I shall continue giving a survey of current efforts to bring these two subjects together in the context of finite spectral triples.

The associahedron is a well-known polytope, defined by Stasheff in 1963 as a cell complex, and given its first polytopal realizations by Haiman and by Lee in the 1980s. The past twenty years have seen ongoing interest in associahedra and their generalizations because of their connection to cluster algebras; the 1-skeleton of an associahedron is the exchange graph of a type A cluster algebra. More recently, associahedra have attracted attention in physics: Arkani-Hamed, Bai, He, and Yan showed that a particular realization of the associahedron encodes the scattering amplitudes for a particular quantum field theory. It turns out that the most natural way to construct the associahedron which the physicists needed is via the theory of quiver representations. I will explain this approach, and also how it can be generalized to obtain many other polytopes, some of which are also of interest to physicists. No knowledge of quiver representations is assumed; I will explain what I need.

Matroids are versatile combinatorial objects with deep connections to many subjects, including chromatic polynomials of graphs, arrangements of vectors and hyperplanes, and the geometry of Grassmannians. I will discuss joint work with Dhruv Ranganathan in which we use Gromov-Witten theory to associate new invariants to matroids, and I will explain how these new invariants are related to counting curves in certain geometric spaces.

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Integrable systems first came to prominence as a geometric abstraction of structure in classical mechanics. Despite these origins, integrable systems have been found to interact meaningfully with pure mathematics. Modern examples include the role such systems play in toric geometry, the Langlands program, mirror symmetry, and quantum cohomology. At the same time, Mishchenko-Fomenko systems represent another celebrated paradigm of integrable systems in pure mathematics. They exhibit the kind of Lie-theoretic symmetry that allows difficult geometric problems to be posed and solved entirely in algebraic terms. Techniques in commutative algebra thereby yield new examples of integrable systems, as well as deeper insights into existing ones.

I will give a non-technical overview of the themes mentioned above. Some emphasis will be placed on my contributions to the subject, and on directions for future research.

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3:30-4pm Meet and Greet https://gather.town/app/QpSa5CyNCP4WrNKm/MathTea

The notion of a valuation on an algebra is an abstraction of “order of divisibility of an integer by a prime”. Valuations with values in integers classically appear in number theory and algebraic geometry (dating back to late 19th century). But the combinatorial aspects of valuations are much more recent. In this talk we will see how the general notion of a valuation with values in an ordered group such as $\mathbb{Z}^n$ gives a powerful gadget to translate algebra/geometric notions to combinatorial and convex geometric notions. This has far reaching consequences and is the common thread behind several areas such as Gr\”obner basis theory, tropical geometry and non-Archimedean geometry, theory of Newton-Okounkov bodies and Khovanskii bases. I will give a tour of some of these ideas with emphasis on my previous and recent work. Towards the end, I will introduce the notion of a valuation with values in a semi-field and mention recent results on classifying toric principal bundles and toric families of algebraic varieties and connection with combinatorial objects such as linear subspace arrangements and buildings.

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Meeting ID: 920 2823 2306 Passcode: talks One tap mobile +16475580588,,92028232306#,,,,*742760# Canada

3:30-4pm Meet and Greet https://gather.town/app/QpSa5CyNCP4WrNKm/MathTea