WW-Infinity Ward Identities and Correlation Functions in the C=1C=1 Matrix Model

We explore consequences of $W$-infinity symmetry in the fermionic field theory of the $c=1$ matrix model. We derive exact Ward identities relating correlation functions of the bilocal operator. These identities can be expressed as equations satisfied by the effective action of a {\it three} dimensional theory and contain non-perturbative information about the model. We use these identities to calculate the two point function of the bilocal operator in the double scaling limit. We extract the operator whose two point correlator has a {\it single} pole at an (imaginary) integer value of the energy. We then rewrite the \winf~ charges in terms of operators in the matrix model and use this derive constraints satisfied by the partition function of the matrix model with a general time dependent potential.Comment: 17 page

Gauge Theory Formulation of the c=1c=1 Matrix Model: Symmetries and Discrete States

We present a non-relativistic fermionic field theory in 2-dimensions coupled to external gauge fields. The singlet sector of the $c=1$ matrix model corresponds to a specific external gauge field. The gauge theory is one-dimensional (time) and the space coordinate is treated as a group index. The generators of the gauge algebra are polynomials in the single particle momentum and position operators and they form the group $W^{(+)}_{1+\infty}$. There are corresponding Ward identities and residual gauge transformations that leave the external gauge fields invariant. We discuss the realization of these residual symmetries in the Minkowski time theory and conclude that the symmetries generated by the polynomial basis are not realized. We motivate and present an analytic continuation of the model which realises the group of residual symmetries. We consider the classical limit of this theory and make the correspondence with the discrete states of the $c=1$ (Euclidean time) Liouville theory. We explain the appearance of the $SL(2)$ structure in $W^{(+)}_{1+\infty}$. We also present all the Euclidean classical solutions and the classical action in the classical phase space. A possible relation of this theory to the $N=2$ string theory and also self-dual Einstein gravity in 4-dimensions is pointed out.Comment: 35 page

Genome-wide prediction of synthetic rescue mediators of resistance to targeted and immunotherapy

Most patients with advanced cancer eventually acquire resistance to targeted therapies, spurring extensive efforts to identify molecular events mediating therapy resistance. Many of these events involve synthetic rescue (SR) interactions, where the reduction in cancer cell viability caused by targeted gene inactivation is rescued by an adaptive alteration of another gene (the rescuer). Here, we perform a genome-wide in silico prediction of SR rescuer genes by analyzing tumor transcriptomics and survival data of 10,000 TCGA cancer patients. Predicted SR interactions are validated in new experimental screens. We show that SR interactions can successfully predict cancer patients\u27 response and emerging resistance. Inhibiting predicted rescuer genes sensitizes resistant cancer cells to therapies synergistically, providing initial leads for developing combinatorial approaches to overcome resistance proactively. Finally, we show that the SR analysis of melanoma patients successfully identifies known mediators of resistance to immunotherapy and predicts novel rescuers

