June  3, 2020

Thematic Program on The Geometry of String Theory
2004 - 2005

A joint program of the Fields Institute, Toronto &
the Perimeter Institute for Theoretical Physics, Waterloo

Graduate Courses

Winter Semester 2005
Course on Introduction to Homological Algebra, Instructor: R. Buchweitz

Course on Symplectic Geometry and Topology, Instructor: B. Khesin

Mini-Course on Generalized Geometries in String Theory (at Perimeter Institute) |
Accommodation support is available for participants to the Mini-Course at Perimeter to stay overnight in Waterloo on the Tuesday or Wednesday. Please indicate when registering if you require accommodation and for which dates.

February 15-17, 2005 -- Marco Gualtieri (Fields):
February 15 -17, 2005 -- Yi Li (Caltech):
March 1-3, 2005 -- 11 a.m. Mariana Grana (Ecole Polytechnique & Ecole Normale Superieure)

Mini-Course: "Toda lattices: basics and perspectives" (at Fields Institute)

Fall Semester 2004
Course on Mirror Symmetry, Instructor: K. Hori

Course on String Theory, Instructor: A. Peet

Mini-Course on Frobenius Manifolds and Integrable Hierarchies (November 8-12, 2004),
: Boris Dubrovin (SISSA, Trieste)

Minicourse: "Toda lattices: basics and perspectives"
Speakers: B.Khesin (Toronto), A.Marshakov (Lebedev and ITEP, Moscow), M.Gekhtman (Notre Dame, IN)

Thursday, March 31, 2005
Lecture 1 - 10:10-11:00
Speaker: B.Khesin (Toronto)

Lecture 2 - 11:10-12:00
Speaker: A.Marshakov (Lebedev and ITEP, Moscow)

We discuss several versions of the Toda lattice, one of the most popular integrable systems. The first lecture provides the necessary background and discusses the simplest Toda models along with the necessary tools. The second lecture describes the rather unexpected appearence of Toda lattices in the Dijkgraaf-Vafa theory of matrix integrals. The third lecture (to be given in May) describes integrability of the Toda lattices on arbitrary orbits and discusses the corresponding open problems.

Friday, May 27, 2005
Lecture 3 - 11:00-12:00
Speaker: M.Gekhtman (Notre Dame, IN)

I will describe basic constructions of the nonperiodic Toda lattice theory, including interplay between the commutative and non-commutative integrability and Hamiltonian and graduate behaviour. A particular emphasis will be put on Toda flows on generic co-adjoint orbits of the Borel subgroup and integrable lattices associated with so-called minimal Toda orbits. If time permits, a connection of the latter with biorthogonal Laurent polynomials will also be discussed. This lecture is the third part of the minicourse on Toda theory, but starts with all necessary preliminaries and is completely selfcontained.

Mini-Course on Generalized Geometries in String Theory (to register)
Accommodation support is available for participants to the Mini-Course at Perimeter to stay overnight in Waterloo on the Tuesday or Wednesday. Please indicate when registering if you require accommodation and for which dates.

**Note: There is additional shuttle service available during the mini-courses.

February 15-17, 2005 -- Marco Gualtieri (Fields):
'Generalized geometric structures'
Generalized complex geometry is a unification of complex and symplectic geometry, and provides a geometrical context for understanding parts of mirror symmetry. In these lectures I will provide an introduction to generalized complex, Kahler, and related geometries, and describe some of their appearances in physics.
Course start times
Feb. 15, 11am
Feb. 16 11am
Feb. 17 11am

February 15 -17, 2005 -- Yi Li (Caltech):
'Twisted Generalized Calabi-Yau Manifolds and Topological Sigma Models with Flux'
In these lectures, we examine how twisted generalized Calabi-Yau (GCY) manifolds arise in the construction of a general class of topological sigma models with non-trivial three-form flux. The topological sigma model defined on a twisted GCY can be regarded as a simultaneous generalization of the more familiar A-model and B-model. Emphasis will be given to the relation between topological observables of the sigma model and a Lie algebroid cohomology intrinsically associated with the twisted GCY. If time permits, we shall also discuss topological D-branes in this more general setting, and explain how the viewpoint from the Lie algebroid helps to elucidate certain subtleties even for the conventional A-branes and B-branes. The lectures will be physically motivated, although I will try to make the presentation self-contained for both mathematicians and physicists.
Course start times
Feb. 15, 2 pm
Feb. 16 3:30pm
Feb. 17 2pm

March 1-3, 2005 -- 11 a.m. Rm 405, Perimeter Institute
Mariana Grana
(Ecole Polytechnique & Ecole Normale Superieure)
'Supergravity Backgrounds from Generalized Calabi-Yau Manifolds'
We will see how generalized Calabi-Yau manifolds as defined by Hitchin emerge from supersymmetry equations in type II theories. In the first lecture, we will review the formalism of G-structures, which is central in the context of compactification with fluxes. In the second lecture we will see how (twisted) generalized Calabi-Yau manifolds emerge from supersymmetry equations using SU(3) structure. In the last lecture, we will discuss special features about compactifications on Generalized Calabi-Yau's.

Date/Time: Mondays, 9:00 - 12:00 a.m.; start date January 17, 2005 (to register)
Course on Introduction to Homological Algebra
MAT 1103HS
Instructor: R. Buchweitz

This course intends to give a concise introduction into the concepts of modern homological algebra and to highlight their applications. To this end, two hours of lectures will be followed by a one hour discussion of examples and applications.

