The moduli space of fundamental group representations into a reductive Lie group modulo conjugation is a central object of interest in geometric topology. Equipping the underlying topological surface with a complex structure permits the introduction of holomorphic and gauge-theoretic methods for its study via nonabelian Hodge theory. Certain gluing techniques can be established using tools from these alternative perspectives which allow us to identify open sets of particular significance in the moduli space. In this lecture series we will survey some of the main aspects of the nonabelian Hodge correspondence and describe gluing techniques in a number of cases.

Venue: Zoom Link (Passcode- ppF7C8)
Time: 3rd April: 12:15 pm Indian Time (9:45 am Greece time); 5th April: 2 pm Indian time (11:30 am Greece time)

In the first talk, I will discuss geometrical aspects of Lie groups and homogeneous manifolds $M=G/H$, where $G$ is a Lie group and $H$ a closed subgroup of $G$ (e.g. adjoint and isotropy representation, curvature). In the sequel, I will give an update of homogeneous Einstein metrics and present some recent results about invariant Einstein metrics on compact simple Lie groups, which are not naturally reductive. Audiences with some background in Riemannian geometry and Lie groups will comfortably follow most parts of these talks.

Venue: Zoom Link (Passcode- 5DE3iC)
Time: 3 pm Indian time (11:30 am Greece time)
Lecture Notes 1
Lecture Notes 2

Many different structures on manifolds can be expressed in the language of principal bundles via the notion of reduction of structure group. It is natural to ask then when we can perform a reduction of structure group in order to equip a manifold with a given structure. This can be addressed by obstruction theory. However, it is often quite difficult to identify what the obstructions are. Using the language of classifying spaces, there is some hope of doing exactly that. I will end by providing examples where these techniques have been successful in my own research.

Venue: Zoom Link (Passcode- Ut6d4q)
Time: 7:30 pm Indian time (9 am Canadian time)
Lecture Notes 1
Lecture Notes 2
Lecture Notes 3
Lecture Notes 4

String topology studies algebraic structures that arise by intersecting loops, where a “loop” can mean something topological or algebraic. For example (on the topological side) on the homology of the free loop space of a closed, oriented manifold, there is a binary operation called the “loop product” and a unary operation called the “BV operator.” These two operations together give the homology of the free loop space the structure of a “BV algebra.” Separately (on the algebraic side) in the presence of an algebraic version of Poincaré duality, there is product and BV operator on the Hochschild cohomology of this algebra. These operations give the Hochschild cohomology of the algebra the structure of a BV algebra as well. Further, when the algebra is the cochain algebra of a closed, oriented, simply connected manifold there is an isomorphism between its Hochschild cohomology and the homology of the free loop space of the manifold. While Cohen and Jones showed that this isomorphism respects the product structure, subsequent work of Menichi suggested that, in the case of the 2-sphere with mod 2 coefficients, it did not respect the BV operator. In this talk, I will describe these operations and show that with an appropriate updated algebraic version of Poincaré duality for algebras–one involving higher homotopies–Hochschild cohomology can be given a BV operator that is, in fact, preserved by the isomorphism from the homology of the free loop space of the 2-sphere with mod 2 coefficients. This is joint work with Thomas Tradler.

Venue: Zoom Link (Passcode- aLajm8)
Time: 6:30 pm Indian time (8 am NYC time)
Lecture Notes 1
Lecture Notes 2
Lecture Notes 3