Abstract: M. Abouzaid and Y. Imagi recently proved that any closed, immersed, unobstructed( in Floer theory) special Lagrangian (SL) which is very close to an embedded SL whose fundamental group has no nonabelian free subgroup is unbranched. In this talk we go thorough the proof of this theorem. The main idea is to use Thomas-Yau uniqueness theorem with inputs coming from Fukaya category of the cotangent bundle of the embedded SL. After a very brief introduction to Fukaya category in general, we talk about the necessary facts from the Fukaya category of the cotangent bundle that are required to apply Thomas-Yau uniqueness theorem.

Abstract: This is a continuation of the previous talk. The invariant of counting special Lagrangian rational homology 3-spheres in an almost Calabi-Yau 3-fold will be interesting if it is stable under deformations of almost Calabi-Yau structures or at least changes in a predictable way in these deformations. Joyce proposed a conjectural weighted counting invariant which might have these properties based on two particular bifurcation phenomenas, one is special Lagrangians with transverse intersections and another is special Lagrangian with conical singularity modeled on a cone over Clifford tori. We will first discuss results about conically singular special Lagrangians and their desingularizations. Later we will talk about the above two bifurcations in detail to justify Joyce’s proposal.

Abstract: We continue previous talk by following Simon Donaldson's paper.

Abstract: We follow Simon Donaldson's paper.

Abstract: We continue previous talk by following D. Baraglia's paper.

Abstract: We follow D. Baraglia's paper.

Abstract: We go through the proof of the Cheeger’s finiteness theorem which says there are only finitely many homeomorphism types of connected, closed manifolds of any fixed dimension admitting a Riemannian metric satisfying a uniform upper bound on diameter and sectional curvature, and uniform lower bound on volume.

Abstract: We go through the proof of the Narasimhan-Seshadri theorem due to Simon Donaldson which says that a holomorphic vector bundle over a Riemann surface admits a Hermitian Yang-Mills connection if and only if it is polystable.

Abstract: The holonomy group of a Riemannian manifold exhibits various geometric structures compatible with the metric. In 1955, M.Berger classified all possible Riemannian holonomy groups. Studying all these are more than one semester subject. So, in this talk after a brief introduction we overview very basics of these holonomy groups.

Abstract: We follow Donaldson's Floer homology book, chapter 4.

Abstract: We follow Donaldson's Floer homology book, chapter 3.

Abstract: Although Seiberg Witten invariant originally introduced for four manifolds, but its three dimensional version is also interesting. After a brief discussion on the definition of the Seiberg Witten invariant on three manifolds we will see some results from literature equating this invariant to some known invariants of three manifolds.

Abstract: We continue previous talk by following Taubes' paper. 

Abstract: We follow Taubes' paper. 

Abstract: After a brief introduction of Seiberg-Witten equations on closed smooth four manifolds, we will see how moduli space of solutions leads to an oriented compact manifold and a topological invariant (Seiberg-Witten Invariant) for the four manifold. Then for the purpose of computation of this invariant on Kähler manifolds, we will rewrite the equation in terms of complex geometry and see for most of the Kähler Surfaces the answer will be in terms of algebraic geometric criterion of the surface. Most of the technical details will be omitted but some brief sketches will be there. I will follow John Morgan's Book on Seiberg Witten equations.

Abstract: The Kempf-Ness theorem is a fundamental result at the intersection of complex algebraic Geometry and Symplectic Geometry .It states the equivalence of Symplectic and Geometric invariant theory quotients. After brief introduction of each of the quotients we will cover the proof of the theorem.

Abstract: In this talk after going through the statements of Hodge decomposition and sketch of proof, we will try to visit some applications of it in Riemannian Geometry and Complex Geometry.