contact Graham for zoom link, thanks.

For a Weyl polytope $P$, there is an associated real toric variety $X_P^{\mathbb{R}}$. The quotient of $X_P^{\mathbb{R}}$ by the Weyl group action is obtained by gluing copies of cubes together. In this lecture, we will see the sufficient condition for the glued subspace of the cube to be contractible, and hence the corresponding gluing operation is a homotopy equivalence.

In 1751, Euler wrote a letter to Goldbach in which he conjectured a formula for the number of triangulations of a regular polygon with n sides. It turns out that the triangulations of a regular polygon are in bijection with many other geometric and combinatorial sets of objects. There is a mythical polytope, called the associahedron, whose vertices correspond to the triangulations of a regular polygon. The associahedron itself has a long mathematical history, starting with work of Tamari and Stasheff. The associahedron today has connections to many diverse areas of mathematics, including moduli spaces and topology, quiver representation theory, cluster algebras and toric varieties. We will discuss the beginnings of this story, starting with cutting small polygons into triangles using non-crossing diagonals.

Building on her seminal work regarding moduli space volumes for Riemann surfaces, Mirzakhani also calculated expected values for various geometric functions on moduli space. Notably, she examined the expected Cheeger constant, the injectivity radius at a random point, and the statistical distribution of different types of curves on surfaces of large genus. We will review several of Mirzakhani's key results, which collectively offer insights into the geometry of random surfaces in high genus. Following this, we will explore some extensions of her findings in the context of translation surfaces.

Representatives from Fields Information Security will present on their Information Security Specialist Program. Email Cassandra Schultz at for more information and program brochure.

This talk is about Kac's famous inverse problem from 1966: "can one hear the shape of a drum?" The question asks whether the frequencies of vibration of a bounded domain determine the shape of the domain. First we present a quick survey on the known results. Then we discuss the key connection between eigenvalues of the Laplacian and the dynamics of the billiard, which is governed by the so called "Poisson Summation Formula". Finally we discuss our main theorem that "one can hear the shape of nearly circular ellipses". This is a joint work with Steve Zelditch (1953-2022).

Homological algebra is a powerful tool to differentiate between structures and study obstructions. For instance, homology of spaces is an invariant that is classically simple to compute. For these reasons, it is desirable to develop these tools constructively, that means without using the law of excluded middle and the axiom of choice. We will discuss the relevance of these axioms, what breaks, and different approaches to fix or side-step the problems all together.