Invited Talks

2. 28 Sep 2022, Geometry seminar, Université libre de Bruxelles 

Title: Associatives in twisted connected sum G_2 manifolds 
Abstract: G_2 manifolds are Riemannian manifolds whose holonomy contained in the exceptional Lie group G_2 and associatives inside them are some 3 dimensional calibrated submanifolds which play a crucial role for defining several enumerative theories of the G_2 manifolds. This motivates us to construct many examples of associatives.  The most effective method to date of constructing G_2 manifolds is the twisted connected sum construction which glues a matching pair of asymptotically cylindrical (ACyl) G_2 manifold or  ACyl Calabi-Yau 3-fold. In this talk we present a method to construct closed rigid (unobstructed) associatives in the twisted connected sum G_2-manifolds by gluing  ACyl associatives in ACyl G_2-manifolds under a hypothesis which can be interpreted as a transverse Lagrangian intersection condition. We rewrite the gluing hypothesis for ACyl associatives obtained from ACyl holomorphic curves or ACyl special Lagrangian 3-folds in ACyl Calabi-Yau 3-folds. This helps us to construct many new associatives which are diffeomorphic to S^3, RP^3 or RP^3#RP^3.
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

26. 26 May 2023, Research Seminar: Gauge theory, Humboldt-Universität zu Berlin 

Title: Coassociative K3 fibrations of compact G2 manifolds.  

25. 05 May 2023, Seminar: Elliptic boundary value problems and applications in geometry, Humboldt-Universität zu Berlin 

Title: The model operator and the trace theorem. 

24. 01 Nov 2022, Gauge theory graduate student seminar , SLMath (MSRI) Berkeley 

Title: On counting associative submanifolds in G_2 manifolds and Seiberg-Witten monopole 

Abstract: In this talk, I will discuss the proposal of Doan and Walpuski to define an invariant of G_2-manifolds based on counting associative submanifolds with Seiberg-Witten monopoles on them.  Their proposal was motivated by the analysis of various transitions that can change the number of associative submanifolds. In the most general case, the invariant they propose are Floer homology groups. Reference: Doan-Walpuski, arXiv:1712.08383

23. 22 Sep 2022, Journal club (5 minute presentation) SLMath (MSRI) Berkeley

Title: G_2 instantons on resolutions of G_2 orbifolds 

22. 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.

21. 7 Feb 2022, Research Seminar: Symplectic Geometry (WS 2021/22), Humboldt-Universität zu Berlin  

Title: On counting special Lagrangian homology 3-spheres

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.

20. 31 Jan 2022, BMS student seminar, Humboldt-Universität zu Berlin  

Title: Gluing of Morse flow lines 

19. 29 Nov 2021, BMS student seminar, Humboldt-Universität zu Berlin  

Title: Analytic set up for Morse homology

18. 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.

17. 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.

16. 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.

15. 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.

14. 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.

13. 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.

12. 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.

11. 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.

10. 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.

9. 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.

8. 25 Mar 2019, Geometric Analysis Seminar, Michigan State University

Title: Taubes' approach to Casson invariant

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

7. 11 Mar 2019, Geometric Analysis Seminar, Michigan State University

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

Abstract: We follow Taubes' paper. 

6. 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.

5. 14 Nov 2018, Geometric Analysis Seminar, Michigan State University

Title: Federer's dimension reduction argument, II

4. 7 Nov 2018, Geometric Analysis Seminar, Michigan State University

Title: Federer's dimension reduction argument, I

3. 10 Oct 2018, Geometric Analysis Seminar, Michigan State University

Title: Morrey's theorem: Regularity of minimizing harmonic maps in dimension 2

2. 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.