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A group of nine mathematicians has proved the geometric Langlands conjecture, a key component of one of the most sweeping paradigms in modern mathematics. The proof represents the culmination of three decades of effort. The Langlands program, originated by Robert Langlands in the 1960s, is a vast generalization of Fourier analysis, a far-reaching framework in which complex waves are expressed in terms of smoothly oscillating sine waves. The Langlands program holds sway in three separate areas of mathematics: number theory, geometry and something called function fields.

In this third and final article in a series on philosophy and science, we look at how modern science shows the validity of dialectical processes and how this can help guide science. The article touches on the idea that consciousness might be an emergent property of the brain's complexity and the chemical interactions between neurons. This perspective places consciousness within a materialist and dialectical framework, suggesting that it arises from the material conditions and interactions within the brain.

In a remarkable discovery that could reshape our understanding of Earth's history and the origins of life, scientists have found evidence of oxygen production in the lightless depths of the Pacific Ocean. This finding, published in Nature Geoscience, challenges the long-held belief that all Earth's oxygen is generated through photosynthesis.