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The seminar will be about our recent thermodynamic analysis which lends support to the observation of negative absolute temperatures in spin systems. These experimental results are challenged by theorists who argue that negative temperatures violate basic thermodynamic principles. We will show that these arguments are not logically compelling and lead to inconsistencies. Our work proves that negative temperatures are hotter than any positive temperature and that it is possible to generate a Carnot cycle with 100% efficiency. According to Einstein, ‘the edifice of thermodynamics will never be overthrown.’ Would he have thought that negative absolute temperatures were inconsistent with the unassailable foundations of this edifice? Our conclusion supports consistency but different valid choices in defining entropy lead to controversy.