This talk explores how image processing can be achieved through classical, non-AI methods rooted in partial differential equations (PDEs) and the calculus of variations. From smoothing and denoising to segmentation, we will see how mathematical structure and physical intuition guide the design of algorithms—without learning from data. From there, we will also discover how machine learning enters naturally—not to replace these models, but to enhance them, closing the loop between theory and data.

Flow polytopes are a family of beautiful mathematical objects that have connections to many areas in mathematics including optimization and representation theory. Finding the volumes and enumerating lattice points of some flow polytopes turns out to be a combinatorially interesting problem that involves beautiful enumeration formulas and many familiar combinatorial objects. Baldoni and Vergne find a series of formulas, which they call Lidskii formulas, that are combinatorially pleasing. Together with Benedetti et al., we provide combinatorial interpretations for the Lidskii formulas in terms of familiar combinatorial objects similar to parking functions. A more recent proof of the Lidskii formulas has been achieved by Mészáros and Morales, following the ideas of Postnikov and Stanley, by using polytopal subdivisions. For a smaller class of flow polytopes, these subdivisions are triangulations that coincide with a family of framed triangulations defined by Danilov, Karzanov, and Koshevoy. These triangulations turn out to have interesting hidden combinatorial structures. In joint work with von Bell, Mayorga, and Yip, we characterize the combinatorial structures arising from two triangulation strategies on a family of polytopes, which provide a surprising unifying framework for the Young lattice and the Tamari lattice. In work with Morales, Philippe, Tamayo Jiménez, and Yip, we use similar techniques to provide a geometric realization of the s-weak order answering a conjecture of Ceballos and Pons.

Given an algebraic dynamical system, can we bound the complexity of the simplest nonobvious invariant subvarieties which appear? Why might we want to do this? In this talk, we'll answer these question while also explaining the connection to difference equations, model theory, and the classification of finite simple groups.

Could similar structures help complete the mathematics of quantum physics as well as make neural networks more interpretable? In this talk we will learn a little combinatorial game theory and explore these membership-augmented set theoretic foundations in the context of quantum mechanics, topological quantum field theory, and some important symmetries and groups therein. We will hopefully have time to touch on pressing challenges in neural-net-based machine learning, towards improving AI with these very same game structures as well. The hope will be to generate some mathematically-based inspiration and insight on a few cherished concepts, like the complex symmetries of spacetime, while also reflecting on both chance and choice.

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