12/13/06: String field theory. [Eliashberg's plenary talk at ICM-2006]
12/13/06: String field theory. [Eliashberg's plenary talk at ICM-2006]
12/11/06: Brane topology: a higher-dimensional generalization of
string topology. [ST: Sections 5.1-3]
12/08/06: Floer homology of the cotangent bundle and string
topology. [ST: Section 4.2]
12/06/06: Morse theory on loop spaces: Floer homology. [M. Schwarz's
book on Morse theory; ST: Section 4.2]
12/04/06: Shortened class because of the potluck: Morse theory on loop
spaces. [M. Schwarz's book on Morse theory; ST: Section 4.1]
12/01/06: Morse-Bott theory. [M. Schwarz's book on Morse theory]
11/29/06: Review of Morse theory [J. Milnor's, M. Schwarz's, or
Yu. Matsumoto's book on Morse theory or a section from
Dubrovin-Novikov-Fomenko's "Modern Geometry"]
11/27/06: Open-closed ribbon graphs and string topology. [ST: Section
3.2; Eric Harrelson's paper on the ArXiv]
11/22/06: Have a happy Thanksgiving!
11/20/06: Open-closed oriented surfaces, TFTs, and string
topology. [ST: Section 3.3]
11/17/06: Higher genus string topology. [ST: Section 3.2]
11/15/06: Ribbon graphs and admissible graphs and relation to moduli
space. Strebel differentials. [ST: Section 3.2]
11/13/06: Ribbon graphs and Sullivan chord diagrams. [ST: Section
3.2]
11/10/06: Motivic TCFTs. Problem: define a reasonable motivic category
of topological spaces or manifolds. Ribbon graphs. [ST: (the rest of)
Section 3.1]
11/08/06: Examples: Gromov-Witten theory, String Topology, Eric's
theorem. [ST: (more of) Section 3.1]
11/06/06: CFTs, VOAs, CohFTs, quantum gravity, and TCFTs. Review of
moduli spaces of Riemann surfaces (algebraic curves). [ST: part of
Section 3.1]
11/03/06: Dijkgraaf-Witten's toy model. CFTs. [ST: part of Section
3.1]
11/01/06: TFTs and Frobenius algebras: the exercise in linear algebra:
the existence of \psi: k --> V \tensor V and (,): V \tensor V --> k
and the plumbing property that the elbow equals the cylinder imply dim
V < infty and the symmetric bilinear form (,) is nonsingular. The
"hard" part of the proof: given a Frobenius algebra, define a
TFT. [ST: Section 3.1.1]
10/30/06: TFTs and Frobenius algebras: the easy part of the
correspondence. [ST: part of Section 3.1.1]
10/27/06: The cactus operad and string topology: compatibility of the
BV structure derived from the cacti with string topology; the
Salvatore-Wahl recognition principle. The notion of an n-dimensional
TFT [ST: Sections 2.2, 2.3.2, and the first page of 3.1.1]
10/25/06: The cactus operad and string topology: an action of the
cacti on the loop spaces, continued. [ST: Sections 2.2 and 2.3.1]
10/23/06: The cactus operad and string topology: an action of the
cacti on the loop spaces. [ST: Section 2.3]
10/20/06: The cactus operad. [ST: Section 2.2, leaving the proof of
Theorem 2.2.1 for later time]
10/18/06: Proof of F. Cohen-Getzler's theorem (Part III: checking the
induction claim on the H_* (fD) side). [ST: pp. 34-35 (part of Section
2.1.4)]
10/16/06: Free operads, operad ideals, operads defined by generators
and relations. The little disks operad and BV algebras: proof of
F. Cohen-Getzler's theorem (Part II: setting up the induction to show
that the operad morphism BV --> H_* (fD) which comes from Part I is an
isomorphism; checking the induction claim on the BV operad side). [ST:
pp. 36-37 (Section 2.1.5) and pp. 34-35 (part of Section 2.1.4)]
10/13/06: The little disks operad and BV algebras: proof of
F. Cohen-Getzler's theorem (Part I: constructing a BV algebra
structure on an algebra over the homology framed little disks
