arXiv subjects

Samik Basu

Publications and source records attributed to Samik Basu.

At least 37 records · Page 2Linked to original sources

arXiv2023

Representing and Reasoning with Multi-Stakeholder Qualitative Preference Queries

Many decision-making scenarios, e.g., public policy, healthcare, business, and disaster response, require accommodating the preferences of multiple stakeholders. We offer the first formal treatment of reasoning with multi-stakeholder qualitative preferences in a setting where stakeholders express their preferences in a qualitative preference language, e.g., CP-net, CI-net, TCP-net, CP-Theory. We introduce a query language for expressing queries against such preferences over sets of outcomes that satisfy specified criteria, e.g., $\mlangpref{ψ_1}{ψ_2}{A}$ (read loosely as the set of outcomes satisfying $ψ_1$ that are preferred over outcomes satisfying $ψ_2$ by a set of stakeholders $A$). Motivated by practical application scenarios, we introduce and analyze several alternative semantics for such queries, and examine their interrelationships. We provide a provably correct algorithm for answering multi-stakeholder qualitative preference queries using model checking in alternation-free $μ$-calculus. We present experimental results that demonstrate the feasibility of our approach.

cs.AI↗

arXiv2023

Equivariant Cohomology of Projective Spaces

We compute the equivariant homology and cohomology of projective spaces with integer coefficients. More precisely, in the case of cyclic groups, we show that the cellular filtration of the projective space $P(kρ)$, of lines inside copies of the regular representation, yields a splitting of $H\underline{\mathbb{Z}}\bigwedge P(kρ)_+$ as a wedge of suspensions of $H\underline{\mathbb{Z}}$. This is carried out both in the complex case, and also in the quaternionic case, and further, for the $C_2$ action on $\mathbb{C} P^n$ by complex conjugation. We also observe that these decompositions imply a degeneration of the slice tower in these cases. Finally, we describe the cohomology of the projective spaces when $|G|=p^m$ of prime power order, with explicit formulas for $\underline{\mathbb{Z}_p}$-coefficients. Letting $k=\infty$, this also describes the equivariant homology and cohomology of the classifying spaces of $S^1$ and $S^3$.

math.AT↗

arXiv2023

Inertia Groups and Smooth Structures on Quaternionic Projective Spaces

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia group and its analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

math.GT↗

arXiv2022

$p$-local decompositions of projective Stiefel manifolds

The main objective of this paper is to analyze the $p$-local homotopy type of the complex projective Stiefel manifolds, and other analogous quotients of Stiefel manifolds. We take the cue from a result of Yamaguchi about the $p$-regularity of the complex Stiefel manifolds which lays down some hypotheses under which the Stiefel manifold is $p$-locally a product of odd dimensional spheres. We show that in many cases, the projective Stiefel manifolds are $p$-locally a product of a complex projective space and some odd dimensional spheres. As an application, we prove that in these cases, the $p$-regularity result of Yamaguchi is also $S^1$-equivariant.

math.AT↗

arXiv2022

The index of equidimensional flag manifolds

In this paper, we consider the flag manifold of $p$ orthogonal subspaces of equal dimension which carries an action of the cyclic group of order $p$. We provide a complete calculation of the associated Fadell-Husseini index. This may be thought of as an odd primary version of the computations of Baralić et al [Forum Math., 30 (2018), pp. 1539--1572] for the Grassmann manifold $G_n(\mathbb{R}^{2n})$. These results have geometric consequences for $p$-fold orthogonal shadows of a convex body.

math.AT↗

arXiv2021

Equivariant homology decompositions for cyclic group actions on definite 4-manifolds

In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular filtrations on the connected sum using spheres inside unitary representations. A judicious choice of filtration implies a splitting on equivariant homology for general cyclic groups under a divisibility hypothesis, and in all cases for those of prime power order.

math.AT↗

arXiv2021

BP-cohomology of projective Stiefel manifolds

In this paper, we compute the BP-cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and BP-operations to prove new results about equivariant maps between Stiefel manifolds.

