Open problems in Physical Mathematics

Abstract: Part of our larger Assumptions of Physics research program, the aim of Physical Mathematics is to develop a set of basic mathematical starting points with a clear and tight physical justification that can serve as a foundation for all physical theories. The idea is to have a shared notion of states and processes that can generalize features and theorems common to all physical theories, while finding the specific ones through addition of further physical assumptions. We will give a short overview of the approach and then focus on open problems and mathematical conjectures to foster discussion. In particular, we will see how a physical theory must at least provide a space of ensembles that must have a T_0 second countable topology, a convex structure and an entropy. We will see how we can already show that the interplay between this structure constrains the space, leading to constructions that find connection to many areas (e.g. measure theory, Choquet theory, information geometry, orthogonality spaces, quantum logic, ...) and we conjecture whether these axioms can already show that an ensemble space is a bounded subset of a locally convex metrizable topological vector space, or further conditions must be found.

Speaker bio: Gabriele Carcassi is a researcher in the physics department of the University of Michigan. Together with prof. Christine A. Aidala, he leads a research program called Assumptions of Physics ( assumptionsofphysics.org ) that aims to find a minimum set of physical assumptions from which the laws can be rigorously rederived.

Moderator: Ted Theodosopoulos. Ted is a mathematician who, after working for years in academia and industry, transitioned to teaching at the pre-college level sixteen years ago, the last eight at Nueva, where he teaches math and economics. Ted’s research background is in the area of interacting stochastic systems, with particular applications in biology and economics.

Computer scienceMathematics

Audience: researchers in the topic

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Relatorium seminar

Series comments: The name "Relatorium" combines "relator" with the Latin root "-ium," meaning "a place for activities" (as in "auditorium" or "gymnasium"). This seminar series is a platform to relate ideas, interact with math, and connect with each other.

In this series, we explore math beyond what we usually hear in standard talks. These sessions fall somewhere between a technical talk and a podcast: moderately formal, yet conversational. The philosophy behind the series is that math is best learned by active participation rather than passive listening. Our aim is to “engage and involve,” inviting everyone to think actively with the speaker. The concepts are accessible, exploratory, and intended to spark questions and discussions.

The idea of relatability has strong ties to compassion — creating space for shared understanding and exploration - which is the spirit of this seminar! This is a pilot project, so we’re here to improvise, learn, and evolve as we go!

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