Rahul Dutta | University of New Mexico (original) (raw)
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Papers by Rahul Dutta
Theory and Applications of Graphs, 2021
In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The se... more In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.
In this article I have presented a brief introduction to algebraic geometry, with more emphasis o... more In this article I have presented a brief introduction to algebraic geometry, with more emphasis on Algebraic Curves and Riemann Surfaces. Moreover I have provided theorems and lemmas with less proofs, and along with it many examples that allow one to gain a deeper intuitive understanding of the material. I have tried to keep the prerequisites required to understand this article at a minimum level. However, this paper demonstrates an elementary proof of the Riemann-Roch Theorem, which is a vital tool to the fields of complex analysis and algebraic geometry. It is used for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It also relates the complex analysis of a compact, connected Riemann surface with the surface's purely topological genus, in a way that can be carried over into purely algebraic settings. Finally I have added the part on Sheaves and Sheaf Cohomology as an introduction to the Hirzebruch Riemann-Roch Theorem. 3 Contents 4 3 Contents 4. Introduction 5. Preliminaries 6. Some Category Theory 7. An overview on Differential Forms and Integration 8. Divisors and Meromorphic functions 9. Riemann-Roch for Algebraic Curve 10.Applications of Riemann-Roch 11.Sheaves and Sheaf Cohomology 12.Invertible Sheaves 13.Conclusion 14.Bibliography
In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The se... more In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.
Drafts by Rahul Dutta
In this memoire we present the theory of symplectic quotients and the interactions with the theor... more In this memoire we present the theory of symplectic quotients and the interactions with the theory of kahler quotients. The starting point is the following natural question: in the presence of a symplectic action of a Lie group G on a symplectic manifold pM, ωq can one define in a coherent way the quotient of M by G (so that the result will have a natural symplectic structure)? Even if we assume that G is compact and the action is free, we see that in general M {G cannot be symplectic (for dimension reasons). In order to obtain a symplectic quotient, one needs a new ingredient: a moment map. The fundamental theorem of the theory states that, assuming that G acts freely and properly around the zero locus of the moment map, then the G-quotient of this zero locus has a natural structure of a symplectic manifold. We will solve in detail the existence and unicity problems for moment maps. The main tool needed here is Lie algebra cohomology. We will continue with explicit computations of moment maps and explicit descriptions of symplectic quotients. We will prove that many interesting manifolds (projective spaces, Grassman manifolds, flag manifolds) can be obtained as symplectic quotients. We will conclude with a general principle which emphasizes an interesting relation between symplectic geometry and complex geometry: Let X be a Kähler manifold, K be a compact Lie group and G " K CˆX Ñ X be a holomorphic action on X which restricts to a Hamiltonian symplectic action of K which is free around µ´1p0q. Then the corresponding symplectic quotient has a natural structure of complex manifold. We will illustrate this principle in the examples we study.
Theory and Applications of Graphs, 2021
In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The se... more In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.
In this article I have presented a brief introduction to algebraic geometry, with more emphasis o... more In this article I have presented a brief introduction to algebraic geometry, with more emphasis on Algebraic Curves and Riemann Surfaces. Moreover I have provided theorems and lemmas with less proofs, and along with it many examples that allow one to gain a deeper intuitive understanding of the material. I have tried to keep the prerequisites required to understand this article at a minimum level. However, this paper demonstrates an elementary proof of the Riemann-Roch Theorem, which is a vital tool to the fields of complex analysis and algebraic geometry. It is used for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It also relates the complex analysis of a compact, connected Riemann surface with the surface's purely topological genus, in a way that can be carried over into purely algebraic settings. Finally I have added the part on Sheaves and Sheaf Cohomology as an introduction to the Hirzebruch Riemann-Roch Theorem. 3 Contents 4 3 Contents 4. Introduction 5. Preliminaries 6. Some Category Theory 7. An overview on Differential Forms and Integration 8. Divisors and Meromorphic functions 9. Riemann-Roch for Algebraic Curve 10.Applications of Riemann-Roch 11.Sheaves and Sheaf Cohomology 12.Invertible Sheaves 13.Conclusion 14.Bibliography
In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The se... more In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.
In this memoire we present the theory of symplectic quotients and the interactions with the theor... more In this memoire we present the theory of symplectic quotients and the interactions with the theory of kahler quotients. The starting point is the following natural question: in the presence of a symplectic action of a Lie group G on a symplectic manifold pM, ωq can one define in a coherent way the quotient of M by G (so that the result will have a natural symplectic structure)? Even if we assume that G is compact and the action is free, we see that in general M {G cannot be symplectic (for dimension reasons). In order to obtain a symplectic quotient, one needs a new ingredient: a moment map. The fundamental theorem of the theory states that, assuming that G acts freely and properly around the zero locus of the moment map, then the G-quotient of this zero locus has a natural structure of a symplectic manifold. We will solve in detail the existence and unicity problems for moment maps. The main tool needed here is Lie algebra cohomology. We will continue with explicit computations of moment maps and explicit descriptions of symplectic quotients. We will prove that many interesting manifolds (projective spaces, Grassman manifolds, flag manifolds) can be obtained as symplectic quotients. We will conclude with a general principle which emphasizes an interesting relation between symplectic geometry and complex geometry: Let X be a Kähler manifold, K be a compact Lie group and G " K CˆX Ñ X be a holomorphic action on X which restricts to a Hamiltonian symplectic action of K which is free around µ´1p0q. Then the corresponding symplectic quotient has a natural structure of complex manifold. We will illustrate this principle in the examples we study.