ATMW Algebraic Structures on Manifolds - (2016) Speakers and Syllabus

A) A brief description of the Workshop :- The ATM workshop aims to understand certain algebraic constructions on free loop space of manifolds. These constructions have algebraic counterpart in Hochschild and cyclic homology of associative algebras. The workshop is broadly divided into four sections:

  1.  Preliminaries: Manifolds, Poincar ́e duality, homotopy theory, classifying spaces, bar and cobar constructions.
  • Possible speakers:
  • Amiya Mukherjee (Professor, ISI, Kolkata),
  • Debasis Sen (Assitant Professor, IIT, Kanpur).
  • Hochschild Homology: Hochschild (co) homology of associative algebras, Gerstenhaber and BV algebras, deformation of algebras, Poisson manifolds.
    • Possible speakers:
    • Alice Fialowski (Professsor, University of P ́ecs, Institute of Mathematics and Informatics,/ E ̈otv ̈os lor’and university, Budapest, Hungary),
    • Anita Naolekar (Assitanat Professor, ISI, Bangalore),
    • Ashis Mandal (Assitant Professor, IIT, Kanpur),
    • Mahuya Datta (Professor, ISI, Kolkata).
  • Cyclic Homology: Cyclic homology and S1-equivariant homology.
    • Possible speakers:
    • Debasis Sen (Assitanat Professor, IIT, Kanpur),
    • Goutam Mukherjee (Professor, ISI, Kolkata).
  • Loop Spaces: Loop product, Goldman bracket, homological conformal field theory.
    • Possible speakers:
    • Samik Basu (Assitant Professor, RKMVU, Belur),
    • Somnath Basu (Assitanat Professor, IISER, Kolkata).

     The speakers themselves will conduct the tutorial classes.

    B) Target Audience:
    PhD students working in Geometry and Topology.

    c) Prerequisites:
    Acquainted with the topology syllabus of AFS-II and AFS-III. More specifically, some basic knowledge of smooth manifolds, and homology and cohomology will be assumed.

    1. Preliminaries: We will discuss manifolds, Poincar ́e duality, cap product. We will also review basic homotopy theory, including fibrations and simplicial objects. We will review Milnor’s construction of classifying spaces and then discuss bar and cobar constructions of algebras and coalgebras respectively.
      References: Algebraic Topology by Hatcher, Construction of universal bundles I, II by Milnor, The geometric realization of a semi-simplicial complex by Milnor, Algebraic Operads by Loday, Valette.
    2.  Hochschild homology: We will discuss Hochschild (co)homology of associative algebras. We will study deformations of algebras leading to Gerstenhaber algebras. This further leads to Batalin-Vilkovisky algebra. The algebra of smooth differential forms on a Poisson manifold provides an example of Gerstenhaber algebra.
      References: On the cohomology groups of an associative algebra by Hochschild, On the deformation of rings and algebras I, II by Gerstenhaber, From Poisson algebras to Gerstenhaber algebras by Kosmann-Schwarzbach.
    3.  Cyclic homology: We will discuss cyclic sets and circle action leading to cyclic homology of algebras as well as S1 -equivariant homology of spaces on which the circle acts. We will discuss the connection of cyclic homology of cochains on a manifold with the S1-equivariant homology of the free loop space of the manifolds.
      References: Cyclic homology by Loday, Free loop space and homology by Loday, Cyclic homology and equivariant homology by Jones, Cyclic homology, derivations and the free loop space by Goodwillie.
    4. Loop spaces: We will describe loop product on the homology of free loop space of a manifold. We will discuss how this homology becomes a BV algebra. The case of surfaces and Goldman bracket will be discussed. We will finish by describing the connection of these with topological conformal field theory.
      References: String Topology by Chas, Sullivan, A homotopy theoretic realization of string topology by Cohen, Jones, String topology and cyclic homology by Cohen, Hess, Voronov, Open and closed string field theory interpreted in classical algebraic topology by Sullivan.

     D) Time-Table

    There will be 9 days of lectures excluding Sunday, 18th December, 2016.

    For every day of the workshop, there will be 4 hours of lectures together with 1.5 hours of discussion and/or lectures by participants. The suggested schedule on a typical day is the following:

    Lecture 1 : 10am-12:15pm with a 15 minute break;

    Discussion 1 : 12:30-1:00pm;

    Lecture 2 : 2:30-4:45pm with a 15 minute break;

    Discussion 2/participants lecture : 5 - 6pm.

     Date 10 - 12:15 12:30 - 1 1 - 2:30 2:30 - 4:45 5 - 6
    13/12/16 Lecture A.1 Discussion L Lecture A.2 Discussion
    14/12/16 Lecture A.3 Discussion U Lecture A.4 Discussion
    15/12/16 Lecture B.1 Discussion N Lecture B.2 Discussion
    16/12/16 Lecture B.3 Discussion C Lecture B.4 Discussion
    17/12/16 Lecture C.1 Discussion H Lecture C.2 Discussion
    18/12/16 B R E A K
    19/12/16 Lecture C.3 Discussion L Lecture C.4 Discussion
    20/12/16 Lecture C.5 Discussion U Lecture D.1 Discussion
    21/12/16 Lecture D.2 Discussion N Lecture D.3 Discussion
    22/12/16 Lecture D.4 Discussion CH Lecture D.5 Discussion

     

    The lecture titled Lecture X.n means nth lecture on topic X, where X is one of the topics listed in the syllabus.