Prof. Pavel Mnev, Petersburg Department of Steklov Institute of Mathematics
Title: Mini Course ”On simplicial BF theory”
Abstract: I will discuss the construction of a "simplicial BF theory", the field theory with finite-dimensional space of fields, associated to a triangulated manifold, that is in a sense equivalent to topological BF theory on the manifold (with infinite-dimensional space of fields). This is done in framework of simplicial program - program of constructing discrete topological field theories. I will also discuss the relation of these constructions to homotopy algebra.
Date: March 7, 2007
Speaker: Pavel Mnev, Petersburg Department of Steklov Institute of Mathematics
Time: 2pm
Title: Mini course on simplicial BF theory
Abstract: I will discuss the construction of a "simplicial BF theory", the field theory with finite-dimensional space of fields, associated to a triangulated manifold, that is in a sense equivalent to topological BF theory on the manifold (with infinite-dimensional space of fields). This is done in framework of simplicial program - program of constructing discrete topological field theories. I will also discuss the relation of these constructions to homotopy algebra.
Date: February 28, 2007
Speaker: Prof. Alex Bene, University of Southern California
Time: 2-3:30pm
Title: Canonical Lifts of the Johnson homomorphisms to the Torelli Groupoid.
Abstract: The Torelli group of a surface S with one boundary component is defined as the subgroup of the mapping class group of S which acts trivially on the first homology of S. This is in fact just the first of a sequence of nested "higher Torelli" subgroups which serves as an approximation to the mapping class group itself. The study of the Torelli groups often involves analysis of the Johnson homomorphisms which are certain abelian quotients of the Torelli groups. Morita has shown that the first Johnson homomorphism lifts to a crossed homomorphism of the whole mapping class group, while recently Morita and Penner have shown by use of homology marked fatgraphs that it in fact lifts canonically to the Torelli groupoid. In this talk, I will report on recent work (joint with R. Penner and N. Kawazumi) in which canonical lifts of the higher Johnson homomorphisms to the Torelli groupoid have been constructed. The construction relies on Kawazumi's interpretation of the Johnson homomorphisms in terms of Magnus expansions adapted to the Morita-Penner perspective using fatgraphs.
Speaker: Pavel Mnev, Petersburg Department of Steklov Institute of Mathematics
Time: 4pm-until
Title: Mini course on simplicial BF theory
Abstract: I will discuss the construction of a "simplicial BF theory", the field theory with finite-dimensional space of fields, associated to a triangulated manifold, that is in a sense equivalent to topological BF theory on the manifold (with infinite-dimensional space of fields). This is done in framework of simplicial program - program of constructing discrete topological field theories. I will also discuss the relation of these constructions to homotopy algebra.
Date: February 21, 2007
Special All day conference on Quantum Theory, Manifolds and Moduli spaces
1-2pm Prof. John Terilla, Queens College
Title: "Deformations of algebraic quantum field theories"
Abstract: Using Park's Quantum Background formalism for quantum field theories, I will describe a mathematical theory of their deformations. In the case that the moduli space is smooth---a condition guaranteed if the number of classical parameters equals the number of quantum parameters---I will describe an algebraic structure on the tangent bundle of the moduli space which is relevant to correlation functions.
2:30-4:15pm Prof. Maxim Kontsevich, Institut des Hautes Études Scientifiques
Title: "Deformations of quantum field theories"
Abstract: Usually physicists describe quantum field theories in terms of classical lagrangians. In particular, the number of parameters is the same for the quantum and for the classical worlds, but the exact correspondence depends on the choice of so called regularization scheme. On another hand, conformal field theories in 2 dimensions provide examples of quantum field theories with no classical lagrangian attached. The picture of the parameter space was proposed many years ago by A. Zamolodchikov.
I will describe a general formalism for deformation theory of a quantum field theory which gives an algebra over chains in little discs operad in the translation invariant case.
No prior knowledge of quantum field theory is assumed. I will give relevant definitions and examples.
5-6pm Prof. Thomas Tradler, NYC College of Technology
Title: "A TCFT on the Hochschild complex of an Calabi-Yau elliptic space"
Abstract: We describe the construction of a topological conformal field theory (TCFT) on the Hochschild complex of a Calabi-Yau elliptic space. A main example of a Calabi-Yau elliptic space comes from the de Rham complex of differential forms on a compact oriented manifold M. In fact, in this case, the above TCFT gives rise to the Chas-Sullivan string topology operations on a chain model of the free loop space, when M is simply connected. This is a joint work with Kevin Costello and Mahmoud Zeinalian.