operad). [ST: pp. 34-35 (part of Section 2.1.4)]
10/11/06: Lemma on identifying the operad using the notion of a free
algebra. [ST: pp. 34-35 (part of Section 2.1.4)]
10/09/06: The little disks operad (framed, nonframed, action on the
double loup space, the double loop space recognition principle) and
relation to BV algebras. [ST: pp. 32-35 (part of Section 2.1.4)]
10/06/06: The associative and the Lie operads. Free algebras over an
operad and recovering the operad from them. [ST: pp. 29-30 (part of
Section 2.1.4)]
10/04/06: Why operads (and PROPs)? Why the commutative operad
describes graded commutative algebras. The associative operad. [ST:
pp. 28-30 (part of Section 2.1.4)]
10/02/06: Scott Wilson took over over that class meeting. According to
his taste, he discussed some more details about what was known and
what was open in relation between the BV structures on the Hochschild
cohomology and/or the homology of the loop space in a manifold. He
also gave a nice survey of PROPs and operads. [ST: pp. 21-23 (Section
1.5) and pp. 25-28 (part of Section 2.1)]
09/29/06: The Hochschild chain model of the free loop space. The BV
structures on the homology of the loop space and the Hochschild
cohomology (open problems). [ST: pp. 19-23 (Section 1.5)]
09/27/06: Simplicial and cosimplicial sets. Geometric realization. The
simplicial models of the point and the circle. [P. Selick,
Introduction to Homotopy Theory: Section 8.1; J. P. May, A Concise
Course in Algebraic Topology: Sections 16.1-2; ST: p. 19 (Section
1.5)]
09/25/06: Equivariant String Topology: the string bracket. [ST:
pp. 13-14 (Section 1.3)]
09/22/06: Equivariant homology: definition and properties. The
"unparameterized" loop space - what's that? [ST: pp. 13 (Section 1.3)]
09/20/06: The BV structure on the shifted homology of a loop space LM:
the BV operator and the BV bracket. Equivariant homology: the rough
idea. [ST: pp. 11-13 (part of Section 1.3)]
09/18/06: More ingredients of String Topology. BV algebras. [ST:
pp. 10-11 (part of Section 1.3) and 33-34 (part of Section 2.1.4 after
Theorem 2.1.1)]
09/15/06: The proof of the fact that M^8 --> LMxLM is an embedding of
manifolds of codimension d. Chas-Sullivan's loop product: the
algebraic part. Theorem: (1) the loop product makes the shifted
homology of LM a graded commutative algebra; (2) the evaluation map
induces on homology an algebra homomorphism from that algebra to the
intersection algebra of the target space M. [ST: pp. 9-10 (Section
1.2)]
09/13/06: Recollection of the Pontryagin-Thom collapse. Intersection
theory on loop spaces: Chas-Sullivan's loop product: the geometric
part. [ST: pp. 8-9 (Section 1.2)]
09/11/06: Proof of the Frobenius algebra statement. The
Pontryagin-Thom construction: Thom collapse. [ST: pp. 6-7 (Section
1.1)]
09/08/06: Intersection theory on manifolds. Frobenius
algebras. Intersection theory provides a (graded) Frobenius algebra
structure on homology. [ST: pp. 6-7 (Section 1.1)]
09/06/06: Overview of the course. Who is E. Witten and what he
predicted for the development of mathematics in the 21st century. What
is String Topology? How it relates to Stringy Topology and the other
two topics courses offered this semester by Ionut Ciocan-Fontanine and
Tian-Jun Li. Motivation of the study of loop spaces from the point of
view of classical algebraic topology and string theory. [ST (this will
refer to my paper Notes on String
Topology with R. Cohen, which is part of the book String
Topology and Cyclic Homology): pp. 3-4 (Introduction)]