IMSc Theses/ Dissertations

Exotic Smooth Structures on Products of Certain Smooth Manifolds with Spheres [HBNI Th282]

2026-01-01T00:00:00Z

Exotic Smooth Structures on Products of Certain Smooth Manifolds with Spheres [HBNI Th282] Ankur Sarkar The classification of smooth structures on manifolds is a central problem in differential topology, particularly for manifolds sharing the same underlying topological type. This thesis studies this problem for product manifolds of the form M x S^k, where 1 <= k <= 10 and M is a closed, oriented, connected smooth 4-manifold, a closed simply connected smooth 5- or 6-manifold, or a closed, oriented, 3-connected smooth 8-manifold. We first study smooth structures on manifolds up to concordance. For a smooth manifold N of dimension at least 5, Kirby and Siebenmann identified the set of concordance classes of smooth structures C(N) with the set of homotopy classes of maps from N to Top/O. From this correspondence, we show that the concordance inertia group Ic(M x S^k) is determined by the stable top-cell attaching map of M. In particular, when the stable homotopy type of M is known, the group Ic(M x S^k) can be computed explicitly. By analyzing the stable cell structures of the above-mentioned manifolds M and using known computations of the stable homotopy groups of spheres, we compute Ic(M x S^k) for all 1 <= k <= 10 and classify smooth structures on M x S^k up to concordance for certain values of k. The second part of the thesis addresses the classification of smooth structures up to diffeomorphism. Let S(N) denote the set of orientation-preserving diffeomorphism classes of smooth manifolds that are homeomorphic to a given smooth manifold N. The computation of C(N) plays a key role in this classification. The group of self-homeomorphisms Homeo(N) acts on C(N), and this action induces a one-to-one correspondence between S(N) and the orbit space C(N) / omega_0(Homeo(N)). Equivalently, there is a bijection: C(N) / omega_0(Homeo(N)) <-> S(N) Building on the computations of C(M x S^k) and using surgery theory, we analyze this action and determine the inertia group for the manifolds CP^2 x S^k for 4 <= k <= 6, CP^3 x S^k for 2 <= k <= 7, and HP^2 x S^1, where HP^2 denotes a projective plane-like smooth 8-dimensional manifold. Finally, we obtain a complete diffeomorphism classification of all smooth manifolds homeomorphic to these product spaces, including the case of CP^3 x S^1.

The many facets of stochastic resetting in first-passage and non-equilibrium processes [HBNI Th281]

