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<jats:title>Abstract</jats:title><jats:p>The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures<jats:sup>1</jats:sup>, most famously in the Birch and Swinnerton-Dyer conjecture<jats:sup>2</jats:sup>, a Millennium Prize Problem<jats:sup>3</jats:sup>. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups<jats:sup>4</jats:sup>. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.</jats:p>

<jats:title>Abstract</jats:title><jats:p>During the last glacial–interglacial cycle, Arctic biotas experienced substantial climatic changes, yet the nature, extent and rate of their responses are not fully understood<jats:sup>1–8</jats:sup>. Here we report a large-scale environmental DNA metagenomic study of ancient plant and mammal communities, analysing 535 permafrost and lake sediment samples from across the Arctic spanning the past 50,000 years. Furthermore, we present 1,541 contemporary plant genome assemblies that were generated as reference sequences. Our study provides several insights into the long-term dynamics of the Arctic biota at the circumpolar and regional scales. Our key findings include: (1) a relatively homogeneous steppe–tundra flora dominated the Arctic during the Last Glacial Maximum, followed by regional divergence of vegetation during the Holocene epoch; (2) certain grazing animals consistently co-occurred in space and time; (3) humans appear to have been a minor factor in driving animal distributions; (4) higher effective precipitation, as well as an increase in the proportion of wetland plants, show negative effects on animal diversity; (5) the persistence of the steppe–tundra vegetation in northern Siberia enabled the late survival of several now-extinct megafauna species, including the woolly mammoth until 3.9 ± 0.2 thousand years ago (ka) and the woolly rhinoceros until 9.8 ± 0.2 ka; and (6) phylogenetic analysis of mammoth environmental DNA reveals a previously unsampled mitochondrial lineage. Our findings highlight the power of ancient environmental metagenomics analyses to advance understanding of population histories and long-term ecological dynamics.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Interactions between the mediodorsal thalamus and the prefrontal cortex are critical for cognition. Studies in humans indicate that these interactions may resolve uncertainty in decision-making<jats:sup>1</jats:sup>, but the precise mechanisms are unknown. Here we identify two distinct mediodorsal projections to the prefrontal cortex that have complementary mechanistic roles in decision-making under uncertainty. Specifically, we found that a dopamine receptor (D2)-expressing projection amplifies prefrontal signals when task inputs are sparse and a kainate receptor (GRIK4) expressing-projection suppresses prefrontal noise when task inputs are dense but conflicting. Collectively, our data suggest that there are distinct brain mechanisms for handling uncertainty due to low signals versus uncertainty due to high noise, and provide a mechanistic entry point for correcting decision-making abnormalities in disorders that have a prominent prefrontal component<jats:sup>2–6</jats:sup>.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Symbiotic N<jats:sub>2</jats:sub>-fixing microorganisms have a crucial role in the assimilation of nitrogen by eukaryotes in nitrogen-limited environments<jats:sup>1–3</jats:sup>. Particularly among land plants, N<jats:sub>2</jats:sub>-fixing symbionts occur in a variety of distantly related plant lineages and often involve an intimate association between host and symbiont<jats:sup>2,4</jats:sup>. Descriptions of such intimate symbioses are lacking for seagrasses, which evolved around 100 million years ago from terrestrial flowering plants that migrated back to the sea<jats:sup>5</jats:sup>. Here we describe an N<jats:sub>2</jats:sub>-fixing symbiont, ‘<jats:italic>Candidatus</jats:italic> Celerinatantimonas neptuna’, that lives inside seagrass root tissue, where it provides ammonia and amino acids to its host in exchange for sugars. As such, this symbiosis is reminiscent of terrestrial N<jats:sub>2</jats:sub>-fixing plant symbioses. The symbiosis between <jats:italic>Ca</jats:italic>. C. neptuna and its host <jats:italic>Posidonia oceanica</jats:italic> enables highly productive seagrass meadows to thrive in the nitrogen-limited Mediterranean Sea. Relatives of <jats:italic>Ca</jats:italic>. C. neptuna occur worldwide in coastal ecosystems, in which they may form similar symbioses with other seagrasses and saltmarsh plants. Just like N<jats:sub>2</jats:sub>-fixing microorganisms might have aided the colonization of nitrogen-poor soils by early land plants<jats:sup>6</jats:sup>, the ancestors of <jats:italic>Ca</jats:italic>. C. neptuna and its relatives probably enabled flowering plants to invade nitrogen-poor marine habitats, where they formed extremely efficient blue carbon ecosystems<jats:sup>7</jats:sup>.</jats:p>