# Research

## Differential Geometry

### Friday, November 22, 2019

Title: Geodesics in the Heisenberg group revisited.
Speaker: Cezar Lupu, Texas tech University
Time: 11:00am‐12:00pm
Place: NES 104

#### Abstract

In this talk, we discuss some recent work of Hajlasz and Zimmerman on the geodesics in the Heisenberg group. Firstly, we show that the isoperimetric inequality proved in the paper for curves in $$R^{2n}$$ can be derived directly from the strong form of Wirtingerï¿½s inequality. The structure of the geodesics follows immediately from the equality case. Secondly, based on the Wirtingerï¿½s inequality idea, we prove some Bonnesen-type inequalities and stability results for curves in $$R^{2n}$$ .

Last but not least, if time allows, we would like to generalize the result to closed curves in $$R^3$$, and perhaps to give an interpretation in terms of Sobolev-type inequalities.

### Friday, November 8, 2019

Title: Weights (continued)
Speaker: Thomas Bieske
Time: 11:00am‐12:00pm
Place: NES 104

#### Abstract

We continue our discussion on $$A_p$$ weights.

### Friday, October 11, 2019

No seminar this week.

### Friday, October 4, 2019

Title: TBA
Speaker: TBA
Time: 11:00am‐12:00pm
Place: NES 104

TBA

### Friday, September 20, 2019

Title: On Level Sets in the Heisenberg Group
Speaker: Zachary Forest
Time: 11:00am‐12:00pm
Place: NES 104

#### Abstract

It is known that in Euclidean geometry the $$p$$-Laplace Equation supports a level-set removability property for weak solutions; however, such results are still largely unknown in the class of sub-Riemannian manifolds. Even in spaces such as the Heisenberg group, the simplest nontrivial sub-Riemannian manifold, the question of removability has remained open for years. In this talk we ask the question, "can level-set removability be extended for the Heisenberg group $$\mathbb{H}_1$$?", and thanks to papers by Juutinen and Lindqvist (2005) and Franci-Serapioni-Serra Cassano (2001) can answer it in the affirmative for viscosity solutions to the horizontal $$p$$-Laplacian and horizontal $$p(x)$$-Laplacian under certain regularity conditions.

### Friday, September 13, 2019

Title: Rado-type Removability Results for $$p$$-Laplace and $$p(x)$$-Laplace in the Heisenberg grou
Speaker: Bob Freeman
Time: 11:00am‐12:00pm
Place: NES 104

#### Abstract

We recall the Rado-type removability results for $$p$$-harmonic functions in the plane (Kilpelainen 1994) and the extension to quasi-linear functions in Euclidean space (Juutinen, Lindqvist 2005). We discuss new removability results for solutions to the $$p$$-Laplace and the $$p(x)$$-Laplace equations in the Heisenberg group.