Some New Results in Geometric AnalysisThis thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving flow on space curves. Lastly, we present an abridgment of our work on a family of three-dimensional Lie groups, which, when equipped with canonical left-invariant metrics, interpolate between Sol and hyperbolic space.Coiculescu, Matei P.creatorSchwartz, Richardthesis advisorPausader, BenoitreaderTshishiku, BenareaderBrown University. Mathematicssponsor2021text_resourcesborn digitalEnglishSenior thesis (ScB)--Brown University, 2021MathematicsthesesGeodesics (Mathematics)Geometric analysisDifferential equations, PartialDifferential equations, ParabolicDifferential equations, Partial--Numerical solutionsLie groupsGeometry, DifferentialHyperbolic SpaceAll rights reservedIn CopyrightAll Rights Reserved10.26300/6fjz-ax05