Invited Talks

  1. 13 Sep 2022, Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems, Simons center for Geometry and Physics

Title: Deformations and gluing of asymptotically cylindrical associatives

Abstract: Given a matching pair of asymptotically cylindrical (Acyl) G_2 manifolds the twisted connected sum construction produces a one parameter family of closed G_2 manifolds. We describe when we can construct closed rigid associatives in these closed G_2 manifolds by gluing suitable pairs of Acyl associatives in the matching pair of Acyl G_2 manifolds. The hypothesis and analysis in the gluing theorem requires some understanding of the deformation theory of Acyl associatives which will also be discussed. At the end we will describe examples of closed associatives coming from Acyl holomorphic curves or special Lagrangians.

Voluntary Talks

  1. 22 Sep 2022, Journal club (5 minute presentation) SLMath (MSRI) Berkeley

Title: G_2 instantons on resolutions of G_2 orbifolds

Abstract: A short overview of Daniel Platt's paper .

  1. 18 Jul 2022, Research Seminar: Symplectic Geometry (SS 2022), Humboldt-Universität zu Berlin

Title: Abouzaid-Imagi’s theorem on nearby special Lagrangians

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.

  1. 7 Feb 2022, Research Seminar: Symplectic Geometry (WS 2021/22), Humboldt-Universität zu Berlin

Title: On counting special Lagrangian homology 3-spheres, II

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.

  1. 31 Jan 2022, BMS student seminar, Humboldt-Universität zu Berlin

Title: Gluing of Morse flow lines

  1. 29 Nov 2021, BMS student seminar, Humboldt-Universität zu Berlin

Title: Analytic set up for Morse homology

  1. 14 Jul 2021, Informal seminar, Humboldt-Universität zu Berlin

Title: Adiabatic limits of co-associative Kovalev-Lefschetz fibrations, II

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

  1. 7 Jul 2021, Informal seminar, Humboldt-Universität zu Berlin

Title: Adiabatic limits of co-associative Kovalev-Lefschetz fibrations, I

Abstract: We follow Simon Donaldson's paper.

  1. 30 Jun 2021, Informal seminar, Humboldt-Universität zu Berlin

Title: Moduli of Coassociative Submanifolds and Semi-Flat Coassociative Fibrations, II

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

  1. 23 Jun 2021, Informal seminar, Humboldt-Universität zu Berlin

Title: Moduli of Coassociative Submanifolds and Semi-Flat Coassociative Fibrations, I

Abstract: We follow D. Baraglia's paper.

  1. 21 Apr 2021, Seminar: Riemannian Convergence Theory (SS 2021), Humboldt-Universität zu Berlin

Title: Cheeger’s finiteness theorem

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.

  1. 31 Jan 2021, Seminar: Gauge Theory (WS 2020/21), Humboldt-Universität zu Berlin

Title: The Narasimhan-Seshadri theorem

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.

  1. 4 Nov 2019, Student Geometry & Topology Seminar, Michigan State University

Title: Introduction to Riemannian Holonomy

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.

  1. 29 Oct 2019, Geometric Analysis Seminar, Michigan State University

Title: Gauge theory and tubular ends

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

  1. 22 Oct 2019, Geometric Analysis Seminar, Michigan State University

Title: Linear analysis on cylindrical end manifolds

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

  1. 17 Apr 2019, Student Geometry & Topology Seminar, Michigan State University

Title: Introduction to Seiberg-Witten invariants on three manifolds

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.

  1. 25 Mar 2019, Geometric Analysis Seminar, Michigan State University

Title: Taubes' approach to Casson invariant

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

  1. 11 Mar 2019, Geometric Analysis Seminar, Michigan State University

Title: Taubes' approach to the Casson invariant, an overview

Abstract: We follow Taubes' paper.

  1. 21 Nov 2018, Student Geometry & Topology Seminar, Michigan State University

Title: Introduction to Seiberg-Witten invariants on four manifolds

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.

  1. 14 Nov 2018, Geometric Analysis Seminar, Michigan State University

Title: Federer's dimension reduction argument, I

  1. 7 Nov 2018, Geometric Analysis Seminar, Michigan State University

Title: Federer's dimension reduction argument, II

  1. 28 Mar 2018, Student Geometry & Topology Seminar, Michigan State University

Title: Symplectic Quotients and GIT Quotients : The Kempf-Ness Theorem

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.

  1. 15 Nov 2017, Student Geometry & Topology Seminar, Michigan State University

Title: Hodge theory and its applications

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.