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We compute the $\text{PSL}(2,\mathbb{C})$ Chern-Simons partition function of a closed 3-manifold obtained from Dehn fillings of the link complement $\mathbf S^3\backslash {\mathcal{L}}$, where $\mathcal{L}=\mathcal{K}# H$ is the connected sum of the knot $\mathcal {K}$ with the Hopf link $H$. Motivated by our earlier work on topological entanglement and the reduced density matrix $\sigma$ for such link complements, we wanted to determine a choice of Dehn filling so that the trace of the matrix $\sigma$ becomes equal to the $\text{PSL}(2,\mathbb{C})$ partition function of the closed 3-manifold. We use the SnapPy program and numerical techniques to show this equivalence up to the leading order. We have given explicit results for all hyperbolic knots $\mathcal{K}$ up to six crossings., Comment: 25 pages, 5 figures, 5 tables

Reward allocation, also known as the credit assignment problem, has been an important topic in economics, engineering, and machine learning. An important concept in reward allocation is the core, which is the set of stable allocations where no agent has the motivation to deviate from the grand coalition. In previous works, computing the core requires either knowledge of the reward function in deterministic games or the reward distribution in stochastic games. However, this is unrealistic, as the reward function or distribution is often only partially known and may be subject to uncertainty. In this paper, we consider the core learning problem in stochastic cooperative games, where the reward distribution is unknown. Our goal is to learn the expected core, that is, the set of allocations that are stable in expectation, given an oracle that returns a stochastic reward for an enquired coalition each round. Within the class of strictly convex games, we present an algorithm named \texttt{Common-Points-Picking} that returns a point in the expected core given a polynomial number of samples, with high probability. To analyse the algorithm, we develop a new extension of the separation hyperplane theorem for multiple convex sets.

We study data corruption robustness for reinforcement learning with human feedback (RLHF) in an offline setting. Given an offline dataset of pairs of trajectories along with feedback about human preferences, an $\varepsilon$-fraction of the pairs is corrupted (e.g., feedback flipped or trajectory features manipulated), capturing an adversarial attack or noisy human preferences. We aim to design algorithms that identify a near-optimal policy from the corrupted data, with provable guarantees. Existing theoretical works have separately studied the settings of corruption robust RL (learning from scalar rewards directly under corruption) and offline RLHF (learning from human feedback without corruption); however, they are inapplicable to our problem of dealing with corrupted data in offline RLHF setting. To this end, we design novel corruption robust offline RLHF methods under various assumptions on the coverage of the data-generating distributions. At a high level, our methodology robustifies an offline RLHF framework by first learning a reward model along with confidence sets and then learning a pessimistic optimal policy over the confidence set. Our key insight is that learning optimal policy can be done by leveraging an offline corruption-robust RL oracle in different ways (e.g., zero-order oracle or first-order oracle), depending on the data coverage assumptions. To our knowledge, ours is the first work that provides provable corruption robust offline RLHF methods.

In 1998, Wang constructed an ergodic action of the compact quantum group $U_n^+$ (free unitary quantum group) on the Cuntz algebra $\mathcal{O}_n$. Later, in 2018-2019, S. Joardar and A. Mandal showed that the quantum automorphism group of the Cuntz algebra $\mathcal{O}_{n}$ (as a graph $C^*$-algebra) is $U_n^+$ in the category introduced by them. In this article, we explore the quantum symmetry of the direct sum of Cuntz algebras viewing them as a graph $C^*$-algebra in the category as mentioned before. It has been shown that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras $\{\mathcal{O}_{n_i}\}_{i=1}^{m}$ is ${U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}$ for distinct $n_i$'s, i.e. if $ L_{n_i}$ (the graph contains $n_i$ loops based at a single vertex) is the underlying graph of $\mathcal{O}_{n_i}$, then \begin{equation*} Q_{\tau}^{Lin}(\sqcup_{i=1}^{m} ~ L_{n_i}) \cong *_{i=1}^{m} ~~ Q_{\tau}^{Lin}(L_{n_i}) \cong {U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}. \end{equation*} Moreover, the quantum symmetry of the direct sum of $m$ copies of isomorphic Cuntz algebra $\mathcal{O}_n$ (whose underlying graph is $L_n$) is $U_n^+ \wr_* S_m^+,$ i.e. \begin{equation*} Q_{\tau}^{Lin}(\sqcup_{i=1}^{m} ~ L_n) \cong Q_{\tau}^{Lin}(L_n) \wr_* S_m^+ \cong U_n^+ \wr_* S_m^+. \end{equation*} On the other hand, it is known that the quantum automorphism group of $m$ disjoint copies of a simple, connected graph $\Gamma$ is isomorphic to the free wreath product of the quantum automorphism group of $\Gamma$ with $S_m^+$. Though an analoguous result is true for $\mathcal{O}_{n}$ (as a graph $C^*$-algebra), we have provided a counter-example to show that this result is not in general true for an arbitrary graph $C^*$-algebra., Comment: A new remark has been added. Few stylistic changes have been done. Comments are welcome

