## Speaker:

## Institution:

## Time:

## Location:

A useful tool to study a 3-manifold is the space of the

representations of its fundamental group, a.k.a. the 3-manifold group, into

a Lie group. Any 3-manifold can be decomposed as the union of two

handlebodies. Thus, representations of the 3-manifold group into a Lie group

can be obtained by intersecting representation varieties of the two

handlebodies. Casson utilized this observation to define his celebrated

invariant. Later Taubes introduced an alternative approach to define Casson

invariant using more geometric objects. By building on Taubes' work, Floer

refined Casson invariant into a graded vector space whose Euler

characteristic is twice the Casson invariant. The Atiyah-Floer conjecture

states that Casson's original approach can be also used to define a graded

vector space and the resulting invariant of 3-manifolds is isomorphic to

Floer's theory. In this talk, after giving some background, I will give an

exposition of what is known about the Atiyah-Floer conjecture and discuss

some recent progress, which is based on a joint work with Kenji Fukaya and

Maksim Lipyanskyi. I will only assume a basic background in algebraic

topology and geometry.