Seminars

In this talk, we investigate the degree and cardinality of Grobner bases under various orderings. We identify a family of polynomials and a criterion for monomial orderings such that the reduced Grobner basis is double-exponential in both degree and cardinality. We also show that this criterion is satisfied by standard orderings such as lexicographic, degree lexicographic, and weighted orderings.

Human health is shaped by both genetics and lifetime environmental exposures. The exposome framework systematically captures such environmental exposures and links them to health outcomes. The chemical exposome is of special concern, as thousands of new chemicals are introduced into the global market annually, with many of them inevitably finding their way into the environment, yet their effects on human health remain poorly understood. Traditional toxicity testing relied heavily on slow and expensive animal studies, resulting in only a fraction of these chemicals being evaluated. In recent years, computational approaches have emerged as vital alternatives. They help organize toxicity data to prioritize chemicals for testing and provide mechanistic insights into their adverse effects. Therefore, in this thesis, we aim to link the chemical exposome to human health by systematically characterizing the chemical exposome and gaining mechanistic insights into chemical-induced adverse effects through diverse computational approaches. As the first objective, we focus on characterizing the chemical exposome associated with specific human health outcomes. We investigate the chemical exposome of vitiligo by curating a dedicated knowledgebase of its environmental chemical triggers. We next explore the chemical space to assess regulatory coverage, and utilize skin-specific transcrip- tomic signatures to gain mechanistic insights into the otherwise less understood etiology of this disease. Building on this, we extend our focus to endocrine disrupting chemicals (EDCs), a broader class of environmental chemicals associated with a wide range of adverse health effects. We update the Database of Endocrine Disrupting Chemicals and their Toxicity profiles (DEDuCT) by integrating data from diverse toxicological resources, providing a single window access to all EDC-relevant information. We further map these chemicals to Adverse Outcome Pathways (AOPs), and organize all compiled data into a large-scale toxicology knowledge graph, termed DEDuCT-KG, which we utilize to gain mechanistic insights into EDC-associated obesity and neurodegenerative disorders. As the second objective, we expand the scope of our computational investigation beyond human health, employing diverse network toxicology frameworks to assess the impacts of inorganic arsenic and cadmium on human and ecosystem health from a One Health perspective. We construct Aggregate Exposure Pathway (AEP) networks for the first time in the Indian context, and integrate them with stressor-AOP networks to trace complete source-to-outcome pathways. We compute Species Sensitivity Distributions (SSDs) from aquatic toxicity data to estimate hazard concentrations, and integrate them into stressor-species networks to identify particularly vulnerable species and aid in ecological risk assessment. Finally, we calculate Risk Quotients (RQs) to gauge the ecological risks associated with arsenic and cadmium in Indian rivers. Overall, the resources developed and frameworks utilized in this thesis provide results and insights that are easily accessible to the scientific community, laying the groundwork for future efforts towards a more complete characterization of the chemical exposome. The structured network toxicology investigation provides a basis for exploring the links between environmental chemical stressors and human and ecosystem health from a One Health perspective. ************* Zoom Link: Meeting ID: 907 155 2168 Passcode: 813471 *************

For a prime $p$, let $f_n(p^e)$ denote the number of unital subrings of index $p^e$ in $\mathbb Z^n$, with coordinatewise multiplication. A natural question in the study of subring zeta functions is whether, for fixed $n$ and $e$, this count is a polynomial in $p$. I show that this fails: there is no polynomial $P\in\mathbb Q[X]$ such that $f_{10}(p^{17})=P(p)$ for every sufficiently large prime $p$. The proof relates subring counts to point counts on varieties and detects a contribution from Ramanujan's weight-$12$ cusp form $\Delta$ in their cohomology. www.imsc.res.in/~anupdixit/IMSc-CMI-NT-seminar.html

In this talk, we will discuss a systematic approach to gravitational waves generated by localized sources in de Sitter spacetime, starting from quadrupolar radiation and moving toward a general multipolar framework. We will first review the geometry of de Sitter spacetime and explain how it modifies the standard flat-space description of radiation. Next, we will discuss de Sitter Teukolsky waves and demonstrate their relation, through explicit gauge transformations, to other quadrupolar solutions in the literature. We will explain how a consistent truncation of the source multipole expansion can be performed while preserving stress-tensor conservation and the generalized harmonic gauge conditions, alongside its asymptotic Bondi description. Going beyond the quadrupole, we will describe the construction of the source-generated metric through octupolar order, including its electric and magnetic ($\ell=3$) sectors. Finally, we will present recent work extending this construction to arbitrary multipole order using an exact map to an auxiliary flat-space problem. The resulting framework provides a systematic starting point for studying gravitational radiation and its observables at arbitrary multipole order in an accelerating background.

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