Definitions, structure, and representation theorems for Abelian Categories. Resolutions by projectives, injectives and cohomology theories. Hochschild Cohomology, Gerstenhaber Algebras, Super Poisson and Super Lie algebras. Triangulated Categories, exact functors Fourier-Mukai transforms and other equivalences. Stable theories: Tate cohomolgy, periodicity. The homological mirror conjectures.

Prerequisite: Core course in Algebra required, interest in algebraic geometry helpful.

Bass, Hyman . Algebraic $K$-theory.(W. A. Benjamin, Inc., New York-Amsterdam 1968 xx+762 pp.)
Bourbaki, Nicolas . Algebre. Chapitre 10. Algebre homologique. (Masson, Paris, 1980. vii+216 pp. ISBN: 2-225-65516-2)
Cartan, Henri; Eilenberg, Samuel . Homological algebra.(With an appendix by David A. Buchsbaum.Reprint of the 1956 original.
Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. ISBN: 0-691-04991-2)
Gelfand, Sergei I.; Manin, Yuri I. Methods of homological algebra. 2nd ed. Springer Monographs in Mathematics. (Springer-Verlag, Berlin, 2003. xx+372 pp. ISBN: 3-540-43583-2)
Hilton, P. J. ; Stammbach, U. A course in homological algebra. 2nd ed. (Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp. ISBN: 0-387-94823-6)
Mac Lane, Saunders. Homology. Reprint of the 1975 edition.Classics in Mathematics. (Springer-Verlag, Berlin, 1995. x+422 pp. ISBN: 3-540-58662-8)
Mac Lane, Saunders . Categories for the working mathematician. 2nd ed. Graduate Texts in Mathematics, 5. (Springer-Verlag, New York, 1998. xii+314 pp. ISBN: 0-387-98403-8)
Rotman, Joseph J. An introduction to homological algebra.Pure and Applied Mathematics, 85. (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. xi+376 pp. ISBN: 0-12-599250-5)
Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. (Cambridge University
Press, Cambridge, 1994. xiv+450 pp. ISBN: 0-521-43500-5; 0-521-55987-1)

Date/Time: Wednesdays, 10:00 - 1:00 a.m.; start date January 19, 2005 (to register)
Topics in Symplectic Geometry and Topology: Symplectic Topology and Integrable Systems
MAT 1347HS
Instructor: B. Khesin

This is a course on the main notions and basic facts of symplectic topology. The topics to be covered include: symplectic and contact spaces, Morse theory, generating functions for symplectomorphisms, symplectic fixed point theorems (Arnold's conjectures), and invariants of legendrian knots. We also plan to touch on almost complex structures, groups of symplectomorphisms, definitions of symplectic capacities and Floer homology. In the second part of the course we are going to cover various constructions of completely integrable systems in finite and infinite dimensions. An acquaintance with basic symplectic geometry (e.g. covered by MAT 1344HF, Introduction to Symplectic Geometry) is advisable.

S.Tabachnikov: Introduction to symplectic topology (Lecture notes, PennState U.)
D.McDuff and D.Salamon: Introduction to symplectic topology (Oxford Math. Monographs, 1998)
A.Perelomov: Integrable systems of classical mechanics and Lie algebras (Birkhauser, 1990)

Date/Time: Tuesdays, 1:00 - 4:00
Course on Mirror Symmetry
MAT 1739F
Instructor: K. Hori

Mirror symmetry plays a central role in the study of geometry of string theory.
In mathematics, it reveals a surprising connection between symplectic geometry and algebraic geometry.
In physics, it provides a conceptual guide as well as powerful computational tools, especially in compactifications to four-dimensions.

1. Background:
Supersymmetry and homological algebra
Non-linear sigma models (NLSM)
Landau-Ginzburg models
topological field theory and topological strings
2. Linear sigma models, moduli space of theories
3. Mirror Symmetry
4. Mirror Symmetry involving D-branes
* B-branes in NLSM - holomorphic bundles, coherent sheaves
* B-branes in LG models - level sets, matrix factorizations
* A-branes in NLSM - Lagrangian submanifolds, Floer homology
* A-branes in LG models - vanishing cycles and Picard-Lefschetz monodromy

The course does not follow a textbook but the following may be useful.
1. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil and E. Zaslow, ``Mirror Symmetry'' Clay Mathematics Monographs Vol 1 (AMS, 2003).
2. P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, E. Witten, ``Quentum Fields and Strings: A Course for Mathematicians'' (AMS 1999).

Date/Time: Fridays, 10:00 - 12:00
Course on String Theory
PHY 2406F
Instructor: A. Peet

Please see the course website for up-to-date information regarding the course at:

November 8-12, 2004, 4:00 - 5:00 (no talk Wed., Nov. 10)
Mini-Course on Frobenius Manifolds and Integrable Hierarchies
Instructor : Boris Dubrovin (SISSA, Trieste)

Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.

Financial Assistance

As part of the Affiliation agreement with some Canadian Universities, graduate students are eligible to apply for financial assistance to attend graduate courses.
Two types of support are available:

  • Students outside the greater Toronto area may apply for travel support. Please submit a proposed budget outlining expected costs if public transit is involved, otherwise a mileage rate is used to reimburse travel costs. We recommend that groups coming from one university travel together, or arrange for car pooling (or car rental if applicable).

  • Students outside the commuting distance of Toronto may submit an application for a term fellowship. Support is offered up to $1000 per month.

    For more details on the thematic year, see Program Page or contact thematic(PUT_AT_SIGN_HERE)