Scalar Inert models are well motivated as they provide the much needed dark matter and make the electroweak vacuum stable. These scalar extensions are popular to counter the negative effects of the fermionic extensions viz. Seesaw scenarios, and the larger Higgs portal couplings also ensure the first order electroweak phase transitions. This further motivates the electroweak baryogenesis and that produces gravitational waves. However, an interesting phenomenon happens if we see these models as an extension of $\Phi^4$ theory. In $\Phi^4$ theory, for every odd-loop of its beta function hits the Landau pole (LP) and every-even-loop beta-function gives rise to fixed point (FP). The Standard Model (SM) breaks this pattern due to restriction from the electroweak measurements along with constraints from Gauge and Yukawa sectors. Nevertheless, when we extend the SM with an inert, singlet, doublet or triplet, the Fixed points may appear depending on the parameter space at two-loop(verified). We notice that in one hand the self couplings and terms without residual phases enhance such appearance of FP, the terms with residual phases oth the other hand decrease such possibility. Finally, we put bounds from perturbative unitary on these scalar quartic couplings at one- and two-loop levels. Ref: 2504.18086 [hep-ph] ( will appear in PRD)

My lecture will focus on a very classical subject: Can a non-trivial solution to a PDE vanish with all its derivatives at a point? I will start with a brief historic overview. ThenI will talk about a new sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. I will then show that an application of such an estimate for real-analytic leading coefficients leads to a novel parabolic Donnelly-Fefferman type nodal set estimate. This is based on joint work with Vedansh Arya and Nicola Garofalo.

The IMSc Foundation Series 2025 in Theoretical Physics is a week-long academic initiative organized by the Outreach Team at the Institute of Mathematical Sciences (IMSc), Chennai. The program is scheduled from 23rd to 27th June 2025 and will offer three intensive short-term courses at the MSc Physics level tailored for early-stage physics students from Chennai and nearby regions. The courses will be delivered by renowned academicians from IMSc Chennai, aiming to bridge the gap between coursework and research. Through these engaging lectures, participants will gain insights into the Master's level topics from a research perspective. Our goal is to inspire the next generation of physicists and encourage them to pursue careers in academic research. https://indico.imsc.res.in/e/fstp2025

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