Speaker: Bob Robey, AMD Inc.
Title: A Challenge for our Time
Abstract: Computer precision has seen little change for two decades. As a result, computer precision has been given little attention. Now we have the emergence of very different precision on HPC and AI platforms. What does this mean for the future and what skills are needed to navigate this new world? What should future computing platforms look like? There is no single, simple answer. We need math and the scientific approach to help give direction at this important time.
Speaker: Frank Sottile, Texas A&M University
Title: Algebraic Geometry and Discrete Periodic Operators
Abstract: Spectral theory of operators on a periodic medium is a topic in mathematical physics having deep interactions with analysis. Its discretization, operators on periodic graphs, additionally communicates with discrete mathematics and algebraic geometry. Many natural aspects of discrete periodic operators are directly understood in terms of algebraic geometry and some basic questions are answered using standard techniques from algebraic geometry. My talk will explain some algebraic aspects of discrete periodic operators, illustrating how the interactions flow in both directions, to the benefit of both disciplines.
Speaker: Guo-Wei Wei, University of Georgia
Title: Transforming Artificial Intelligence Through Modern Mathematics
Abstract: Despite the tremendous success of artificial intelligence (AI) in science, engineering, and technology in the past decade, its explainability, generalizability, and reliability have been a major concern. The solution to these challenges holds the future of AI. Topological deep learning (TDL), a new paradigm in rational learning introduced by us in 2017, offers interpretable and generalized AI approaches. TDL utilizes topological data analysis (TDA), which is originally rooted in persistent homology, an algebraic topology technique. However, persistent homology has many limitations, including the lack of localization, being restricted to point cloud data, and the inability to represent non-topological information. To address these challenges, we generalized TDA to combinatoric spectral theory (e.g. Topological Laplacian and Dirac), differential topology (e.g. de Rham-Hodge theory), and geometric topology (e.g. Khovanov homology) to handle data on graphs, differentiable manifolds, and curves embedded in 3-space, respectively (see arXiv:2507.19504 for a review). To further advance AI through modern mathematical theories, we introduced commutative algebra as a new frontier in data science and machine learning.
Speaker: Ann Almgren, Lawrence Berkeley National Laboratory
Title: Adaptive Mesh Refinement: Algorithms, Applications and AI
Abstract: Adaptive mesh refinement (AMR) is a technique we use as a numerical microscope to zoom in on areas of interest, or to represent multiple length and time scales, in a single computer simulation . With more resolution we get more accurate answers and can see in more detail what is happening in the simulation, but AMR can also introduce quite a bit of complexity in the algorithm and implementation. In this talk I will describe in general terms how AMR works and how we use it effectively on today's supercomputers, and I will present examples from a wide range of applications. I will also briefly discuss how we are using AI to write and debug code as well as in orchestrating our applications.