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 prnciple, 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: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: 04/25/2007
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