String Topology and Geometry

           SUNY-Stony Brook

          August 11 - 15,  2003

 

 

Tentative Program

 

 

Survey and General Talks

 

1.  Survey of questions in topology and algebraic geometry motivated by string theory.

 

2.  Survey  of  the use of graphs in the study of  moduli spaces

 

3.  The generalized Mumford conjecture and its proof

 

4.  State of the art and open conjectures about the homology of moduli space of curves

 

5.  Automorphisms of free groups and mapping class groups - computational techniques

 

6.  Conformal Field theory, spin bundles on loop spaces,  and elliptic objects

 

7.  Gromov-Witten theory of open and closed strings

 

8.  Gerbes and Duality

 

9.  Compactifications of moduli space and potential use in string topology

 

 

 

 

 

Focused Talks

 

1.  String topology operations on the homology of the free loop space.

Discussion of basic operations, BV structure, cacti, homotopy realization

 

References 

1. Chas and Sullivan,   String Topology,   preprint: math.GT/9911159,  

2. R.L. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology , Math. Annalen, published online: DOI

10.1007/s00208-002-0362-0 (2002).    Preprint: math.GT0107187.

3.  A. Voronov, Notes on universal algebra, to appear in proceedings of Stony Brook conf. on ``Graphs and Patterns in

Mathematics and Physics", June 2001.  Preprint: math.QA/0111009

 

2.  Closed string (equivariant) operations

Discussion of Lie bialgebras, higher string algebras

 

References

1. Chas and Sullivan,   String Topology,   preprint: math.GT/9911159, 

2. M.Chas and D. Sullivan, Closed string operators in topology leading to Lie bialgebras and higher string algebra, preprint:

Math.GT/0212358, (2002).

 

3.  Open string topology

Category of open string states,  operations on path spaces

 

References

1.  math.QA/0302332 Open and Closed String field theory interpreted in classical Algebraic Topology. Dennis Sullivan . 19 pages. QA (GT ).

2.  Segal,  Topological structures in string theory,  R. Soc. Lond. Philos. Trans. Ser A Math. Phys. Eng. Sci. 359 (2001), 1389-1398,

 

 

 

4.  Homotopy theoretic aspects of string topology

Ring spectrum models for the free loop space and other mapping spaces, string topology operations in generalized homology,

applications to other mapping spaces, 

 

References.

1 R.L. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology , Math. Annalen, published online: DOI

10.1007/s00208-002-0362-0 (2002).    Preprint: math.GT0107187.

2. D. Chataur,  A bordism aproach to string topology,  math. AT/0306080

3. Recent work of W. Dwyer, J. Klein, P. Hu,  S.Kallel  and P. Salvatore

 

 

5.  Use of operads in moduli spaces

Use of operads in the structure of moduli of curves, to understand infinite loop structure of stable moduli space,

automorphisms of free groups

 

References

1. U. Tillmann,   Higher genus surface operad detects infinite loop spaces, Math. Ann. 317 (2000), 613-628

2. N. Wahl, Infinite loop space structures on the the stable mapping class group,  math.AT/0204169

3.Ralph M. Kaufmann , Muriel Livernet , R. C. Penner,  Arc Operads and Arc Algebras.   math.GT/0209132

 

 

6. Calculations in the stable homology of moduli space

Calculations using operad and infinite loop space structures,  applications of Madsen-Weiss theorem

 

References

1. Ib Madsen , Michael S. Weiss . math.AT/0212321 The stable moduli space of Riemann surfaces: Mumford's conjecture.

2.  Madsen and Tillmann,  The stable mapping class group and Q(CP), Invent. Math. 145 (2001) 509-544

3. S. Galatius-Smith,  The mod p homology of \Omega^\infty CP^\infty _1.

 

7.  Stability theorems in mapping class groups and related groups

Harer - Ivanov stability theorems,  Hatcher stability for automorphisms of free groups,

 

References

1. J. Harer, Stability of the homology of the maping class groups of oriented surfaces, Ann. Math. 121 (1985) 215-249

2. N. Ivanov Stabilization ofthe homology of the Teichmuller modular groups, Algebra i Analiz 1 (1989) 120-12, translation in:

Leningrad math. Jour. 1 (1990), 675-691

 

  

8.  Spaces of graphs and their applications to string topology

Spaces of chord diagrams, categories of fat graphs, moduli spaces of bordered surfaces in a manifold.

 

References

1.  Cohen and Godin,  A polarized view of string topology,  math.AT/0303003

2.  A. Voronov, Notes on universal algebra, to appear in proceedings of Stony Brook conf. on ``Graphs and Patterns in

Mathematics and Physics", June 2001.  Preprint: math.QA/0111009

 

9. Hochshield homology and loop spaces

Hochshield homology of cochains, and of chains of the based loop space, and their relationship

to the homology of loop spaces, string topology operations, deformation quantization.

 

References

1 R.L. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology , Math. Annalen, published online: DOI

10.1007/s00208-002-0362-0 (2002).    Preprint: math.GT0107187.

2. T. Tradler,  The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products. math.QA/0210150

3. Yves Felix , Jean-Claude Thomas,   Spaces of self-equivalences and free loops spaces, math.AT/0204152

4. Yves Félix , Luc Menichi , Jean-Claude Thomas,  Duality in Gerstenhaber algebras. math.AT/0211229

 

 

10.  Differential topology of loop spaces

Tangent bundle of loop space, polarizations,  bordism,  spin structures

 

References.

1. R.L. Cohen and A. Stacey, Fourier decompositions of loop  bundles   preprint: math/0210351,  2002

2. R. Cohen and V. Godin,  A polarized view of string topology,  math.AT/0303003

3. David Chataur,  A bordism approach to string topology. math.AT/0306080

4. P. Teichner, and S. Stolz,  What is an elliptic object,  to appear in Segal proceedings, see Peter Teichner's home page

 

 

11.  Conformal field theories and elliptic objects

Spinors on  loop spaces, conformal connections, relation to elliptic cohomology

 

References

1. P. Teichner, and S. Stolz,  What is an elliptic object,  to appear in Segal proceedings,see Peter Teichner's home page

2. P. Hu and I. Kriz, conformal field theory and elliptic cohomology,  see Igor Kriz's home page

3. G. Segal, Elliptic cohomology, Seminaire Bourbaki 695 (1988) 187-201

4. G. Segal,  The definition of conformal field theory,  to appear in Segal proceedings.

 

 

12.  Gromov-Witten theory of open and closed strings

Definition of the invariants, computational techniques, D-branes,  Gopakumar-Vafa conjectures,

 

References

1. R.Gupakumar and C.Vafa, On the Gauge Theory/Geometry Correspondence, hep-th/9811131,

Adv. Theor. Math. Phys. 3 (1999) 1415

2.  M. Liu, Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and

Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair, Math-SG/0210257

3.  S. Hosono, M-H. Saito, A. Takahashi, Relative Lefschetz Action and BPS State Counting,

math.AG/0105148, Internat. Math. Res. Notices, (2001), No. 15, 783-816.