The Standard Model (SM), crowned in 2012 with the discovery of the Higgs boson, exhibits remarkable predictive power. However, several phenomena remain unexplained and evidence for physics beyond the SM continues to emerge. The Higgs boson appears at the center of many of these pressing issues, making its study one of the top priorities at the Large Hadron Collider (LHC). To extend its discovery potential, the LHC will undergo a major upgrade that will bring a ten-fold increase in integrated luminosity and increase the center-of-mass energy to $14 \TeV$. Extracting relevant physics in these unprecedented extreme conditions will require an upgrade of the detector and trigger system, as well as innovative analysis techniques to enhance signal-to-background discrimination. The research presented in this work followed these new directions and challenges on two parallel fronts, with the shared goal of improving our understanding of the scalar sector, and with a common focus on the development of new machine learning methods.On one front, this work contributed to a search for new heavy resonances decaying to two SM bosons (using the full Run 2 ATLAS dataset). Models that predict such particles are often interpreted in the context of two general frameworks -- the Heavy Vector Triplet and the two-Higgs-doublet models -- and address important open questions related to the Higgs sector: the naturalness problem and the possibility of an extended scalar sector. In addition, this work presents the development of a new multi-class deep neural network (DNN) jet tagger strategy to compete with traditional analysis techniques. The development of the tagger as a standalone tool, as well as the deployment within the analysis workflow to improve analysis sensitivity are presented.On the other front, this work made several contributions to the High-Luminosity LHC upgrade of the ATLAS hardware-based trigger. These started from the development of the software simulation framework for trigger performance studies, and proceeded to focus on the development of new jet triggers, targeting in particular $HH \to b \bar{b}b \bar{b}$, an important signature for the measurement of the Higgs self-coupling. This work presents the development, benchmarking, and preliminary firmware simulation of a new jet reconstruction and triggering strategy, as well as the development and performance of a new DNN for pileup mitigation, with both algorithms designed for deployment on fast FPGA hardware.
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