String Theory, General Relativy and PDE's, May/June, 2009

"FRG Workshop: Manifolds, Strings and 2D Quantum Field Theory

Aug 24 - Aug 28, 2009, Organizers: Mike Hopkins, Jacob Lurie, Mike Hill

Harvard University, SC 507

This is a workshop for the NSF funded Focused Research Groups FRG. "Collaborative Research: How the Algebraic Topology of Closed Manifold Relates to Strings and 2D Quantum Field Theory". Organized by Mike Hopkins (mjh@math.harvard.edu), Jacob Lurie (lurie@math.harvard.edu) Mike Hill (mikehill@virginia.edu)

Monday, August 24th, 2009, 1:30PM - 5PM Science Center 507

1:30       PM   Welcome & Coffee Common Room on 4th Floor
2-3        PM   David Nadler, Northwestern   Representation theory via topological field theory
3-4        PM   Tea Common Room on 4th Floor
4-5        PM   Curt McMullen, Harvard, Perspectives on moduli space from topology, geometry and analysis

Tuesday, August 25th, 2009, 9AM - 5PM Science Center 507

9-9:30     AM   Breakfast
9:30-10:30 AM   Dennis Sullivan, Stony Brook   Stratification of String Space
10:30-11   AM   Coffee 5th Floor
11-12      AM  Kate Poirier, CUNY String diagrams and a compactification of moduli spaces of Riemann surfaces
12  -1:30  PM   Lunch
1:30-2     PM   Coffee 5th Floor
2   -3     PM   David Nadler, Northwestern Representation theory via topological field theory (II)
3   -4     PM   Tea 5th Floor
4   -5     PM   Nathaniel Rounds, Stony Brook     Local Poincare Duality
     5     PM   Reception & Dinner in the Common Room on the 4th floor. BBQ from Redbones.  Beer and Wine.

Wednesday, August 26th, 2009, 9AM-5PM Science Center 507

9    -9:30 AM   Breakfast
9:30 -10:30AM   David Nadler, Northwestern Representation theory via topological field theory (III)
10:30-11   AM   Coffee Common Room on 4th Floor
11   -12   AM   Joe Harris, Harvard TBA 
12   -1:30 PM   Lunch
1:30 -2    PM   Coffee Common Room on 4th Floor
2    -3    PM   Group Discussion
3    -4    PM   Tea Common Room on 4th Floor
4    -5    PM   Group Discussion

Thursday, August 27th, 2009, 9AM-5PM, Science Center 530

9    -9:30  AM   Breakfast
9:30 -10:30 AM   David Ayala, Harvard University
10:30-11    AM   Coffee Common Room on 4th Floor
11   -12    AM   Chris Schommer-Pries A Generators and Relations Approach to Local Topological Field Theories
12   -1:30  PM   Lunch
1:30 -2     PM   Coffee Common Room on 4th Floor
2    -3     PM   Orlando Alvarez TBA
3    -4     PM   Tea Common Room on 4th Floor
4    -5     PM   Kevin Costello, Northwestern  A mathematical approach to Bosonic string theory

Friday, August 28th, 2009, 9AM-Noon, Science Center 507

9    -9:30  AM   Breakfast
9:30 -10:30 AM   Mark Behrens, MIT TBA
10:30-11    AM   Coffee Common Room on 4th Floor
11   -12    AM   Orlando Alvarez TBA

Abstracts

David Nadler: Representation theory via topological field theory
I'll describe how the representation theory of Lie groups is organized by the formalism of topological field theory. In particular, I'll discuss character sheaves and Langlands duality in 3d TFT, and relations with the 4d Geometric Langlands TFT. This is joint work with David Ben-Zvi (Texas).
Kate Poirier String diagrams and a compactification of moduli spaces of Riemann surfaces
It is known that the original space of diagrams governing string topology operations and the moduli space of Riemann surfaces are not homotopy equivalent. We define a completion of the space of diagrams (over which string topology operations extend) and a compactification of the moduli space that is homotopy equivalent to the completed space of diagrams.
Nathaniel Rounds Local Poincare Duality
Closed oriented manifolds satisfy Poincare Duality, but not every space satisfying Poincare Duality has the homotopy type of a closed manifold. We represent a Poincare Duality space as a chain complex with a fixed basis satisfying certain axioms. We use this basis to define an algebraic notion of locality, allowing us to characterize closed manifolds as based chain complexes with local Poincare Duality.
David Ayala Harvard Geometric and singular cobordism categories
We'll consider cobordism categories whose morphisms are, possibly singular, manifolds endowed with a geometric structure such as a metric, a complex structure, or a gauge theoretic connection. We will see in what sense the classifying space of such a geometric cobordism category approximates moduli spaces of such geometric structures; some relevant results will be discussed. A case study of particular interest to Gromov-Witten theory will be elaborated upon which will require a gentle account of manifolds with prescribed singularities.
Chris Schommer-Pries A Generators and Relations Approach to Local Topological Field Theories
There are multiple approaches to the classification of topological field theories. In this talk we will explain one approach based on explicit generators and relations presentations. Such presentations can be obtained by essentially elementary cerf-theoretic techniques. This approach has several advantages, both from its conceptual simplicity and its wide applicability. It can be adapted to include field theories with a large range of topological structures as well as to partially extended field theories. We hope to illustrate this with several examples.
Kevin Costello, Northwestern A mathematical approach to Bosonic string theory
I'll discuss how one can construct factorization algebras encoding the bosonic string theory, and certain related theories, using obstruction theory.

Science Center, Room 507