Date: February 7, 2007
2-3:30pm: Prof. Jean-Michel Bismut, Universite Paris-Sud
Title: The hypoelliptic Dirac operator
Abstract: If X is a compact spin or spin-c manifold, the Dirac operator is an elliptic first order differential operator acting on smooth twisted spinors. All the operators coming from de Rham or holomorphic Hodge theory are Dirac operators.
The purpose of the lecture will be to explain the construction of a hypoelliptic Dirac operator on the total space X of the tangent bundle of X. It depends on a parameter b> 0. As bà 0, the operator converges in the proper sense to the classical Dirac operator on X, and as bà 8, it ‘converges’ to the geodesic flow on TX. In earlier work, we had obtained a deformation of the classical Hodge-de Rham theory. As a special case, we obtain here a deformatio of the Dolbeault-Hodge theory.
Applications to holomorphic Ray-Singer torsion will be outlined. The R genus of Gillet and SoulLe reappears in this context.
4pm-until: Prof. John Terilla, Queens College (CUNY)
Title: Minicourse on deformation theoretic ideas related to quantum objects
Abstract: This week I will discuss L_infty algebras and the relationship between versal solutions to the Maurer-Cartan equation, minimal models, and Massey products. This is the classical preparation for the introduction of the "h-bar direction" into these settings.
Date: January 31, 2007
Time: 2:00 pm
Speaker: Prof. John Terilla, Queens College, CUNY
Title: In the direction of h-bar
Abstract: When Witten gave the J.W. Gibbs lecture in 1998, he emphasized that string theory relies on deformations of mathematical structures in two very different directions and he emphasized that mathematicians were proficient in handling only one of these directions.
Similarly, given some initial BV data, there are two, substantially different, deformation directions. I will call these two directions the "t-direction" and the "h-bar direction." Deformations in the t-direction are the traditional ones, but in this setting they are trivial. Deformations in h-bar direction are less well understood, but contain the key to unlocking physical-type correlation functions. In this introductory lecture I will introduce this h-bar direction and try to illustrate its importance in quantum invariants. I will expand on this idea and provide details and context in a mini-course.
Tea
Time: 4:15 pm
Speaker: Nathalie Wahl, University of Copenhagen
Title: Stabilizing mapping class groups of 3-manifolds
Abstract: (joint work with Allen Hatcher) Let M be a compact, connected 3-manifold with a fixed boundary sphere d_0M. For each prime manifold P, we consider the mapping class group of the manifold M_n^P obtained from M by taking a connected sum with n copies of P. We prove that the ith homology of this mapping class group is independent of n in the range n>2i+1. Our theorem moreover applies to certain subgroups of the mapping class group and include, as special cases, homological stability for the automorphism groups of free groups and of other free products, for the symmetric groups and for wreath products with symmetric groups.
Date: December 13, 2006
Special All day conference on Quantum Theory and Manifolds
Speaker: Prof. Graeme Segal, All Souls College/Oxford University
Title: Locality in quantum field theory
Abstract: Quantum field theory adds to quantum mechanics the idea that phenomena are localized in a given space-time. There have been a variety of attempts to axiomatize the subject. I shall discuss a version which is motivated by the path-integral point of view. On one hand, it starts from a partition function defined globally for the whole space-time and analyses how the function factorizes when the space-time is split into pieces; on the other, it can be viewed as assembling the partition function from local contributions. I shall relate these two approaches to some well-known phenomena in algebraic topology on one side, and in operator-algebra theory on the other.
Speaker: Prof. Kevin Costello, Northwestern University
Title: Some remarks on renormalisation and the Batalin-Vilkovisky formalism.
Abstract: I'll talk about some work in progress on this topic, which gives an approach to the renormalisation of certain gauge theories on compact manifolds.
Date: November 8, 2006
Speaker: Prof. Andrei Pajitnov, Universite de Nantes
Title: “Circle-valued Morse functions and dynamics of their gradient flows”
Abstract: The Morse theory relates the critical points of a real-valued function on a manifold to the topology of the manifold. A generalization of this theory to the case of circle-valued Morse functions was initiated by S.P. Novikov in the early 80s.
The central geometric construction of the circle-valued Morse theory is the Novikov complex - a generalization of its classical predecessor, the Morse complex. The Novikov complex of a circle-valued Morse function f on a manifold M is freely generated by the critical points of f over a certain Laurent series ring and computes the completed homology of a suitable covering of M.
This algebraic object carries a lot of information about the dynamics of the gradient flow of f. We introduce this construction and then discuss the Novikov exponential growth conjecture and recent developments of the circle-valued Morse theory for 3-dimensional manifolds.
