Mainak Poddar

Bio: Mainak Poddar is an academic researcher from Indian Institute of Science Education and Research, Pune. The author has contributed to research in topics: Toric variety & Equivariant map. The author has an hindex of 9, co-authored 43 publications receiving 304 citations. Previous affiliations of Mainak Poddar include University of Los Andes & Middle East Technical University Northern Cyprus Campus.

Holomorphic orbi-discs and Lagrangian Floer cohomology of symplectic toric orbifolds

Abstract: We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential.We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta, and Ono.

On quasitoric orbifolds

Abstract: Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen-Ruan cohomology groups of an almost complex quasitoric orbifold.

The global mckay¿ruan correspondence via motivic integration

Abstract: The purpose of this paper is to show how the motivic integration methods of Kontsevich, Denef-Loeser and Looijenga can be adapted to prove the McKay-Ruan correspondence, a generalization of the McKay-Reid correspondence to orbifolds that are not necessarily global quotients.

Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

Abstract: We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk-deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential. We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta and Ono.

A class of torus manifolds with nonconvex orbit space

Abstract: We study a class of smooth torus manifolds whose orbit space has the struc- ture of a simple polytope with holes. We prove that these manifolds have stable almost complex structure and give combinatorial formula for some of their Hirzebruch genera. They have (invariant) almost complex structure if they admit positive omniorientation. In dimension four, we calculate the homology groups, construct symplectic structure on a large class of these manifolds, and give a family which is symplectic but not complex.

Orbifolds and Stringy Topology

Abstract: An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role. The subject is first developed following the classical description analogous to manifold theory, after which the book branches out to include the useful description of orbifolds provided by groupoids, as well as many examples in the context of algebraic geometry. Classical invariants such as de Rham cohomology and bundle theory are developed, a careful study of orbifold morphisms is provided, and the topic of orbifold K-theory is covered. The heart of this book, however, is a detailed description of the Chen-Ruan cohomology, which introduces a product for orbifolds and has had significant impact. The final chapter includes explicit computations for a number of interesting examples.

The interior-point revolution in optimization: History, recent developments, and lasting consequences

Abstract: Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interior-point techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded, nearly to the point of oblivion, by newly emerging and seemingly more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization. This picture changed dramatically in 1984, when Narendra Karmarkar announced a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have continued to transform both the theory and practice of constrained optimization. We present a condensed, unavoidably incomplete look at classical material and recent research about interior methods.

Transformation of Groups

Abstract: In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Wall-crossings in toric Gromov–Witten theory I: crepant examples

Abstract: Let X be a Gorenstein orbifold with projective coarse moduli space X and let Y be a crepant resolution of X . We state a conjecture relating the genus-zero Gromov‐ Witten invariants of X to those of Y , which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan‐Graber, and prove our conjecture when XD P.1;1;2/ and XD P.1;1;1;3/. As a consequence, we see that the original form of the Bryan‐Graber Conjecture holds for P.1;1;2/ but is probably false for P.1;1;1;3/. Our methods are based on mirror symmetry for toric orbifolds. 53D45; 14N35, 83E30