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.
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"
Click here for
a list of all previous seminars.
Last Modified on: 1/26/2010
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