WEDNESDAY, SEPT. 27TH
Einstein Chair Seminar in Topology & Quantum Objects
Website: http://math.gc.cuny.edu/seminars/einsteinchair.html
Meets 2:00PM – “until”
Rm. 6417 [Note new meeting day for fall '06]
Speaker: Stavros Garoufalidis (Georgia Institute of Technology)
Title: "Resurgence and asymptotic expansions in quantum topology"
Abstract: A resurgent function (due to J. Ecalle) is an analytic function in a neighborhood of zero that admits an endless analytic continuation at the complex numbers minus a discrete set of singularities (but no natural boundaries). Examples of resurgent functions are meromorphic functions, algebraic functions, or more generally solutions of differential or difference equations (linear or not). An elementary (but unnoticed) observation is that the Taylor series of a resurgent function have asymptotic expansions, with exponentially small terms included. What does resurgence have to do with quantum topology? The latter associates formal power series to knotted 3-dimensional objects, and numerous numerical invariants. The existence of asymptotic expansions is a well-prized problem that includes, among other things the Volume Conjecture. We will formulate a general resurgence conjecture of the formal power series of knotted objects, and give two crucial test cases of it: a proof for the two simplest knots: the trefoil, and the figure-eight.
We will also discuss the relation between resurgence and the Gromov norm of graph-valued invariants, such as the Kontsevich integral.
April 25, 2006
Speaker: Prof. Jae-Suk Park, Yonsei
University, Seoul
Title of talk 1: "Obstruction of Symmetries."
Abstract: Obstruction occurs both in
classical and quantum cases. In classical case, extended deformation problem of
certain mathematical structure may be obstructed in a process of solving
classical master equation (Maurer-Cartan equation of deformation complex). This
does not imply that the associated moduli space should be bad but implies that
the space has some additional structures, which I will discuss about.
The simplest example leads to certain Lie group G action on total space of a
vector bundle E with certain G-equivariant section S over certain manifold, as a
smooth model for the moduli space, which may be "identified" with S^{-1}(0)/G in
a favorable situation only.
In quantum case, the prominent form of obstruction is so called "anomaly" in
Physicist term which obstructs extension of a solution to classical master
equation to that of quantum master equation. This does not imply that quantum
version of deformation theory is ill defined but implies that there are some
hidden structures in the theory. Those structures will be characterized by
notion of minimal quantum L-infinity algebra and give certain relations among
"correlation functions", thus to be regarded as quantum symmetry.
Title of talk 2: "Quantization of DGA"
Abstract: Differential Graded Commutative
Algebra was one of favored gadget of Dennis Sullivan in 1970's. I will define
the notion of quantization of differential graded associative algebra
(A-infinity algebra in general), which involves another quantum A-infinity
structure (playing the role of L-infinity algebra in quantum deformation
theory).
I will also discuss briefly on generalization to quantization of differential
graded category (A-infinity category in general).
March 28, 2006
Speaker: Prof. Kevin Costello, University of Chicago
Title: "String topology, Chern-Simons theory and supersymmetric quantum
mechanics."
Abstract: The Laplacian and the d and d* operators on the forms on a
Riemannian manifold can be interpreted as defining a kind of supersymmetric
quantum mechanics. Consideration of the Feynmann diagrams for this
theory leads to a construction of forms on the moduli space of Riemann surfaces
with boundary. These forms yield the cohomology classes in moduli space
constructed from higher-genus string topology.This theory can be coupled to a
flat U(n) bundle. The integral of the corresponding forms on moduli space, if it
converged, would yield the partition function for Chern-Simons theory.These
forms on moduli space satisfy the gluing conditions necessary to define an open
topological conformal field theory. I'll discuss how this construction fits into
the general framework of open-closed TCFTs, and hopefully say about
renormalisation of Chern-Simons theory and Sen-Zwiebach's BV formalism.
October 25
Speaker: Jean-Michel Bismut (Université Paris-Sud)
Title: “The hypoelliptic Laplacian”
Abstract: We construct a deformation of Hodge theory, whose corresponding
a Laplacian is a hypoelliptic operator on the cotangent bundle. This Laplacian
interpolates between classical Hodge theory and the geodesic flow, and more
generally any Hamiltonian dynamical system. This Laplacian should be thought of
as a semiclassical limit of the Witten deformation of the Laplacian on the loop
space, associated to the energy functional on the loop space. Applications to
the Ray-Singer analytic torsion will be presented.
October 11
Speaker: Janko Latschev (Humboldt University)
Title: “Symplectic field theory and string topology”
Abstract: Symplectic field theory constructs algebraic invariants for
contact manifolds by counting suitable punctured pseudo-holomorphic curves.
Naively one can view the relevant curves as cobordisms between collections of
closed Reeb orbits.