We investigate the black hole search problem by a set of mobile agents in a dynamic torus. Black hole is defined to be a dangerous stationary node which has the capability to destroy any number of incoming agents without leaving any trace of its existence. A torus of size $n\times m$ ($3\leq n \leq m$) is a collection of $n$ row rings and $m$ column rings, and the dynamicity is such that each ring is considered to be 1-interval connected, i.e., in other words at most one edge can be missing from each ring at any round. The parameters which define the efficiency of any black hole search algorithm are: the number of agents and the number of rounds (or \textit{time}) for termination. We consider two initial configurations of mobile agents: first, the agents are co-located and second, the agents are scattered. In each case, we establish lower and upper bounds on the number of agents and on the amount of time required to solve the black hole search problem.

Let $ K$ be a field admitting a Galois extension $L$ of degree $n$, denoting the Galois group as $G = \gal(L/K)$. Our focus lies on the space $\sym_K(L)$ of symmetric $K$-bilinear forms on $L$. We establish a decomposition of $\sym_K(L)$ into direct sum of $K$-subspaces $A^{\sigma_i}$, where $\sigma_i \in G$. Notably, these subspaces $ A^{\sigma_i}$ exhibit nice constant rank properties. The central contribution of this paper is a decomposition theorem for $\sym_K(L)$, revealing a direct sum of $\frac{(n+1)}{2}$ constant rank $n$-subspaces, each having dimension of $n$. This holds particularly when $G$ is cyclic, represented as $G = \gal(L/K) = \langle\sigma\rangle$. For cyclic extensions of even degree $n = 2m$, we present slightly less precise but analogous results. In this scenario, we enhance and enrich these constant results and show that, the component $ A^{\sigma}$ often decomposes directly into a constant rank subspaces. Remarkably, this decomposition is universally valid when $-1 \notin L^{2}$. Consequently, we derive a decomposition of $\sym_K(L)$ into subspaces of constant rank under several situations. Moreover, leveraging these decompositions, we investigate the maximum dimension of an $n$-subspace inside $M(n,K)$ and $S(n,K)$ for various field $K$ where $M(n,K)$ and $ S(n,K)$ denote the vector spaces $(n \times n)$ matrices and symmetric matrices over $K$, respectively., Comment: 15 pages.All comments are welcome

In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, have been studied at roots of unity. In this context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed., Comment: 22 pages. All comments are welcome

The occurrence of thermal transport phenomena is widespread, exerting a pivotal influence on the functionality of diverse electronic and thermo-electric energy-conversion devices. The traditional first-principles theory governing the thermal and thermodynamic characteristics of insulators relies on the perturbative treatment of interatomic potential and ad-hoc displacement of atoms within supercells. However, the limitations of these approaches for highly anharmonic and weakly bonded materials, along with discrepancies arising from not considering explicit finite temperature effects, highlight the necessity for a well-defined quasiparticle approach to the lattice vibrations. To address these limitations, we present a comprehensive numerical framework in this study, designed to compute the thermal and thermodynamic characteristics of crystalline semiconductors and insulators. The self-consistent phonon renormalization method we have devised reveals phonons as quasiparticles, diverging from their conventional characterization as bare normal modes of lattice vibration. The extension of the renormalization impact to interatomic force constants (IFCs) of third and fourth orders is also integrated and demonstrated. For the comprehensive physical insights, we employed an iterative solution of the Peierls-Boltzmann transport equation (PBTE) to determine thermal conductivity and carry out Helmholtz free energy calculations, encompassing anharmonicity effects up to the fourth order. In this study, we utilize our numerical framework to showcase its applicability through an examination of phonon dispersion, phonon linewidth, anharmonic phonon scattering, and temperature-dependent lattice thermal conductivity in both highly anharmonic materials (NaCl and AgI) and weakly anharmonic materials (cBN and 3C-SiC).