2026-01-01T00:00:00Z

The many facets of stochastic resetting in first-passage and non-equilibrium processes [HBNI Th281] Arup Biswas The focus of this thesis was threefold: i) application and effects of stochastic resetting on physical in a variety of processes, namely, chemical reactions, operations research and viscoelastic media, ii) a unified renewal approach to experimental realisation of resetting, and iii) proposing a novel FP optimization strategy based on threshold resetting. In the following, we summarise the primary results emanated out of this thesis. The Chapters 2–4 are dedicated to the understanding of applications of resetting in several fields. In Chapter 2, we study how a gated chemical reaction can be facilitated with resetting. Resetting a chemical reaction refers to the unbinding of the enzyme-substrate complex, after which the reaction is started afresh. As a primary result, we find, a certain unbinding rate can indeed expedite the product formation of a chemical reaction, especially when the target is ‘gated’ in nature, which models the active-inactive phase of the substrate. We modelled the chemical reaction with a one-dimensional drift-diffusion process with a gated target. By doing so, we find a parameter space in terms of diffusion and drift coefficients where the reaction rate can be enhanced with stochastic resetting. In Chapter 3, we discuss how service resetting can reduce the queue length. Considering the vast fields of applications of queuing theory, strategies to effectively manage a queue remain a pivotal challenge. With analytical formulations, we explicitly find the mean queue length of an M/G/1 queue subjected to the service resetting strategy, where resetting consumes a random period of overhead time. We show that depending on the fluctuations of the overhead time, one can suitably choose an optimal resetting rate that can reduce the mean queue length to its best. Furthermore, we find that periodic resetting does a better optimization in contrast to resetting at random times. In Chapter 4 we study the effects of resetting in a particle embedded in a viscoelastic medium. Viscoelastic medium are different from the traditional viscous medium since the collisions between the particles are not forgotten instantaneously. Due to the memory dependence of the particle’s trajectory, the system is highly non-Markovian in nature. Our study reveals that, resetting a particle in such a medium generates non-trivial non-equilibrium steady states and correlations. It also gives rise to a resetting rate-dependent timescale in the system that can help the external agent to gain control over the dynamics of the particle in such a highly non-Markovian system. In Chapter 5, we propose an experimentally amenable approach to stochastic resetting. Although the idea of resetting being a subject to extensive studies, a majority of them (along with the models considered in the previous section) consider resetting to be an instantaneous event. In reality, a resetting event is space-time coupled so that a finite time is consumed when the particle heads back to the starting position. We propose a universal framework of stochastic resetting where the agent returns to the starting position in a space-time coupled non-instantaneous fashion. By considering the paradigmatic example of diffusion, we show that a stochastic component in the return motion of the particle can indeed enhance the search efficiency in a FP process. Furthermore, we corroborate our theory with experimental parameters and identify suitable parameter regimes where stochastic return can indeed be beneficial than the classical instantaneous return. By adding a drift to the particle dynamics, we also show that stochastic return can outperform the underlying reset-free process as well as the optimal instantaneous return protocol. In summary, we demonstrate that search with stochastic returns provides not only a physically realizable approach to resetting but also can be more advantageous. In Chapter 6, we propose a novel approach to the collective search process optimization based on threshold resetting. Under this set-up, the system resets only when any one of many searchers hits a pre-defined threshold value. Although the resetting epochs are still stochastic in time, it is now governed by some space-dependent threshold, leading to an event-driven process resetting strategy. We explicitly analyse the effect of such a resetting strategy in a multi-agent search process. Note that all the search agents are reset to the initial position at each resetting epoch, which introduces long-range correlations among the searchers. We show that the mean FPT of the process shows rich optimisation features with respect to the threshold distance and the number of searchers. We also quantify the cost of maintaining such a threshold-dependent search process and find an interesting optimisation of the cost function. Despite the extensive progress achieved through the studies presented in this thesis, stochastic resetting continues to be a fertile ground for exploration. Its remarkable ability to induce nonequilibrium steady states, optimise search and reaction efficiencies, and generate novel dynamical correlations underscores its fundamental and applied importance. Future studies could explore resetting in systems with spatial or temporal disorder, active matter, and biological networks, where feedback-controlled or adaptive resetting rules may yield richer dynamics. Moreover, extending resetting concepts to quantum systems, machine learning algorithms, or complex supply-chain models could open exciting interdisciplinary avenues. The insights obtained here not only advance our understanding of resetting as a universal control mechanism but also lay the groundwork for its potential exploitation in experimental and technological contexts—from microscale transport processes to large-scale optimisation problems.

Problems on Hilbert schemes and quiver bundles [HBNI Th280]