math.AT↗

arXiv2021

Non-trivial extensions in equivariant cohomology with constant coefficients

In this paper, we prove some computational results about equivariant cohomology over the cyclic group $C_{p^n}$ of prime power order. We show that there is an inductive formula when the dimension of the $C_p$-fixed points of the grading is large. Among other calculations, we also show the existence of non-trivial extensions when $n\geq 3$.

math.AT↗

arXiv2020

Measuring the Impact of Influence on Individuals: Roadmap to Quantifying Attitude

Influence diffusion has been central to the study of propagation of information in social networks, where influence is typically modeled as a binary property of entities: influenced or not influenced. We introduce the notion of attitude, which, as described in social psychology, is the degree by which an entity is influenced by the information. We present an information diffusion model that quantifies the degree of influence, i.e., attitude of individuals, in a social network. With this model, we formulate and study attitude maximization problem. We prove that the function for computing attitude is monotonic and sub-modular, and the attitude maximization problem is NP-Hard. We present a greedy algorithm for maximization with an approximation guarantee of $(1-1/e)$. Using the same model, we also introduce the notion of "actionable" attitude with the aim to study the scenarios where attaining individuals with high attitude is objectively more important than maximizing the attitude of the entire network. We show that the function for computing actionable attitude, unlike that for computing attitude, is non-submodular and however is \emph{approximately submodular}. We present approximation algorithm for maximizing actionable attitude in a network. We experimentally evaluated our algorithms and study empirical properties of the attitude of nodes in network such as spatial and value distribution of high attitude nodes.

cs.SI↗

arXiv2020

Bredon cohomology of finite dimensional $C_p$-spaces

For finite dimensional free $C_p$-spaces, the calculation of the Bredon cohomology ring as an algebra over the cohomology of $S^0$ is used to prove the non-existence of certain $C_p$-maps. These are related to Borsuk-Ulam type theorems, and equivariant maps related to the topological Tverberg conjecture. For certain finite dimensional $C_p$-spaces which are formed out of representations, it is proved that the cohomology is a free module over the cohomology of a point. All the calculations are done for the cohomology with constant coefficients $\mathbb{Z}/p$.

math.AT↗

arXiv2020

Equivariant cohomology for cyclic groups of square-free order

The main objective of this paper is to compute $RO(G)$-graded cohomology of $G$-orbits for the group $G=C_n$, where $n$ is a product of distinct primes. We compute these groups for the constant Mackey functor $\underline{Z}$ and for the Burnside ring Mackey functor $\underline{A}$. Among other things, we show that the groups $\underline{H}^α_G(S^0)$ are mostly determined by the fixed point dimensions of the virtual representations $α$, except in the case of $\underline{A}$ coefficients when the fixed point dimensions of $α$ have many zeros. In the case of $\underline{Z}$ coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain $G$-complexes.

math.AT↗

arXiv2020

Note on Toda brackets

We provide a general definition of Toda brackets in a pointed model categories, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.

math.AT↗

arXiv2019

Computations in $C_{pq}$-Bredon cohomology

In this paper, we compute the $RO(C_{pq})$-graded cohomology of $C_{pq}$-orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group $C_p$. The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group $C_p$ was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem.

math.AT↗

arXiv2018

The homotopy type of the loops on $(n-1)$-connected $(2n+1)$-manifolds

For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.

math.AT↗

arXiv2018

Nambu Structures And Associated Bialgebroids

This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.

math.DG↗

arXiv2017

Generalized Thom spectra and their topological Hochschild homology

We develop a theory of R-module Thom spectra for a commutative symmetric ring spectrum R and we analyze their multiplicative properties. As an interesting source of examples, we show that R-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on R. We apply the general theory to obtain a description of the R-based topological Hochschild homology associated to an R-algebra Thom spectrum.

math.AT↗

arXiv2017

Higher structure in the unstable Adams spectral sequence

We describe a variant construction of the unstable Adams spectral the sequence for a space $Y$, associated to any free simplicial resolution of $H^*(Y;R)$ for $R=\mathbb{F}_p$ or $\mathbb{Q}$. We use this construction to describe the differentials and filtration in the spectral sequence in terms of appropriate systems of higher cohomology operations.

math.AT↗

arXiv2016

Equivariant maps related to the topological Tverberg conjecture

Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of Özaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.

math.AT↗