A long-standing problem in metric geometry is to recognize those metric spaces that admit a “good” parametrization by the Euclidean space or the unit sphere. In this talk, we will discuss the quasisymmetric uniformization problem and some recent advancements in terms of the Menger curvature, a curvature suited for Euclidean sets with no a priori smooth structure. The talk is based on a joint work with G. C. David (Ball State University).

In this talk, we study the problem of minimizing the distance from a data point to a subvariety of the Grassmannian from the perspective of metric algebraic geometry. An important invariant in this line of work is the number of critical points of this optimization problem for general data. When the data is a general point in the ambient space, this number is called the Euclidean distance (ED) degree of the variety. However, we focus mainly on the case when the data is a general point in the Grassmannian, and call the number of critical points the Grassmann distance (GD) degree. As we will see, the GD degree is more subtle than the ED degree. We give formulas for the ED and GD degrees of prominent subvarieties of the Grassmannian, including Schubert and matroid varieties.

David and Semmes proved that, for an Ahlfors regular measure, (L^2)-boundedness of a sufficiently rich class of singular integral operators implies uniform rectifiability. I will discuss analogues in parabolic space and the Heisenberg group. This is joint work with John Hoffman (EHU, University of the Basque Country).

The mean width of a set, and more generally, the volume of its typical projections, are fundamental quantities which arise in various high-dimensional problems. We present new, sharp estimates for the mean width, quermassintegrals, and metric entropy of convex bodies. Joint work with Grigoris Paouris.

Consider a finite subgroup of the general linear group, with its natural action on a polynomial ring.  We discuss how the local cohomology module of the invariant ring compares with the invariant part of the local cohomology of the polynomial ring.  This has various consequences, such as for the \(a\)-invariant and for the Hilbert series.  This is joint work with Kriti Goel and Jack Jeffries.

Given an ideal \(I\) in a regular local ring \(A\), the cohomological dimension of \(I\) in \(A\) is the index of the highest non-vanishing local cohomology of \(A\) supported at \(I\). Determining effective upper bounds on the cohomological dimension in terms of topological invariants of \(\textrm{Spec}(A/I)\) is a central problem in commutative algebra: foundational results include the Hartshorne--Lichtenbaum Vanishing Theorem and the Second Vanishing Theorem.

In equal characteristic, Faltings established in 1980 a general bound on the cohomological dimension of an ideal in terms of its “big height”. In this talk, we extend Faltings’ results to the unramified mixed characteristic setting and show that the resulting bound is sharp.

The Busemann intersection inequality is a fundamental isoperimetric principle for the volume of slices of a convex body. It is closely connected to boundedness of the Radon and k-plane transforms.  

The original proof of the Busemann intersection inequality relied on symmetrization but rather indirectly, as monotonicity under a single symmetrization was not addressed.  More recent proofs by Adamczak--Paouris--Pivovarov--Simanjuntak and Milman--Yehudayoff--Shabelman explicitly treat such monotonicity.    

I will discuss a new form of the Busemann intersection inequality. It avoids working with slices, and instead relies on enumerating random points in slabs.  I will explain how this "empirical" version of the Busemann intersection inequality is well-suited to direct analysis under symmetrization.  

Based on joint work with  G. Paouris (Texas A\&M) and P. Simanjuntak (Texas A\&M).