Jonathan and Yano will talk a bit about what it's like to do research in algebraic geometry. Younger grad students are especially encouraged to come.

Algebraic geometry open house

Thu, May. 4 3:35pm (MATH 3…

Probability

Oanh Nguyen (Brown University)

X

The Kac polynomial is one of the most studied models of random polynomials. It has the form ${f}_{n}(x)=\sum _{i=0}^{n}x\phantom{\rule{0}{0ex}}{i}_{i}\phantom{\rule{0}{0ex}}{x}^{i}.$

It is known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. Disproving a natural conjecture for these holes, we will show that all the hole radii are of the same order. We will also discuss open problems and recent fascinating conjectures concerning the flow of roots.

Joint work with Hoi Nguyen.

Hole radii for the Kac polynomials

Thu, May. 4 4pm (MATH 220)

Init Conditions

Howy Jordan

X

A category of sets is foundational to mathematics, and a category of complete inner product spaces is foundational to modern physics. The former has been captured by categorical language through the work of topos theory. As for the latter, the culmination of reconstruction programs led to a categorical characterization in 2022 by Heunen and Kornell. This could be summarized by saying that the category of Hilbert spaces and continuous linear maps is the dagger-abelian dagger monoidal category, with a simple monoidal separating unit and with directed colimits of dagger monics.

In the first part of this talk, we'll consider the axioms of elementary topoi, which capture Set-like categories, and see how to distinguish Set among these. We'll introduce the idea of subobjects in a category, and see that their classifying objects correspond to truth values giving us classifying functions of subsets. As time permits, we'll encounter related objects, like power objects and exponentials.

In the second part, we'll consider axioms for the category of Hilbert spaces. In particular, we'll introduce dagger and monoidal structures on categories, overview how a monoidal unit can obtain the structure of a field, and see how the axioms impose an orthomodular structure on the subobjects.

The Category of Sets, The Category of Hilbert Spaces