# Video Lectures

Endow the edges of the ZD lattice with positive weights, sampled independently from a suitable distribution (e.g., uniformly distributed on [a,b] for some b greater than a greater than 0). We wish to study the geometric properties of the resulting...

p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov...

Any non-negative univariate polynomial over the reals can be written as a sum of squares. This gives a simple-to-verify certificate of non-negativity of the polynomial. Rooted in Hilbert's 17th problem, there's now more than a century's work that...

We discuss the shape invariant, a sort of set valued symplectic capacity defined by the Lagrangian tori inside a domain of R4. Partial computations for convex toric domains are sometimes enough to give sharp obstructions to symplectic embeddings...

In distributed certification, our goal is to certify that a network has a certain desired property, e.g., the network is connected, or the internal states of its nodes encode a valid spanning tree of the network. To this end, a prover generates...

I'll give an exposition of the theory of "multiplicative polynomial laws," introduced by Roby, and how (following a suggestion of Scholze) they can be applied to the theory of commutative (flat) group schemes. This talk will feature more questions...

I will discuss how much the choice of coefficients impacts the quantitative information of Floer theory, especially spectral invariants. In particular, I will present some phenomena that are specific to integer coefficients, including an answer to a...

The key principle in Grothendieck's algebraic geometry is that every commutative ring be considered as the ring of functions on some geometric object. Clausen and Scholze have introduced a categorification of algebraic and analytic geometry, where...

For a reductive group G, its BdR+-affine Grassmannian is defined as the étale (equivalently, v-) sheafification of the presheaf quotient LG/L+G of the BdR-loop group LG by the BdR+ -loop subgroup L+G. We combine algebraization and approximation...