Abstract: 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.

Join Zoom Meeting https://westernuniversity.zoom.us/j/92028232306

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