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).

To a triple of a ring \(R\), an \(R\)-module \(M\), and an ideal \(I\) we can associate the local cohomology modules \(H^n_I(M)\). One classical problem dating back to a question of Grothendieck is to identify when these vanish for all \(n > i\) and all modules \(M\), the smallest such \(i\) we denote the cohomological dimension of \(R\) with respect to \(I\). Grothendieck showed that this is bounded by \(d = \dim R\), while a very simple condition on \(\widehat{R}\) and \(I\widehat{R}\) controls the vanishing at \(d\). A more subtle topological condition controls the vanishing at \(d-1\) for regular local rings as posed by Hartshorne in the late 60s, and was proven in equicharacteristic in the early 70s. After 50 years it was shown to be true in unramified mixed characteristic, and the main result of this talk is the ramified mixed characteristic case.

Discriminant is a basic invariant associated with an algebraic number field. Its notion was
first introduced by Dedekind in 1871. The problem of effective computation of discriminant as
well as an integral basis of an infinite family of algebraic number fields which are defined over
the field $\mathbb{Q}$ of rational numbers by certain types of irreducible polynomials has been tackled
by several mathematicians. In this lecture we shall review the progress made regarding this problem in the case of pure fields. We shall present a formula for the exact power of any prime $p$ dividing the discriminant of the field  $K_n=\mathbb{Q}(\alpha_n),$ where $\alpha_n\in\mathbb{C}$ is a root of the $n$th exponential Taylor polynomial
$\frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+\cdots+\frac{x^2}{2!}+\frac{x}{1!}+1,
$ in terms of the $p $-adic expansion of positive integer $n$. We also describe an explicit $p$-integral basis of $K_n$ for each prime $p$. These local bases lead naturally to the construction of an integral basis of $K_n$.