In the special case that the contact manifold is the unit cotangent bundle of a
closed oriented Riemannian manifold Q, these invariants depend only on the
underlying smooth structure of Q. As the Reeb orbits are in bijective
correspondence with non-constant closed geodesics on Q in this case, one is lead
to suspect a close relationship to the topology of the free loop space of Q.
I will start my talk by giving a short account of some of the algebraic
structures on the contact homology side (due to Eliashberg, Givental and Hofer)
and on the loop space side (due to Chas and Sullivan).
Then I will sketch a program to construct chain level homomorphisms that
intertwine these structures (this is joint work in progress with Kai Cieliebak
from Munich).
May 17
Speaker: Robert Penner
1st talk Title: "The cell decomposition of Riemann's moduli space"
2nd talk Title: "Compactifying the cell decomposition of Riemann's moduli
space"
April 19
Speaker:
Stephan Stolz (University of Notre Dame)
Title: "Elliptic cohomology via 2-dimensional conformal field theories"
Abstract: This is a report on a joint project with Peter Teichner (Berkeley)
with the eventual goal of relating elliptic cohomology to 2-dimensional
conformal field theories. We conjecture that the spaces CFT_n of `super
symmetric conformal field theories of degree n' fit together in a spectrum
homotopy equivalent to the `topological modular form spectrum' TMF of
Miller-Hopkins.
Evidence is provided by the fact that the 'partition function' of a
supersymmetric conformal field theory is an integral modular form.
Furthermore, as Elke Markert will explain in her talk, the analogous spaces
EFT_n of 'supersymmetric 1-dimensional euclidean field theories of degree n'
fit together in an Omega-spectrum homotopy equivalent to the K-theory
spectrum.
Speaker:
Elke Markert (University of Notre Dame)
Title: "K-Theory and 1-dimensional euclidean field theories."
Abstract: In this talk I will show that the space EFT_n of 'supersymmetric
1-dimensional euclidean field theories of degree n' can be interpreted as a
'configuration space'. This can be used to show that the loop space of EFT_n
is homotopy equivalent to EFT_{n+1}; in other words, the spaces EFT_n fit
together in an Omega-spectrum. It turns out that this spectrum is homotopy
equivalent to the usual K-theory spectrum. Furthermore, one can define
'connective' versions of field theories leading to spaces eft_n forming a
spectrum homotopy equivalent to the connective K-theory spectrum.
March 15
Speaker:
Yakov Eliashberg (Stanford University)
Title: " Madsen-Weiss theorem for not-so-algebraic topologists"
Abstract: Madsen-Weiss's proof of Mumford conjecture about the stabilized
mapping class group cohomology is a tour de force in algebraic topology. I
will explain how using a bit more advanced singularity theory one can make the
proof more accessible for algebraically challenged topologists. This approach
allows one to also get some information about the topology of the
Deligne-Mumford compactification of the moduli space of Riemann surfaces. This
is a joint work with S. Galatius and N. Mishachev.
March 8, 2005
Speaker:
Paul Seidel (University of Chicago)
Title: "Khovanov cohomology and symplectic geometry"
Abstract: Khovanov cohomology is a knot
invariant which refines the Jones polynomial.
After a brief introduction,
I will explain the(conjectural) interpretation of this invariant in terms of symplectic geometry.
This involves symplectic Floer cohomology.
We will then look at a \Z/2-equivariant version, and explain how that explains the relationship
between Khovanov cohomology and Ozsvath-Szabo cohomology of double branched covers. Joint work with Ivan Smith (Cambridge).
February 22, 2005
Speaker:
Peter S. Ozsvath (Columbia University)
Tite: "Holomorphic disks and Heegaard diagrams"
Abstract: Heegaard Floer homology is an invariant for
three-manifolds constructed using a suitable construction of Lagrangian Floer
homology in a symplectic manifold associated to a Heegaard diagram for a
three-manifold. I will describe properties of this invariant, and applications
to Dehn surgery problems for knots in the three-sphere. The material I
describe here is joint work with Zoltan Szabo.
February 8, 2005
Speaker:
Veronique Godin(IAS):
Title:
Bordered fat graphs and operations on the homology of a mapping class
Abstract: Using fat graphs, I will introduce a model for the classifying
space of the bordered mapping class group. I will then introduce the bordered
fat graph complex which computes the integral homology of the mapping class
group. I will finally use this complex to build a Kudo-Araki-Dyer-Lashoff
operation which is induced by Miller's double loop structure.
