Attention layers, as commonly used in transformers, form the backbone of modern deep learning, yet there is no mathematical description of their benefits and deficiencies as compared with other architectures. In this work we establish both positive and negative results on the representation power of attention layers, with a focus on intrinsic complexity parameters such as width, depth, and embedding dimension. On the positive side, we present a sparse averaging task, where recurrent networks and feedforward networks all have complexity scaling polynomially in the input size, whereas transformers scale merely logarithmically in the input size; furthermore, we use the same construction to show the necessity and role of a large embedding dimension in a transformer. On the negative side, we present a triple detection task, where attention layers in turn have complexity scaling linearly in the input size; as this scenario seems rare in practice, we also present natural variants that can be efficiently solved by attention layers. The proof techniques emphasize the value of communication complexity in the analysis of transformers and related models, and the role of sparse averaging as a prototypical attention task, which even finds use in the analysis of triple detection.

Large Language Models (LLMs) have demonstrated impressive capabilities in natural language and code generation, and are increasingly deployed as automatic judges of model outputs and learning activities. Yet, their behavior on structured tasks such as predicting the difficulty of competitive programming problems remains under-explored. We conduct a systematic comparison of GPT-4o, used purely as a natural-language difficulty assessor, against an interpretable Light-GBM ensemble trained on explicit numeric and textual features. On a dataset of 1,825 LeetCode problems labeled Easy, Medium, or Hard, LightGBM attains 86% accuracy, whereas GPT-4o reaches only 37.75%. Detailed analyses, including confusion matrices and SHAP-based interpretability, show that numeric constraints -- such as input size limits and acceptance rates -- play a crucial role in separating Hard problems from easier ones. By contrast, GPT-4o often overlooks these cues and exhibits a strong bias toward simpler categories. We further probe GPT-4o through a synthetic Hard-problem generation protocol. Surprisingly, GPT-4o labels almost all of its own synthetic Hard problems as Medium, contradicting its tendency to downgrade real Hard problems to Easy. Our findings connect to recent work on LLMs-as-judges and automatic difficulty estimation in programming and education, and highlight concrete failure modes that must be addressed before LLM-based judges can be considered trustworthy in competitive programming, educational platforms, or reinforcement-learning pipelines.

Scientific applications continue to rely on legacy Fortran codebases originally developed for homogeneous, CPU-based systems. As High-Performance Computing (HPC) shifts toward heterogeneous GPU-accelerated architectures, many accelerators lack native Fortran bindings, creating an urgent need to modernize legacy codes for portability. Frameworks like Kokkos provide performance portability and a single-source C++ abstraction, but manual Fortran-to-Kokkos porting demands significant expertise and time. Large language models (LLMs) have shown promise in source-to-source code generation, yet their use in fully autonomous workflows for translating and optimizing parallel code remains largely unexplored, especially for performance portability across diverse hardware. This paper presents an agentic AI workflow where specialized LLM "agents" collaborate to translate, validate, compile, run, test, debug, and optimize Fortran kernels into portable Kokkos C++ programs. Results show the pipeline modernizes a range of benchmark kernels, producing performance-portable Kokkos codes across hardware partitions. Paid OpenAI models such as GPT-5 and o4-mini-high executed the workflow for only a few U.S. dollars, generating optimized codes that surpassed Fortran baselines, whereas open-source models like Llama4-Maverick often failed to yield functional codes. This work demonstrates the feasibility of agentic AI for Fortran-to-Kokkos transformation and offers a pathway for autonomously modernizing legacy scientific applications to run portably and efficiently on diverse supercomputers. It further highlights the potential of LLM-driven agentic systems to perform structured, domain-specific reasoning tasks in scientific and systems-oriented applications.

We consider the problem of certified robustness for sequence classification against edit distance perturbations. Naturally occurring inputs of varying lengths (e.g., sentences in natural language processing tasks) present a challenge to current methods that employ fixed-rate deletion mechanisms and lead to suboptimal performance. To this end, we introduce AdaptDel methods with adaptable deletion rates that dynamically adjust based on input properties. We extend the theoretical framework of randomized smoothing to variable-rate deletion, ensuring sound certification with respect to edit distance. We achieve strong empirical results in natural language tasks, observing up to 30 orders of magnitude improvement to median cardinality of the certified region, over state-of-the-art certifications.