2026-01-01T00:00:00Z

Problems on Hilbert schemes and quiver bundles [HBNI Th280] Saurav Holme Choudhury The focus of this thesis was threefold: i) application and effects of stochastic resetting on physical in a variety of processes, namely, chemical reactions, operations research and viscoelastic media, ii) a unified renewal approach to experimental realisation of resetting, and iii) proposing a novel FP optimization strategy based on threshold resetting. In the following, we summarise the primary results emanated out of this thesis. The Chapters 2–4 are dedicated to the understanding of applications of resetting in several fields. In Chapter 2, we study how a gated chemical reaction can be facilitated with resetting. Resetting a chemical reaction refers to the unbinding of the enzyme-substrate complex, after which the reaction is started afresh. As a primary result, we find, a certain unbinding rate can indeed expedite the product formation of a chemical reaction, especially when the target is ‘gated’ in nature, which models the active-inactive phase of the substrate. We modelled the chemical reaction with a one-dimensional drift-diffusion process with a gated target. By doing so, we find a parameter space in terms of diffusion and drift coefficients where the reaction rate can be enhanced with stochastic resetting. In Chapter 3, we discuss how service resetting can reduce the queue length. Considering the vast fields of applications of queuing theory, strategies to effectively manage a queue remain a pivotal challenge. With analytical formulations, we explicitly find the mean queue length of an M/G/1 queue subjected to the service resetting strategy, where resetting consumes a random period of overhead time. We show that depending on the fluctuations of the overhead time, one can suitably choose an optimal resetting rate that can reduce the mean queue length to its best. Furthermore, we find that periodic resetting does a better optimization in contrast to resetting at random times. In Chapter 4 we study the effects of resetting in a particle embedded in a viscoelastic medium. Viscoelastic medium are different from the traditional viscous medium since the collisions between the particles are not forgotten instantaneously. Due to the memory dependence of the particle’s trajectory, the system is highly non-Markovian in nature. Our study reveals that, resetting a particle in such a medium generates non-trivial non-equilibrium steady states and correlations. It also gives rise to a resetting rate-dependent timescale in the system that can help the external agent to gain control over the dynamics of the particle in such a highly non-Markovian system. In Chapter 5, we propose an experimentally amenable approach to stochastic resetting. Although the idea of resetting being a subject to extensive studies, a majority of them (along with the models considered in the previous section) consider resetting to be an instantaneous event. In reality, a resetting event is space-time coupled so that a finite time is consumed when the particle heads back to the starting position. We propose a universal framework of stochastic resetting where the agent returns to the starting position in a space-time coupled non-instantaneous fashion. By considering the paradigmatic example of diffusion, we show that a stochastic component in the return motion of the particle can indeed enhance the search efficiency in a FP process. Furthermore, we corroborate our theory with experimental parameters and identify suitable parameter regimes where stochastic return can indeed be beneficial than the classical instantaneous return. By adding a drift to the particle dynamics, we also show that stochastic return can outperform the underlying reset-free process as well as the optimal instantaneous return protocol. In summary, we demonstrate that search with stochastic returns provides not only a physically realizable approach to resetting but also can be more advantageous. In Chapter 6, we propose a novel approach to the collective search process optimization based on threshold resetting. Under this set-up, the system resets only when any one of many searchers hits a pre-defined threshold value. Although the resetting epochs are still stochastic in time, it is now governed by some space-dependent threshold, leading to an event-driven process resetting strategy. We explicitly analyse the effect of such a resetting strategy in a multi-agent search process. Note that all the search agents are reset to the initial position at each resetting epoch, which introduces long-range correlations among the searchers. We show that the mean FPT of the process shows rich optimisation features with respect to the threshold distance and the number of searchers. We also quantify the cost of maintaining such a threshold-dependent search process and find an interesting optimisation of the cost function. Despite the extensive progress achieved through the studies presented in this thesis, stochastic resetting continues to be a fertile ground for exploration. Its remarkable ability to induce nonequilibrium steady states, optimise search and reaction efficiencies, and generate novel dynamical correlations underscores its fundamental and applied importance. Future studies could explore resetting in systems with spatial or temporal disorder, active matter, and biological networks, where feedback-controlled or adaptive resetting rules may yield richer dynamics. Moreover, extending resetting concepts to quantum systems, machine learning algorithms, or complex supply-chain models could open exciting interdisciplinary avenues. The insights obtained here not only advance our understanding of resetting as a universal control mechanism but also lay the groundwork for its potential exploitation in experimental and technological contexts—from microscale transport processes to large-scale optimisation problems.

An Analytic Study of the Irreducibility, Monogeniety, and Squarefreeness of Certain Polynomials [HBNI Th279]

2025-01-01T00:00:00Z

An Analytic Study of the Irreducibility, Monogeniety, and Squarefreeness of Certain Polynomials [HBNI Th279] Arunabha, Mukhopadhyay This thesis provides a study of problems related to the irreducibility and arithmetic properties of certain families of polynomials. In particular, we emphasize generalized ω-Laguarre polynomials and discriminants of a special class of polynomials. We use some classical analytic methods to approach these problems. The work is divided into two parts. In the first part we establish some results on the irreducibility of generalized ω-Laguerre polynomi- als. The principal tools we applied here are the notion of ω-Newton polygon introduced by Ø. Ore [68] and a generalized version of a lemma of M. Filaseta [19], together with some fundamental results from analytic number theory and the theory of Diophantine equations. The second part of the thesis is based on a quantitative estimate in terms of degree and coe!cients for the number of distinct squarefree parts of discriminants of the monic irreducible polynomials tn +c(atk +b)m → Z[t] of degree n. We study these problems in this part and obtain lower bounds for such quantities, using the square sieve method of D. R. Heath-Brown [28]. Furthermore, assuming the abc- conjecture for number fields, we derive a lower bound for the number of polynomials tn + c(atk + b)m → Z[t] that are monogenic with non-squarefree discriminants or have Galois group Sn . Finally, we conclude the thesis by posing some open problems related to the topics discussed above.