Speaker: Alex Bene(UCLA):
Title:
Moduli space of Riemann surfaces from a combinatorial viewpoint
Abstract: It has long been known that the
moduli space of metric fatgraphs is combinatorial moduli space
, is
homeomorphic to a trivial fiber product over Riemann's moduli space of
n-pointed genus g curves: 
.In
particular, each point gives rise a cell decomposition of where each cell
corresponds to an isomorphism class of fatgraph. Using these cell
decompositions, one can define combinatorial classes which correspond to the
characteristic tautological classes of 
.
In particular, combinatorial cells corresponding to graphs with certain
valency can be shown to be (in some sense) dual to the Mumford kappa classes
ka. In this talk, we discuss the difficulties and advantages
of using these cycles, and homology intersection theory in general, to compute
intersection numbers in the tautological ring
November 16, 2004:
2:00-3:15 pm (Rm C197)
Speaker: Kevin Costello (Imperial College, London)
Title: Topological conformal field theories and Calabi-Yau categories.
Abstract: We study open, closed and open-closed topological conformal
field theories. We show that an open TCFT is the same as a unital
Calabi-Yau A-infinity category. For each such, we show that there is a
universal closed TCFT, which is the initial element in the category of
compatible open-closed TCFTs. The homology of this universal closed TCFT
is calculated to be the Hochschild homology of the category. As a
corollary, part of the higher-genus B model is constructed, and the
higher-genus Deligne conjecture is proved. This result is also used to
relate the Gromov-Witten invariants and the Fukaya category of a
sympectic manifold, assuming the evidence of open-closed Gromov-Witten
invariants.
3:15-4:30 pm (Rm C197)
Speaker: John Terilla (Queens College)
Title: Mathematical foundations of quantum field theory
Abstract quantum field theory, as advanced in recent years by Jae-Suk Park, can be understood from the following background data: a noncommutative ring P, and element m in P satisfying m squared equals 0, a left P-module C, and an element in C which is annihilated by left multiplication by m. I will discuss some of the mathematical foundations of quantum field theory from this point of view. The discussion will be categorical with the goal of capturing the main mathematical inputs and outputs of a quantum field theory as developed from its background data.
April 20th, 2004:
2-3:15pm- Speaker: John Terilla (Stony Brook
University)
Title: Mathematical aspects of Park triples
4pm-Speaker: Jae-Suk Park (Korea Institute
for Advanced Study)
Title: A Mathematical Scheme of Quantum Field
Theory
April 27th, 2004:
2-3:15pm- Speaker: Vasiliy Dolgushev
(Massachusetts Institute of Technology)
Title: Formality theorems for Hochschild (co)
chains of C8(M) on an arbitrary smooth manifold of M
Abstract: Proofs of Tsygan's formality conjectures for chains formulated
in math.QA/9904132 would unlock important algebraic tools which might
lead to new generalizations of the famous index and Riemann-Roch-Hirzebruch theorems.
Despite this pivot role in the traditional investigations and the efforts of various people the most general version
of Tsygan's formality conjecture has not yet been proven. In my recent paper math.QA/0402248 I have proven Tsygan's formality conjecture
for Hochschild chains of the algebra of functions on an arbitrary smooth manifold
M using the Fedosov resolutions proposed in math.QA/0307212 and
the formality quasi-isomorphism for Hochschild chains of ${\Bbb
R}[[y^1, \dots, y^d]]$ proposed in paper math.QA/0010321 by
Shoikhet. In my talk I will explain globalization technique proposed in math.QA/0307212 that allowed me to simplify
Kontsevich's proof of formality theorem for Hochschild cochains of the algebra of functions on an arbitrary
smooth manifold and to prove formality conjecture for Hochschild chains of this algebra. I will also say a couple of words
about applications of these results to equivariant
quantization, computation of Hochschild homology of quantum algebras and description of traces
in deformation quantization.
4pm- Speaker: Thomas Tradler (New York
City College of Technology)
Title: Homotopy Structures and the Hochschild
Complex
Abstract: The Hochschild complex of an
associative algebra has proved to be of great value when
considering all kinds of homotopy structures. In this talk,
I will define chain level operations on the Hochschild
cochain complex of a unital, associative algebra with
non-degenerate invariant inner product, which induce
structures similar to the ones from String Topology. As a
special case, one gets a cell decomposition of the cactus
operad. The homology of the cactus operad is the operad
describing Batalin-Vilkovisky algebras also known as
BV-algebras. The latter introduced these algebras in
an effort to describe algebraically a very general class of
quantum field theories.
May 4th, 2004:
Speaker: Sergei Merkulov (Stockholm
University)
2-3:15pm
Title: Infinity constructions of local
geometries
Abstract: We argue that some classical local geometries are
of infinity origin, i.e. their smooth formal germs are (homotopy)
representations of cofibrant (di)operads in spaces
concentrated in degree zero. In particular, they admit
natural infinity generalizations when one considers homotopy
representations of that (di)operads in generic differential
graded spaces. Poisson and Nijenhuisen geometries provide us
with simplest manifestations of this phenomenon.
