Geometric measure theory provides a useful framework for studying the geometry and regularity of sets and measures in Euclidean and non-Euclidean settings, and has many useful applications to interesting problems in various fields of mathematics. This invited paper session aims to showcase the vibrant interactions between geometric measure theory, partial differential equations, and harmonic analysis. In this session, we will see exciting new developments at the interface of these areas, highlighting the ways in which they intertwine to produce deep insights.

##### Dorronsoro's Theorem and Vertical Versus Horizontal Inequalities on Carnot Groups

*2:00 p.m. - 2:20 p.m.*

**Seung-Yeon Ryoo**, *Princeton University*

The classical Dorronsoro theorem expresses the \(L_1(L_p) (1 < p < \infty )\) norm of the \(\alpha(>0)\) - fractional Laplacian of an \(L^p\) function on a Euclidean space as a certain singular integral measuring the average deviation of the function from being a polynomial around every point and at every scale. We prove a version of Dorronsoro's theorem in the setting of Carnot groups. This extends and strengthens the work of Fässler and Orponen (2020), who proved a one-sided Dorronsoro theorem in the setting of the Heisenberg groups, with a restriction \(\alpha<2\) on the fractional power of the Laplacian. One consequence of the Dorronsoro theorem is that it gives "vertical versus horizontal inequalities" on nonabelian Carnot groups, which quantify the extent to which nonabelian Carnot groups fail to embed into the \(L_1(L_p) (1 < p < \infty )\) spaces.

##### (CANCELED) ~~Tensorization of Sobolev Spaces~~

~~2:30 p.m. - 2:50 p.m.~~

**Silvia Ghinassi**, *University of Washington*

~~A classical - and routinely exploited - fact about Sobolev spaces in \(\R^n\) is that their structure is induced by the structures of Sobolev spaces on their components. This is believed to be true for generalized Sobolev spaces on metric measure spaces, however it is not proven in the full generality. We’ll survey some definitions, their equivalences, and discuss results for Cartesian and warped products of metric measure spaces. The talk is based on a joint upcoming work with V. Giri and E. Negrini.~~

##### Carnot Groups and Bi-Lipschitz Embeddings into \(L^1\)

*3:00 p.m. - 3:20 p.m.*

**Lisa Naples**, *Macalester College*

The classical Dorronsoro theorem expresses the \(L_1(L_p) (1 < p < \infty )\) norm of the \(\alpha(>0)\) - fractional Laplacian of an \(L^p\) function on a Euclidean space as a certain singular integral measuring the average deviation of the function from being a polynomial around every point and at every scale. We prove a version of Dorronsoro's theorem in the setting of Carnot groups. This extends and strengthens the work of Fässler and Orponen (2020), who proved a one-sided Dorronsoro theorem in the setting of the Heisenberg groups, with a restriction \(\alpha<2\) on the fractional power of the Laplacian. One consequence of the Dorronsoro theorem is that it gives "vertical versus horizontal inequalities" on nonabelian Carnot groups, which quantify the extent to which nonabelian Carnot groups fail to embed into the \(L_1(L_p) (1 < p < \infty )\) spaces.

##### Low Dimensional Cantor Sets with Absolutely Continuous Harmonic Measure

*3:30 p.m. - 3:50 p.m.*

**Cole Jeznach**, *University of Minnesota*

The relationship between harmonic measure and surface measure of a domain is largely connected with the geometry of the domain itself. In many fractals (for example, in domains with relatively "large" boundaries, and outside self-similar "enough" cantor sets), these measures are mutually singular, and in fact, have different dimensions. After recalling some of these results I will present joint work with G. David and A. Julia where we demonstrate examples where the exact opposite occurs: we construct Cantor-type sets in the plane that are Ahlfors regular (of small dimension) for which their associated harmonic measure and surface measure are bounded equivalent.

##### Decoupling and Restriction for Ruled Hypersurfaces Generated by a Curve

*4:00 p.m. - 4:20 p.m.*

**Dóminique Kemp**, *Institute for Advanced Study*

In this talk, we shall address the decoupling theory and restriction theory of the ruled Euclidean hypersurfaces generated by a curve. We shall think of these surfaces as "parabolic cylinders of smoothly varying orientation" and see how much mileage this perspective attains for us. In particular, we shall achieve an effective \(\ell^2\) decoupling theorem (of optimal \(L^p\) range) and a reverse square function estimate. Attention shall also be given to possible applications to maximal averages over curves.

##### A Definition of Fractional k-Dimensional Measure

*4:30 p.m. - 4:50 p.m.*

**Cornelia Mihaila**, *Saint Michael’s College*

I will introduce a fractional notion of k-dimensional measure, with \( 0 ≤ k < n \), that depends on a parameter σ that lies between 0 and 1. When k=n−1 this coincides with the fractional notions of area and perimeter, and when k=1 this coincides with the fractional notion of length. We will see that, when multiplied by the factor 1−σ, this σ-measure converges to the k-dimensional Hausdorff measure up to a multiplicative constant. This is based on a joint work with Brian Seguin.

##### A Singular Integral Identity for Surface Measure

*5:00 p.m. - 5:20 p.m.*

**Ryan Bushling**, *University of Washington*

Using only the classical methods of geometric measure theory, we prove that the integral of a certain Riesz-type kernel over \((n-1)\)-rectifiable sets in \(\mathbb{R}^n\) is constant, from which a formula for surface measure immediately follows. Geometric interpretations are given, and the solution to a geometric variational problem characterizing the family of convex sets follows as a corollary.

##### A Local Bernstein Inequality for Laplace Eigenfunctions

*5:30 p.m. - 5:50 p.m.*

**Stefano Decio**, *University of Minnesota*

A powerful heuristic in the study of Laplace eigenfunctions is that they behave like polynomials of degree proportional to the square root of the eigenvalue. We present an instance in which this polynomial behaviour can be made precise, namely we discuss a version of the classical (for polynomials) Bernstein inequality. Along the way we show that analogous inequalities hold for solutions to nice elliptic PDEs, which also resemble polynomials once an appropriate notion of degree is introduced. Based on joint work with Eugenia Malinnikova.