With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of examples so vast that they would be impossible for a single researcher to feasibly review and analyze. In this work, we use graph neural networks to investigate quiver mutation---an operation that transforms one quiver (or directed multigraph) into another---which is central to the theory of cluster algebras with deep connections to geometry, topology, and physics. In the study of cluster algebras, the question of mutation equivalence is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through a sequence of mutations? In this paper, we use graph neural networks and AI explainability techniques to independently discover mutation equivalence criteria for quivers of type $\tilde{D}$. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden representation that allows us to reconstruct known criteria from type $D$, adding to the growing evidence that modern machine learning models are capable of learning abstract and general rules from mathematical data.

In recent years, graph theoretic considerations have become increasingly important in the design of HPC interconnection topologies. One approach is to seek optimal or near-optimal families of graphs with respect to a particular graph theoretic property, such as diameter. In this work, we consider topologies which optimize the spectral gap. We study a novel HPC topology, SpectralFly, designed around the Ramanujan graph construction of Lubotzky, Phillips, and Sarnak (LPS). We show combinatorial properties, such as diameter, bisection bandwidth, average path length, and resilience to link failure, of SpectralFly topologies are better than, or comparable to, similarly constrained DragonFly, SlimFly, and BundleFly topologies. Additionally, we simulate the performance of SpectralFly on a representative sample of micro-benchmarks using the Structure Simulation Toolkit Macroscale Element Library simulator and study cost-minimizing layouts, demonstrating considerable benefit of the SpectralFly topology.

Designing efficient and scalable sparse linear algebra kernels on modern multi-GPU based HPC systems is a challenging task due to significant irregular memory references and workload imbalance across GPUs. These challenges are particularly compounded in the case of Sparse Triangular Solver (SpTRSV), which introduces additional complexity of two-dimensional computation dependencies among subsequent computation steps. Dependency information may need to be exchanged and shared among GPUs, thus warranting for efficient memory allocation, data partitioning, and workload distribution as well as fine-grained communication and synchronization support. In this work, we focus on designing algorithm for SpTRSV in a single-node, multi-GPU setting. We demonstrate that directly adopting unified memory can adversely affect the performance of SpTRSV on multi-GPU architectures, despite linking via fast interconnect like NVLinks and NVSwitches. Alternatively, we employ the latest NVSHMEM technology based on Partitioned Global Address Space programming model to enable efficient fine-grained communication and drastic synchronization overhead reduction. Furthermore, to handle workload imbalance, we propose a malleable task-pool execution model which can further enhance the utilization of GPUs. By applying these techniques, our experiments on the NVIDIA multi-GPU supernode V100-DGX-1 and DGX-2 systems demonstrate that our design can achieve an average of 3.53 × (up to 9.86 ×) speedup on a DGX-1 system and 3.66 × (up to 9.64 ×) speedup on a DGX-2 system with four GPUs over the Unified-Memory design. The comprehensive sensitivity and scalability studies also show that the proposed zero-copy SpTRSV is able to fully utilize the computing and communication resources of the multi-GPU systems.