Applications to extended deformation theory are discussed.
4pm-
Title: De Rham model for string topology
Abstract: We use the theory of iterated integrals to give a
new and short proof of a theorem identifying the
Chas-Sullivan product on the homology (over real numbers) of
the free loop space, LM, of a smooth compact oriented
manifold M with the composition of endomorphisms of the de
Rham algebra of M (viewed as an object of the derived
category of de Rham bimodules). Along the proof we get a new
and sometimes useful algorithm for computing the
Chas-Sullivan product.
May 11th, 2004:
Speaker: Riccardo Longoni (Universita' di
Roma "La Sapienza")
2-3:15pm
Title: "Configuration spaces in quantum
field theory and in algebraic topology"
Abstract: Configuration spaces enter
classically in many constructions in algebraic
topology, and in particular in the study of embeddings and
in the theory of operads. On the other
side, configuration spaces also arise in the "Feynman
diagram" expansion in quantum field theory. In this
talk we will
show how one can successfully combine these
two fields through configuration spaces,
and more specifically how one can apply ideas from quantum
field theory to embeddings (Vassiliev
invariants) and operads (deformation quantization).
4pm-
Title: "Counterexample to the homotopy
invariance of configuration spaces of
manifolds"
Abstract: In this talk we will discuss the
homotopy invariance of configuration
spaces. While homeomorphism invariance for these objects is
obvious, much more complicated is the
homotopy invariance. Following many partial results
obtained in the last two decades, it has been conjectured
that if M and N are homotopy equivalent
closed smooth manifolds, then their
configuration spaces are homotopy equivalent.
We will show that the conjecture is
false by proving that the configuration spaces of the lens
spaces L(7,1) and L(7,2) are not homotopy
equivalent, though L(7,1) is homotopy
equivalent to L(7,2). The main tool used here will be Massey
products.
Special Event: A DAY OF
LECTURES: Combinatorial Approximations of geometry and
analysis
Feb. 10th:
1:00-1:30pm:
Speaker: Jozef Dodziuk -
"Combinatorial approximation to Hodge theory via
Whitney forms"
1:45-2:15pm:
Speaker: Scott Wilson -
"Structures on cochains converging to structures on
forms"
2:30-3pm:
Speaker: Jenny Harrison -
"Discrete exterior calculus with convergence to the
smooth continuum"
-TEA-
3:30-4pm:
Speaker: Robert Kotiuga -
"Whitney Forms: The Enduring Legacy and New
Applications"
4:30-5pm:
Speaker: Dennis Sullivan -
"Combinatorial perturbation of geometry and
analysis"
Oct.14, 2003
Speaker: John Terilla
Title: Deformation Theory and QFT
November 25, 2003
Speaker: Marie Farge
Title: Combinatorial Topology, Mathematical Quantization
& Computational fluid dynamics
March 18th, 2003:
Speaker: Feng
Luo (Rutgers University)
Title:"Simple
loops on surfaces"
April 1st, 2003:
Speaker:
Michael Benedicks (KTH Royal Inst. of Technology, Stockholm)
Title: Parameter selections and ergodic properties for chaotic
Henon maps April 29th:
Speaker: Michael Movshev (Univ. of California at Davis)
Title: D=10,
N=1 Yang Mills Theory and a mathematics related to its
deformations
Dec. 17, 2002
Speaker: Bob Gompf (University of Texas, Austin)
Title: to be announced.
December 3, 2002
SpeakerFeng Luo- Rutgers University
Title: to be announced.
Nov. 19, 2002
To be announced
Nov. 5, 2002
No talk scheduled
Oct. 22nd, 2002:
Speaker: Jae Suk Park- Kaist (Korean Advanced Institute of
Science and Technology)
Title: The mathematics of
formal quantum field theories"
Oct. 8th, 2002:
Speaker: Christiaan Hofman, Rutgers University
Title: "Quantization of Courant Algebroids From
Topological Open Membranes"
September
24, 2002: Ib Madsen (University of Aarhus)
"The stable moduli space
of Riemann surfaces: Mumford's conjecture"
September 20-21, 2002- Special
Event: A Celebration of
the 20th anniversary of Dennis Sullivan's appointment as the Albert Einstein
Chair in Science at The City University of New York at the CUNY Graduate Center.