We present the results from spectral and timing study of the Z source GX 340+0 using AstroSat's SXT and LAXPC data. During the observation the source traced out the complete Z-track, allowing for the spectral evolution study of the Horizontal, Normal and Flaring branches (HB, NB and FB) as well as the hard and soft apexes (HA and SA). The spectra are better and more physically described by a blackbody component and a hot Comptonizing corona with a varying covering fraction, rather than one having a disc component. Along the track, the Comptonized flux (as well as the covering fraction) monotonically decreases. It is the blackbody component (both the temperature and radius) which varies non-monotonically and hence gives rise to the Z-track behaviour. Rapid timing study reveals a prominent Quasi-periodic Oscillation (QPO) at ~ 50 Hz at the HB, HA and upper NB, while a QPO at ~ 6 Hz is seen for the other branches. The fractional r.m.s of the QPOs increase with energy and exhibit soft lags in all branches except SA and FB., Comment: Accepted for publication in MNRAS

Our aim is to study certain algebraic properties of the ring $C(X)_\mathcal{P}$ of real-valued functions on $X$ whose closure of discontinuity set is in an ideal of closed sets. We characterize $\mathcal{P}P$-spaces using $z$-ideals and essential ideals of $C(X)_\mathcal{P}$ and also almost $\mathcal{P}P$-spaces using $z^0$-ideals of $C(X)_\mathcal{P}$ and a topology finer than the original topology on $X$. We deduce that each maximal ideal of $C(X)_F$ \cite{GGT2018} (resp. $T'(X)$ \cite{A2010}) is a $z^0$-ideal. We establish that the notions of clean ring, weakly clean ring, semiclean ring, almost clean ring and exchange ring coincide in the ring $C(X)_\mathcal{P}$. End of this paper, we also characterize $\mathcal{P}P$-spaces and almost $\mathcal{P}P$-spaces using certain ideals having depth zero. We exhibit a condition on $\mathcal{P}$ under which prime and essential ideals of $C(X)_\mathcal{P}$ have depth zero.

A chiral symmetric Su-Schrieffer-Heeger (SSH) chain features topological end states in one of its dimerized configurations. Those mid-gap zero energy states show interesting modifications upon a periodic tuning of the hopping modulations. Besides, more and more in-gap end modes appear at nonzero energies for further partitioning of the Brillouin zone (BZ) due to increased hopping periodicity. The new topological phases are identified with a detailed analysis of the topological invariants namely, winding number and Zak phases. The spectra and topology of these systems with periodically modulated hopping are studied also in the presence of a single static domain wall, separating two topologically inequivalent dimerized structures. The domain wall causes additional in-gap modes in the spectrum as well as zero energy domain wall solitonic states for specific hopping periodicities. We also study the effect of disorder, particularly the chirality breaking onsite ones, on the edge and domain wall states. Other than the SSH type we also consider random, Rice-Mele or AI type disorder to do a comparative analysis of the evolution of chirality and zero energy states as the strength of disorder and hopping periodicity is varied. Our findings can add important feedback in utilizing topological phases in various fields including quantum computations while the results can be easily verified in a cold atom set up within optical lattices., Comment: 14 pages

In this thesis, we have studied the evolution of superconductivity in amorphous Molybdenum Germanium ($\textit{a}$-MoGe) thin films. The work can be broken down into two parts. In the first part, we investigate the effect of decreasing thickness on the suppression of superconductivity in $\textit{a}$-MoGe thin films. Thick $\textit{a}$-MoGe thin film is a typical type-II superconductor and follows the conventional Bardeen-Cooper-Schrieffer (BCS) equation. Conventionally, it is believed that decreasing thickness will decrease the effective attractive pairing interaction because of the gradual loss of screening which holds true for large thicknesses in $\textit{a}$-MoGe. But for lower thicknesses, a new mechanism comes into the picture where superfluid density is suppressed making the superconductor vulnerable to phase fluctuations. This is known as the Bosonic mechanism, where superconductivity is destroyed due to the loss of phase coherence of the superconducting state even though the attractive pairing amplitude remains finite above the transition temperature. In the second part, we explore the electromagnetic response of vortices in $\textit{a}$-MoGe thin film by low-frequency two-coil mutual inductance technique. Penetration depth measured from the two-coil technique was earlier used to determine superfluid density. However in the present work, by analyzing the in-field penetration depth data with the help of a mean-field model proposed by Coffey and Clem, we have demonstrated a procedure of extraction of vortex parameters such as pinning restoring force constant or Labusch parameter, vortex lattice drag coefficient and pinning potential barrier for the thermally activated motion of vortices. The temperature variation of vortex parameters suggests the dominant effect of thermal fluctuations., Comment: 178 pages, a thesis submitted to the Tata Institute of Fundamental Research, Mumbai for the PhD degree in Physics