** **

**Abstract:**

In the late 90's, S. K. Donaldson and R. P.
Thomas

suggested a program on gauge theory in
higher dimensions.

In alignment with this program, Thomas
introduced

the holomorphic Casson invariant for a
smooth projective Calabi-Yau

threefold.

The holomorphic Casson invariant is a
deformation invariant of a smooth

projective Calabi-Yau threefold.

Thomas defined it by constructing
algebro-geometrically a virtual moduli

cycle

of the moduli space of semi-stable sheaves
under some technical assumptions.

This can be viewed as a complex analogy of
the Taubes-Casson invariant

defined by C. H. Taubes.

However, it is not achieved as yet to
construct the invariant

in a "gauge theoretic", or an
analytic way.

In this seminar, we exhibit an analytic
approach to this invariant

by using perturbed Hermitian-Einstein
equations over a compact Kahler

threefold,

which we call Donaldson-Thomas equations.

We begin this seminar with some backgrounds
on the holomorphic Casson

invariant

and the Hermitian-Einstein equations.

Subsequently, we introduce the
Donaldson-Thomas equations

and describe local structures of the moduli
space of solutions

to the Donaldson-Thomas equations.

Furthermore, we discuss analytic results on
solutions to the equations,

aiming at constructing the holomorphic
Casson invariant

for compact Kahler threefolds.