May 7, 2002- Rahul Pandharipande-
The Gromov-Witten/Hurwitz correspondence-Princeton University
April 23, 2002- Uwe Kaiser-"Skein modules and the topology of 3-manifolds"-Boise
State University
April 9, 2002- Kenji Fukaya- "Floer homology & Mirror symmetry"-Kyoto/Institute
of Advanced Study
March 26, 2002-Claude Viterbo-"Symplectic topology and the cohomology of the
loop space"-Polytechnique University
March 12, 2002- Boris Dubrovin-"On
classification of integrable PDEs"- Institute of Advanced Study
February 26, 2002- Stavros Garoufalidis- "Rationality and the Kontsevich integral"-
University of Warwick
February 12, 2002- Michael Polyak- "Cubic spaces and finite type invariants"-
Tel-Aviv University
January 29, 2002- Paul Seidel- "Fukaya categories and deformations"- Institute
of Advanced Study
December 18, 2002 Robbert Dijkgraaf- "String Categories, M-theory"-University
of Amsterdam
December 4, 2001 Eleny Ionel-"Gromov-Witten Invariants of symplectic
glueings"-University
of Wisconsin
November 20, 2001 Yang Geun Oh "Lagranian intersection Floer homology and
filtered A-infinity algebra"-Institute of Advanced Study
October 23, 2001; Mikhail Khovanov; U. of Cal. at Davis;
Homological Algebra of Knot Invariants
October 9, 2001; Edward Witten;
Institute for Advanced Study;
String Theory And Manifolds Of G_2 Holonomy
April 10, 2001; Kenji
Fukaya, Kyoto
University, TBA;
March 27, 2001; Paul Melvin,
Bryn Mawr College, TBA;
December 5,
2000; Dror Bar-Natan, Hebrew
University, TBA;
November 21, 2000; Zhenghan
Wang, Indiana University, TBA;
November
7, 2000; Alberto
Cattaneo, Deformation
Quantization and Poisson Sigma Model;
October
24, 2000; Alexander
A. Voronov; Michigan
State University, String Topology Revisited;
October 10,
2000; Michael
Polyak, Tel
Aviv University, TBA;
September 26,
2000; Jorgen E. Andersen; UC Berkeley; The
Poisson algebra of chord diagrams and quantization
September 13, 2000; Pavel Etingof; Columbia University
May 30, 2000; Abhay Ashtekar;
Penn State
May 23, 2000; Lee Smolin;
Penn State University; M Theory as a Matrix Extension of Chern_Simons Theory
May 16, 2000; Claude Bardos;
Brown University; Reversible and Irreversible Limit of Hamiltonian Systems
May 9, 2000; Mitchell Feigenbaum;
Rockefeller University; Mathematical Physics
May 2, 2000; S. Novikov;
University of Maryland; Topological Phenomena in the Conductivity of Metals
Apr. 25, 2000; Graeme Segal;
Oxford University; Topological Structures in String Theory
Apr. 18, 2000; Michael Mandell;
University of Chicago; Algebraic Structure on Cochains Determining Homotopy
Type
Apr. 11, 2000; Andrew Casson;
Yale University; Linearity of Braid Group
April 10, 2000; Kenji Fukaya, TBA;
Apr. 4, 2000; Alexander Givental;
Berkeley; Gromov-Witten Invariants
March 28, 2000; John Baez;
UC Riverside; Simplicial Quantum Geometry
March 27, 2000; Paul Melvin,
Nov. 20, 2001 Yang Geun Oh "Lagranian intersection Floer homology and filtered
A-infinity algebra"-Institute of Advanced Study Dec. 4, 2001 Eleny Ionel-"Gromov-Witten
Invariants of symplectic glueings"-University of Wisconsin Dec. 18, 2002 Robbert
Dijkgraaf- "String Categories, M-theory"-University of Amsterdam ;
March 21, 2000; Ezra Getzler;
Northwestern University; Frobenius Manifolds in Higher Genus
March 7, 2000; Edward Miller;
Polytechnic University; The SU(3) Casson Invariant
Feb. 29, 2000; Ciprian S. Borcea; Rider University; Special Lagrangians in Calabi-Yau Manifolds
Feb. 15, 2000; Boris Tsygan;
Penn State University; Non Commutative Geometry
Feb. 8, 2000; Carlo Roveli;
University of Pittsburgh; Quantum Gravity
Feb. 1, 2000; Steve Ferry,
Rutgers; Survey of Controlled Topology
Dec. 21,1999; Michael Movshev;
