In the late 19th century, Georg Cantor believed his new theory could help the Church understand the infinite nature of the divine. It might have escaped lay people at the time, but for some observers the ascension of Leo XIV as head of the Catholic Church this year was a reminder that the last time a Pope Leo sat in St. Peter's Chair in the Vatican, from 1878 to 1903, the modern view of infinity was born. Georg Cantor's completely original "naïve" set theory caused both revolution and revolt in mathematical circles, with some embracing his ideas and others rejecting them. Cantor was deeply disappointed with the negative reactions, of course, but never with his own ideas. Because he held firm to the belief that he had a main line to the absolute--that his ideas came direct from (the divine intellect).

Large Language Models (LLMs) have achieved human-level proficiency across diverse tasks, but their ability to perform rigorous mathematical problem solving remains an open challenge. In this work, we investigate a fundamental yet computationally intractable problem: determining whether a given multivariate polynomial is nonnegative. This problem, closely related to Hilbert's Seventeenth Problem, plays a crucial role in global polynomial optimization and has applications in various fields. First, we introduce SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials, along with expert-designed reasoning instructions based on five progressively challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that without structured guidance, all models perform only slightly above the random guess baseline 50%. However, high-quality reasoning instructions significantly improve accuracy, boosting performance up to 81%. Furthermore, our 7B model, SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3 and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation time needed for letters, respectively. Our findings highlight the potential of LLMs to push the boundaries of mathematical reasoning and tackle NP-hard problems.

We study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These metrics originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.

One of the most fundamental conundrums in the philosophy of mathematics is the question of whether mathematics was discovered by humans or invented by them. On one hand, it seems hard to argue that highly sophisticated mathematical objects, such as inaccessible cardinals, were discovered. On the other hand, as Albert Einstein asked, "How can it be that mathematics, being after all a product of human thought, which is independent of experience, is so admirably appropriate to the objects of reality?" The 19th century mathematician Leopold Kronecker offered a compromise, saying "God created the integers, all else is the work of man." So let us consider the natural numbers.

Recently, linear computed tomography (LCT) systems have actively attracted attention. To weaken projection truncation and image the region of interest (ROI) for LCT, the backprojection filtration (BPF) algorithm is an effective solution. However, in BPF for LCT, it is difficult to achieve stable interior reconstruction, and for differentiated backprojection (DBP) images of LCT, multiple rotation-finite inversion of Hilbert transform (Hilbert filtering)-inverse rotation operations will blur the image. To satisfy multiple reconstruction scenarios for LCT, including interior ROI, complete object, and exterior region beyond field-of-view (FOV), and avoid the rotation operations of Hilbert filtering, we propose two types of reconstruction architectures. The first overlays multiple DBP images to obtain a complete DBP image, then uses a network to learn the overlying Hilbert filtering function, referred to as the Overlay-Single Network (OSNet). The second uses multiple networks to train different directional Hilbert filtering models for DBP images of multiple linear scannings, respectively, and then overlays the reconstructed results, i.e., Multiple Networks Overlaying (MNetO). In two architectures, we introduce a Swin Transformer (ST) block to the generator of pix2pixGAN to extract both local and global features from DBP images at the same time. We investigate two architectures from different networks, FOV sizes, pixel sizes, number of projections, geometric magnification, and processing time. Experimental results show that two architectures can both recover images. OSNet outperforms BPF in various scenarios. For the different networks, ST-pix2pixGAN is superior to pix2pixGAN and CycleGAN. MNetO exhibits a few artifacts due to the differences among the multiple models, but any one of its models is suitable for imaging the exterior edge in a certain direction.

Data sets of multivariate normal distributions abound in many scientific areas like diffusion tensor imaging, structure tensor computer vision, radar signal processing, machine learning, just to name a few. In order to process those normal data sets for downstream tasks like filtering, classification or clustering, one needs to define proper notions of dissimilarities between normals and paths joining them. The Fisher-Rao distance defined as the Riemannian geodesic distance induced by the Fisher information metric is such a principled metric distance which however is not known in closed-form excepts for a few particular cases. In this work, we first report a fast and robust method to approximate arbitrarily finely the Fisher-Rao distance between multivariate normal distributions. Second, we introduce a class of distances based on diffeomorphic embeddings of the normal manifold into a submanifold of the higher-dimensional symmetric positive-definite cone corresponding to the manifold of centered normal distributions. We show that the projective Hilbert distance on the cone yields a metric on the embedded normal submanifold and we pullback that cone distance with its associated straight line Hilbert cone geodesics to obtain a distance and smooth paths between normal distributions. Compared to the Fisher-Rao distance approximation, the pullback Hilbert cone distance is computationally light since it requires to compute only the extreme minimal and maximal eigenvalues of matrices. Finally, we show how to use those distances in clustering tasks.

Introducing--artificial intelligence--(AI) to sales training and coaching can provide a more individualized learning experience that can scale across the organization, according to Gartner, Inc. Creating a high-performing sales organization is difficult with traditional training and coaching technology as coaching content and recommendations are generally delivered by role to the sales organization and do not account for individuals-- learning styles. The use of complex--machine learning--algorithms and AI can guide reps and sales managers with recommendations for training and coaching based on their learning style. These technologies utilize branching, a method to guide an individual--s learning through a module based on responses, as well as adaptive learning, where the system directs the learner to appropriate training or coaching based on their interaction with the system. In a--Gartner survey--of organizations that are piloting or deploying AI technologies, 61% of respondents reported the resulting value delivered to the organization as significant. When asked how AI will improve their sales organization, respondents cited increased efficiency, cost reduction and improved revenue streams.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation. We consider discrete as well as continuous distributions, proving convergence rates of the proposed algorithm in both settings. Key elements of our analysis are a new result showing that the Sinkhorn divergence on compact domains has Lipschitz continuous gradient with respect to the Total Variation and a characterization of the sample complexity of Sinkhorn potentials. Experiments validate the effectiveness of our method in practice.

No one with any experience in enterprise IT would suggest that technology itself can boost the bottom line. Emerging AI sales technology, however, can provide a shortcut to success by delivering more leads and making more reliable predictions than sales reps can on their own. Today's sales teams rely on numerous tools to fill out the entire sales value chain, with large organizations reporting the use of 60 or more sales technologies, said Tad Travis, Gartner research director, in an October 2018 panel webinar in which analysts discussed emerging trends in sales technology. Sales force automation (SFA) handles the basics of sales execution programs, like managing accounts, deals, forecasts, sales activity recording and lead management. Then, there are components that integrate with those, such as configure, price, quote tools, master data management, marketing automation and partner relationship management tools.

Long before robots could run or cars could drive themselves, mathematicians contemplated a simple mathematical question. They figured it out, then laid it to rest--with no way of knowing that the object of their mathematical curiosity would feature in machines of the far-off future. Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences. The future is now here. As a result of new work by Amir Ali Ahmadi and Anirudha Majumdar of Princeton University, a classical problem from pure mathematics is poised to provide iron-clad proof that drone aircraft and autonomous cars won't crash into trees or veer into oncoming traffic.