https://dspace.imsc.res.in/xmlui/handle/123456789/2 IMSc Theses/ Dissertations 2026-06-24T21:42:37Z https://dspace.imsc.res.in/xmlui/handle/123456789/921 Problems on Hilbert schemes and quiver bundles [HBNI Th280] Saurav Holme Choudhury This thesis is divided into two distinct projects. (I) Stratified Bundles on Hilbert Scheme of Points on a Surface: Let k be an algebraically closed field of characteristic p>3, and let S be a smooth projective surface over k with a k-rational point x. For n≥2, let S [n] denote the Hilbert scheme of n points on S. We compute the fundamental group scheme π alg (S [n] , n ~ x ​ ), defined by the Tannakian category of stratified bundles on S [n] . (II) Criteria for Rationality of Moduli of Chains: Let X be a compact Riemann surface of genus 2. We study the birational geometry of the moduli of holomorphic chains of type t on X, which are stable with respect to a fixed parameter θ. For suitable t and θ, we establish criteria for the rationality of these moduli spaces. https://dspace.imsc.res.in/xmlui/handle/123456789/920 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. 2025-01-01T00:00:00Z https://dspace.imsc.res.in/xmlui/handle/123456789/919 Utilizing Optimal Transport Theory to Model Peptide Conformational Distributions and Address the Levinthal Problem [HBNI Th275] Vigneshwaran K Peptide conformation studies are essential due to their role in biological functions like cell signaling and drug design, as well as their importance in protein structure prediction. Peptides form secondary structures such as alpha helices and beta hairpins, which can serve as building blocks for predicting three-dimensional protein structures. However, peptides exhibit structural flexibility, adopting a range of conformations, with only specific low-energy conformations being bioactive for particular functions. Constructing conformational distributions for longer peptides is challenging due to limited data from experimental sources like the Protein Data Bank (PDB), which mainly provides information for shorter peptides like dipeptides and tripeptides. In this thesis, we address this challenge by using optimal transport techniques to construct conformational distributions for longer peptides. Starting with dipeptide distributions, we develop a method to generate tetrapeptide conformational distributions by minimizing the expectation value of interaction energy functions. Applying this approach to tetrapeptides composed of alanine and glycine reveals preferences for right-handed alpha helices in alanine-rich sequences (e.g., AAAA, AAAG) and beta turns in glycine-dominated ones (e.g., GGGG, GAGG). Extending this method recursively, we generate conformational probabilities for longer peptides, enabling efficient prediction of their structural behavior. This approach provides an innovative solution for exploring peptide flexibility and bioactive conformations. 2025-01-01T00:00:00Z https://dspace.imsc.res.in/xmlui/handle/123456789/918 Quantified CDCL and Dependency Schemes: A proof-theoretic study [HBNI Th278] Abhimanyu Choudhury Quantified Conflict Driven Clause Learning (QCDCL) is one of the main approaches to solving Quantified Boolean Formulas (QBF). Cube-learning is employed in this approach to ensure that true formulas can be verified. Dependency Schemes help to detect spurious dependencies that are implied by the variable ordering in the quantifier prefix of QBFs but are not essential for constructing (counter)models. This detection can provably shorten refutations in specific proof systems, and is expected to speed up runs of QBF solvers. The simplest underlying proof system QCDCL [BB23a], formalises the reasoning in the QCDCL approach on false formulas, when neither cube-learning nor dependency schemes is used. The work of [BPB24] further incorporates cube-learning. This thesis is the first work that incorporates the dependency scheme heuristic in the QCDCL proof system. The usage of dependency schemes in QCDCL proof system with and with- out cube-learning are formalised and these new family of systems, the D1 + QCDCLORD (ClausePol, CubePol) proof systems, which incorporates dependency schemes into the proof system, and show it to be sound and complete. When the decisions are restricted to follow level order, but dependency schemes are used in propagation and learning, in conjunction with cube-learning, the resulting proof systems using the dependency schemes Dstd and Drrs are investigated in detail and their relative strengths are analysed. 2026-01-01T00:00:00Z