DATA DRIVEN APPROACHES TO IDENTIFY DETERMINANTS OF HEART DISEASES AND CANCER RESISTANCE

Cancer and cardio-vascular diseases are the leading causes of death world-wide. Caused by systemic genetic and molecular disruptions in cells, these disorders are the manifestation of profound disturbance of normal cellular homeostasis. People suffering or at high risk for these disorders need early diagnosis and personalized therapeutic intervention. Successful implementation of such clinical measures can significantly improve global health. However, development of effective therapies is hindered by the challenges in identifying genetic and molecular determinants of the onset of diseases; and in cases where therapies already exist, the main challenge is to identify molecular determinants that drive resistance to the therapies. Due to the progress in sequencing technologies, the access to a large genome-wide biological data is now extended far beyond few experimental labs to the global research community. The unprecedented availability of the data has revolutionized the capabilities of computational researchers, enabling them to collaboratively address the long standing problems from many different perspectives. Likewise, this thesis tackles the two main public health related challenges using data driven approaches. Numerous association studies have been proposed to identify genomic variants that determine disease. However, their clinical utility remains limited due to their inability to distinguish causal variants from associated variants. In the presented thesis, we first propose a simple scheme that improves association studies in supervised fashion and has shown its applicability in identifying genomic regulatory variants associated with hypertension. Next, we propose a coupled Bayesian regression approach -- eQTeL, which leverages epigenetic data to estimate regulatory and gene interaction potential, and identifies combinations of regulatory genomic variants that explain the gene expression variance. On human heart data, eQTeL not only explains a significantly greater proportion of expression variance in samples, but also predicts gene expression more accurately than other methods. We demonstrate that eQTeL accurately detects causal regulatory SNPs by simulation, particularly those with small effect sizes. Using various functional data, we show that SNPs detected by eQTeL are enriched for allele-specific protein binding and histone modifications, which potentially disrupt binding of core cardiac transcription factors and are spatially proximal to their target. eQTeL SNPs capture a substantial proportion of genetic determinants of expression variance and we estimate that 58% of these SNPs are putatively causal. The challenge of identifying molecular determinants of cancer resistance so far could only be dealt with labor intensive and costly experimental studies, and in case of experimental drugs such studies are infeasible. Here we take a fundamentally different data driven approach to understand the evolving landscape of emerging resistance. We introduce a novel class of genetic interactions termed synthetic rescues (SR) in cancer, which denotes a functional interaction between two genes where a change in the activity of one vulnerable gene (which may be a target of a cancer drug) is lethal, but subsequently altered activity of its partner rescuer gene restores cell viability. Next we describe a comprehensive computational framework --termed INCISOR-- for identifying SR underlying cancer resistance. Applying INCISOR to mine The Cancer Genome Atlas (TCGA), a large collection of cancer patient data, we identified the first pan-cancer SR networks, composed of interactions common to many cancer types. We experimentally test and validate a subset of these interactions involving the master regulator gene mTOR. We find that rescuer genes become increasingly activated as breast cancer progresses, testifying to pervasive ongoing rescue processes. We show that SRs can be utilized to successfully predict patients' survival and response to the majority of current cancer drugs, and importantly, for predicting the emergence of drug resistance from the initial tumor biopsy. Our analysis suggests a potential new strategy for enhancing the effectiveness of existing cancer therapies by targeting their rescuer genes to counteract resistance. The thesis provides statistical frameworks that can harness ever increasing high throughput genomic data to address challenges in determining the molecular underpinnings of hypertension, cardiovascular disease and cancer resistance. We discover novel molecular mechanistic insights that will advance the progress in early disease prevention and personalized therapeutics. Our analyses sheds light on the fundamental biological understanding of gene regulation and interaction, and opens up exciting avenues of translational applications in risk prediction and therapeutics

Learning Expressive Prompting With Residuals for Vision Transformers

Prompt learning is an efficient approach to adapt transformers by inserting learnable set of parameters into the input and intermediate representations of a pre-trained model. In this work, we present Expressive Prompts with Residuals (EXPRES) which modifies the prompt learning paradigm specifically for effective adaptation of vision transformers (ViT). Out method constructs downstream representations via learnable ``output'' tokens, that are akin to the learned class tokens of the ViT. Further for better steering of the downstream representation processed by the frozen transformer, we introduce residual learnable tokens that are added to the output of various computations. We apply EXPRES for image classification, few shot learning, and semantic segmentation, and show our method is capable of achieving state of the art prompt tuning on 3/3 categories of the VTAB benchmark. In addition to strong performance, we observe that our approach is an order of magnitude more prompt efficient than existing visual prompting baselines. We analytically show the computational benefits of our approach over weight space adaptation techniques like finetuning. Lastly we systematically corroborate the architectural design of our method via a series of ablation experiments.Comment: Accepted at CVPR (2023

Novel symmetries in N = 2 supersymmetric quantum mechanical models

We demonstrate the existence of a novel set of discrete symmetries in the context of N = 2 supersymmetric (SUSY) quantum mechanical model with a potential function f(x) that is a generalization of the potential of the 1D SUSY harmonic oscillator. We perform the same exercise for the motion of a charged particle in the X-Y plane under the influence of a magnetic field in the Z-direction. We derive the underlying algebra of the existing continuous symmetry transformations (and corresponding conserved charges) and establish its relevance to the algebraic structures of the de Rham cohomological operators of differential geometry. We show that the discrete symmetry transformations of our present general theories correspond to the Hodge duality operation. Ultimately, we conjecture that any arbitrary N = 2 SUSY quantum mechanical system can be shown to be a tractable model for the Hodge theory.Comment: LaTeX file, 23 pages, Title and Abstract changed, Text modified, version to appear in Annals of Physic

Fermionic representation for the ferromagnetic Kondo lattice model -- diagrammatic study of spin-charge coupling effects on magnon excitations

A purely fermionic representation is introduced for the ferromagnetic Kondo lattice model which allows conventional diagrammatic tools to be employed to study correlation effects. Quantum 1/S corrections to magnon excitations are investigated using a systematic inverse-degeneracy expansion scheme which incorporates correlation effects in the form of self-energy and vertex corrections, while explicitly preserving the continuous spin-rotation symmetry. Magnon self-energy is studied in the full range of interaction strength, and shown to result in strong magnon damping and anomalous softening for zone boundary modes, which accounts for several zone-boundary anomalies observed in recent spin-wave measurements of ferromagnetic manganites.Comment: 16 pages, 9 figure