The 2024 SPS Zone 9 + Chicago Area SIAM Student Conference

11:15 - 12:00: Dr. Carlo Graziani, Argonne National Laboratory (Website)

From Heuristics to Algorithmics: Putting Math at the Foundations of AI

Modern AI is heuristic, rather than algorithmic. An algorithm is a computer code representation of a piece of rigorous mathematics.  As such, algorithms are associated with strong expectations with respect to their behavior on input and output, and the falsifiability of those expectations is the basis for classical verification, validation and uncertainty quantification of computer models.  A heuristic is a computer code representation of a collection of non-rigorous, semi-mathematical intuitions, and as such creates no checkable a-priori expectations for its behavior.  Because of this heuristic nature, no AI model is ever "wrong," because no concept of model correctness is available for AI. I will argue that this fact is the origin of many of the intractable pathologies of modern AI. Both notable successes and puzzling failures of AI are traceable to the heuristic choices that pervade deep learning models. I will review difficulties such as the explainability gap and AI hallucinations in the context of this framework.  I will discuss the problematic implications of heuristic AI for science models that adopt AI methods. I will then sketch a program for building a mathematical foundation for AI, so that AI models may gain robustness and verifiability, and begin the process of graduating from heuristics to algorithms.

13:15 - 14:00: Dr. Chris Quigg, Fermi National Accelerator Laboratory (Website)

A Century of Noether’s Theorem

In the summer of 1918, Emmy Noether published the theorem that now bears her name, establishing a profound two-way connection between symmetries and conservation laws. The influence of this insight is pervasive in physics; it underlies all of our theories of the fundamental interactions and gives meaning to conservation laws that elevates them beyond useful empirical rules. Noether’s papers, lectures, and personal interactions with students and colleagues drove the development of abstract algebra, establishing her in the pantheon of twentieth-century mathematicians. This talk traces her path from Erlangen through Göttingen to a brief but happy exile at Bryn Mawr College, illustrating the importance of “Noether’s Theorem” for the way we think today.

15:15 - 16:00: Dr. Robert Eisenberg, Rush University Medical Center (Website)

The Maxwell Equations are Universal and Exact, (therefore) Scary

When the Maxwell equations are written without a dielectric constant, they are universal and exact, from inside atoms to between stars. Dielectric and polarization phenomena need then to be described by phenomenological (constitutive) stress strain relations for charge density, that show how charge redistributes when the electric field is changed, analogous to compressibility relations in fluid mechanics.

Total current (including the ethereal displacement current) then never accumulates at all, independent of the properties of matter, in contrast to electron/conduction current that accumulates as charge. These properties arise from the properties of space time according to the theory of special relativity because of the Lorentz invariance of the elementary charge.

Accumulation of charge contradicts Kirchhoff’s law as stated in texts: “the currents into a node equal those leaving the node.” The universal existence of displacement current forces a generalization of Kirchhoff’s current law. The results can be striking. Hopping phenomena (almost) disappear. In unbranched systems like circuit components or ion channels, total current does not depend on location. Spatial Brownian motion disappears. The infinite spatial variation of a Brownian model of thermal noise becomes the zero spatial variation of total current. Maxwell’s Core Equations become a perfect (spatial) low pass filter.

An Exact and Universal theory of Electrodynamics is a scary challenge to scientists like me, trained to be skeptical of all sweeping claims to perfection.