SUNY at Stony Brook; Topological Open String Theory
Dec. 14,1999; Scott Axelrod;
IBM; Link Invariants
Dec. 7, 1999; Y. Eliashberg;
Stanford University; Symplectic Field Theory
Nov. 30, 1999; Peter Kronheimer;
Harvard; The Seiberd_Witten Equations on 3-Manifolds
Nov. 16, 1999; Alexander Migdal; Metacreations; Large N QCD
Nov. 10, 1999; Victor Kac;
MIT; Fermion Correspondence
Nov. 9, 1999; James Stasheff;
U Penn; Cohomological Physics
Nov. 2, 1999; D. Tamamrkin;
Harvard; On d-Algebras
Oct. 26,1999; M. Douglas;
Rutgers; Branes on Calabi-Yau
Oct. 19,1999; Tom Mrowka;
MIT; Seiberg-Witten and Instanton Homology
Oct. 12,1999; Igor Frenkel;
Yale University; Affine Lie Algebras via Platonic Solids
Oct. 5,1999; Z. Wang; Indiana
University; Geometry of Knots and Plane Curves
Sept. 28, 1999; Boris Tsygan; PSU; Non Commutative Calculus
June 15,1999; Jean-Luc A. Brylinski; PSU; Differential Geometry of Gerbs
June 8, 1999; Ioan James;
Oxford University; Combinatorial Topology versus General Topology
June 1, 1999; M. Khovanov; IAS; The Jones Polynomial
May 25, 1999; G. Mikhalkin;
Harvard University; Real Algebraic Curves and Amoebas
May 18, 1999; Ezra Getzler;
Northwestern University; B-infinity Algebras
May 11, 1999; D. Tamarkin;
Penn State University; Kontsevich’s Formality Theorem
May 4, 1999; N. Reshetikhin,
Berkeley; Survey of Discrete Geometry
Apr. 27, 1999; Slava Krushkal; IAS; 4-Manifolds
Apr. 20, 1999; Etienne Ghys;
Action of Lattices on Circle
Apr. 13, 1999; Mikhail Gromov; IHES; Spaces of Holomorphic Maps
Apr. 6, 1999; Yasha Eliashberg;
Stanford University; Symplectic Field Theory
Mar. 30, 1999; Alexander Voronov; Michigan State University; Homotopy Gerstenhaber Algebras
Mar. 23, 1999; Jack Morava;
Johns Hopkins University; Topological Field Theories
Mar. 16, 1999; Murray Gerstenhaber;
University of Pennsylvania
Mar. 9, 1999; Maxim Kontsevich; IHES; Small Disc Operads and Deformations
Mar. 2, 1999; Gregg Zuckerman;
Yale University; Cohomology and Quantum Field Theory
Mar. 1, 1999; Jochen Bruening;
Institue fur Mathematik; Dirac Operator
Feb. 23, 1999; Boris Khesin;
University of Toronto; Polar homology
Feb. 16, 1999; Ralph Cohen;
Stanford University; Gluing Holomorhic Spheres
Feb. 9, 1999; Siddhartha Sahi; Rutgers University; Deformation Quantization
Feb. 2, 1999; Yakov Pesin;
Penn State University; Hyperbolic Measures
Jan.19, 1999; Shaun Martin; IAS; Symplectic Quotient
Jan 12, 1999; Bryan Clair; CUNY; Torsion
Dec. 8, 1998; Gregg Zuckerman;
Yale University; Cohomology and Quantum Field Theory
Dec. 1, 1998; Stavros Garoufalidis;
Harvard University; Invariants of 3_Manifolds
Nov. 24, 1998; Thomas Schick;
Penn State University; L^2 Betti Numbers and L^2 Reidemeister Torsion
Nov. 17, 1998; Gregg Zuckerman;
Yale University; Deformation Theory
Nov. 10, 1998; Feng Luo;
Rutgers University; Simple Loops on a Surface
Nov. 3, 1998; Adam Sikora;
University of Maryland; Skein Spaces and Their Quantizations
Oct. 28, 1998; Moira Chas;
SUNY at Stony Brook; Fixed-Point Theory and Dynamics
Oct. 27, 1998; S.Matveyev;
SUNY at Stony Brook; 4-Manifolds
Oct. 20, 1998; Jonathan Weitsman;
Lehman College; Moduli Spaces of Vector Bundles
Oct.13, 1998; Kenji Fukaya;
Kyoto University; Quantum Cohomology
Oct.6, 1998; Misha Movshev;
SUNY at Stony Brook; Mirror Symmetry
Sept. 29, 1998; Doug Bullock;
Boise State University; Combinatorial Gauge Theory
Sept. 22, 1998; J. Kania-Bartoszynska;
Boise State University; Combinatorial Gauge Theory
Sept. 15, 1998; Ezra Getzler;
Northwestern University; Gromov_Witten invariants
Sept. 8, 1998; Robin Forman;
Combinatorial Differential Topology
Last Modified on: 04/19/2007
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