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Mathematics & Statistics

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Rigidity theorems of complete Kähler-Einstein manifolds and complex space forms

Tian Chong

2:00pm-3:15pm

CMC 108

**Abstract**

In this talk, we introduce some elliptic differential inequalities from the Weitzenbock formulae for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some \(Lp\) and \(L\infty\) pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among Kähler manifolds with constant scalar curvature.

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Polarizable Carnot groups and Polar Coordinates

Thomas Bieske

2:00pm-3:15pm

CMC 108

**Abstract**

We will explore the geometric issues in establishing polar coordinates in Carnot groups. We will then discuss which groups have polar coordinates and which do not.

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Solutions to \(p\)-Laplacian Equations with Drift Terms in Grushin-type Spaces

Keller Blackwell

2:00pm-3:15pm

CMC 108

**Abstract**

In this talk we introduce sub-Riemannian spaces called Grushin-type spaces. We will then explore past, present, and future work done in solving the \(p\)-Laplacian equation with drift terms in such spaces.

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Equivalence of Weak and Viscosity solutions to the \(p(x)\)-Laplace Equation

Robert Freeman

2:00pm-3:15pm

CMC 108

**Abstract**

We discuss the differences between the \(p\)-Laplace equation and the \(p(x)\)-Laplace equation in Euclidean space and Carnot groups. We mention the different notions of solution to the \(p(x)\)-Laplace equation and outline the proof of the equivalence of the solutions in Carnot groups.

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A \(\mathrm{p}(\cdot)\)-Poincaré Type Inequality and \(\mathrm{p}(\cdot)\)-Capacity in Carnot Groups, Part II

Robert Freeman

2:00pm-3:15pm

CMC 108

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A \(\mathrm{p}(\cdot)\)-Poincaré Type Inequality and \(\mathrm{p}(\cdot)\)-Capacity in Carnot Groups

Robert Freeman

2:00pm-3:15pm

CMC 108

**Abstract**

We discuss the \(\mathrm{p}(\cdot)\)-Capacity and quasi-continuity in the sub-Riemannian setting of Carnot groups. We will use the \(\mathrm{p}(\cdot)\)-capacity and quasi-continuity to show a \(\mathrm{p}(\cdot)\)-Poincaré-type inequality. We will then use this inequality to establish the existence of a minimizer to the \(\mathrm{p}(\cdot)\)-Dirichlet energy integral. Moreover, we use the \(\mathrm{p}(\cdot)\)-Poincaré-type inequality to show the uniqueness of the minimizer, up to a set of zero \(\mathrm{p}(\cdot)\)-capacity, in the Sobolev sense.

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A Theorem of Radó-type in \(R^n\), with possible extensions to Carnot Groups, Part III

Zachary Forrest

2:00pm-3:15pm

CMC 108

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A Theorem of Radó-type in \(R^n\), with possible extensions to Carnot Groups, Part II

Zachary Forrest

2:00pm-3:15pm

CMC 108

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A Theorem of Radó-type in \(R^n\), with possible extensions to Carnot Groups

Zachary Forrest

2:00pm-2:50pm

CMC 108

**Abstract**

We explore a Theorem for Euclidean spaces in the style of Radó for \(p\)-Harmonic functions as presented in a paper of Juutinen & Lindqvist, utilizing techniques of Viscosity Solutions. We consider the connection between the underlying Geometry and the proof of an essential Lemma in the paper, and speculate regarding the adaptability of this proof to the Heisenberg and (more generally) Carnot spaces.

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Regularity of weak solutions of the \(p\)-Laplace equation in the Heisenberg group — Part IV

Diego Ricciotti

2:00pm-2:50pm

CMC 108

**Abstract**

We discuss De Giorgi classes and Holder continuity of the horizontal derivatives of p-harmonic functions in the Heisenberg group.

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Regularity of weak solutions of the \(p\)-Laplace equation in the Heisenberg group — Part III

Diego Ricciotti

2:00pm-2:50pm

CMC 108

**Abstract**

We present a proof of Lipschitz regularity for \(p\)-harmonic functions in the Heisenberg group for \(1

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Regularity of weak solutions of the \(p\)-Laplace equation in the Heisenberg group — Part II

Diego Ricciotti

2:00pm-2:50pm

CMC 108

**Abstract**

Continuing from the regularity results presented last time, we will start discussing the nonlinear case (\(p\) different than \(2\)).

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Regularity of weak solutions of the \(p\)-Laplace equation in the Heisenberg group — Part I

Diego Ricciotti

2:00pm-2:50pm

CMC 108

**Abstract**

I will present some regularity results for weak solutions of the \(p\)-Laplace equation in the Heisenberg group based on difference quotients techniques, starting with the case \(p=2\).