IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

We study the GKM theory for a equivariant stratified space having orbifold structures in its successive quotients. Then, we introduce the notion of an almost simple polytope, as well as a divisive toric variety generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties. © 2020 Elsevier B.V.

Department of Mathematics

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Topology and its Applications

GKM theory for orbifold stratified spaces and application to singular toric varieties

In this paper, we study some properties of retraction sequences, blowups of polytopes and their interrelations. We introduce the blowups of quasitoric orbifolds using the combinatorial data associated with its orbit spaces. We prove that neither new singularity nor new torsion arises in the integral cohomologies of certain blowups of a quasitoric orbifold. Moreover, we show that if a quasitoric orbifold has no p-torsion in its cohomology then the cohomology of its blowup along a fixed point has no p-torsion. As a consequence, we construct infinitely many integrally equivariantly formal quasitoric orbifolds from a given one. We also investigate the torsions in the cohomologies of blowups of a class of simplicial toric varieties. © 2021 Elsevier B.V.

Department of Electrical Engineering

Torsion in the cohomology of blowups of quasitoric orbifolds

We extend the product formula for Lusternik-Schnirelmann category and topological complexity to some classes of fiber bundles. Then we compute and study the Lusternik-Schnirelmann category and topological complexity of Dold manifolds. © 2020 Elsevier B.V.

On LS-category and topological complexity of some fiber bundles and Dold manifolds

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of locally standard torus manifolds whose orbit space may have proper non-acyclic faces. © 2020 American Mathematical Society.

Transactions of the Moscow Mathematical Society

Cohomology rings of a class of torus manifolds

The category of torus orbifolds is a generalization of the category of toric orbifolds which contains projective toric varieties associated to complete simplicial fans. We introduce the concept of “divisive” torus orbifolds following divisive weighted projective spaces. The divisive condition may ensure an invariant cell structure on a locally standard torus orbifold. We give a combinatorial description of equivariant K-theory, equivariant cobordism theory and equivariant cohomology theory of divisive torus orbifolds. In particular, we get a combinatorial description of these generalize cohomology theories for torus manifolds over acyclic polytopes. © Rocky Mountain Mathematics Consortium

Rocky Mountain Journal of Mathematics

Equivariant K-theory of divisive torus orbifolds

The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal. In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold. © 2019 Walter de Gruyter GmbH, Berlin/Boston.

Forum Mathematicum

Infinite families of equivariantly formal toric orbifolds

We introduce the concepts of orbit class, orbit diagram and orbital spaces, and study their properties. Then we apply the results on orbit classes to study equivariant and invariant topological complexity, as well as to give some lower and upper bounds for the equivariant LS-category and A-G-LS-category. © Instytut Matematyczny PAN, 2019

Fundamenta Mathematicae

Orbit class and its applications

Let T n be the real n-torus group. We give a new definition of lens spaces and study the diffeomorphic classification of lens spaces. We show that any 3-dimensional lens space L(p; q) is T 2 -equivariantly cobordant to zero. We also give some sufficient conditions for higher dimensional lens spaces L(p; q1, . . . , qn) to be T n+1-equivariantly cobordant to zero. In 2005, B. Hanke showed that complex equivariant cobordism class of a lens space is trivial. Nevertheless, our proofs are constructive using toric topological arguments.&nbsp;

A new construction of lens spaces

We examine the integral cohomology rings of certain families of 2n-dimensional orbifolds X that are equipped with a well-behaved action of the n-dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs(Q, λ), where Q isasimpleconvex n-polytope and λ a labeling of its facets, and from n-dimensional fans Σ. In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on(Q, λ) or on Σ which ensure that the integral cohomology groups H*(X) of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring H*(X)as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples. © 2017, Mathematical Sciences Publishers. All rights reserved.

Algebraic and Geometric Topology

On the integral cohomology ring of toric orbifolds and singular toric varieties

We first consider some generalities regarding Δ-complexes, in particular, we give a brief history of the category of Δ-complexes, and its relation to the category of semi-simplicial complexes introduced in 1950 by Eilenberg and Zilber. A natural construction of Δ-complexes arising from a group action on a simplicial complex is next considered. Finally, an application of this construction to obtain an elementary explicit computation of the cohomology ring H*(RPn; Z2), based on a Δ-complex structure, is given. © 2017, International Press.

Homology, Homotopy and Applications

Remarks about Δ-complexes and applications

We study the lower bounds and upper bounds for LS-category and equivariant LS-category. In particular we compute both invariants for torus manifolds. There are some examples to show the sharpness of conditions in the theorems. Moreover the equivariant LS-category of the product space is discussed and counterexamples of some previous results are given. © 2015.

Some aspects of equivariant LS-category

We construct orbifolds with quasitoric boundary and show that they have stable almost complex structure. We show that a quasitoric orbifold is complex cobordant to finite disjoint copies of complex orbifold projective spaces. Finally some computations in the complex cobordism ring for manifolds are given.

Complex cobordism of quasitoric orbifolds

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal ℤ22-equivariant triangulations of 2-dimensional small covers. We discuss vertex minimal equilibrium triangulations of ℝℙ3#ℝℙ3, S1 × ℝℙ2 and a nontrivial S1 bundle over ℝℙ2. We construct some nice equilibrium triangulations of the real projective space ℝℙn with 2n + n + 1 vertices. The main tool is the theory of small covers.

Journal of the Ramanujan Mathematical Society

Equilibrium and equivariant triangulations of some small covers with minimum number of vertices

We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch χy genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring. © 2014 American Mathematical Society.

Proceedings of the American Mathematical Society

A class of torus manifolds with nonconvex orbit space

We show the T 2 -cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the T 2 -cobordism classes of CP 2. We construct nice oriented T 2 manifolds with boundary whose boundaries are the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.

T 2 -cobordism of quasitoric 4-manifolds

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.

Osaka Journal of Mathematics

On quasitoric orbifolds

Most of the real life classification problems have ill defined, imprecise or fuzzy class boundaries. Feedforward neural networks with conventional backpropagation learning algorithm are not tailored to this kind of classification problem. Hence, in this paper, feedforward neural networks, that use backpropagation learning algorithm with fuzzy objective functions, are investigated. A learning algorithm is proposed that minimizes an error term, which reflects the fuzzy classification from the point of view of possibilistic approach. Since the proposed algorithm has possibilistic classification ability, it can encompass different backpropagation learning algorithm based on crisp and constrained fuzzy classification. The efficacy of the proposed scheme is demonstrated on a vowel classification problem.

Department of Computer Science and Engineering

Pattern Recognition Letters

Backpropagation learning algorithms for classification with fuzzy mean square error

In this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.