Nassif Ghoussoub, University of British Columbia, Vancouver, BC, CanadaAmir Moradifam, Columbia University, New York, NY, USA

• Hardy type inequalities: Bessel pairs and Sturm's oscillation theory The classical Hardy inequality and its improvements Improved Hardy inequality with boundary singularity Weighted Hardy inequalities The Hardy inequality and second order nonlinear eigenvalue problems Hardy-Rellich type inequalities: Improved Hardy-Rellich inequalities on $H^2_0(\Omega)$ Weighted Hardy-Rellich inequalities on $H^2(\Omega)\cap H^1_0(\Omega)$ Critical dimensions for $4^{\textrm{th}}$ order nonlinear eigenvalue problems Hardy inequalities for general elliptic operators: General Hardy inequalities Improved Hardy inequalities for general elliptic operators Regularity and stability of solutions in non-self-adjoint problems Mass transport and optimal geometric inequalities: A general comparison principle for interacting gases Optimal Euclidean Sobolev inequalities Geometric inequalities Hardy-Rellich-Sobolev inequalities: The Hardy-Sobolev inequalities Domain curvature and best constants in the Hardy-Sobolev inequalities Aubin-Moser-Onofri inequalities: Log-Sobolev inequalities on the real line Trudinger-Moser-Onofri inequality on $\mathbb{S}^2$ Optimal Aubin-Moser-Onofri inequality on $\mathbb{S}^2$ Bibliography

Nassif Ghoussoub, University of British Columbia, Vancouver, BC, CanadaAmir Moradifam, Columbia University, New York, NY, USA

• Hardy type inequalities: Bessel pairs and Sturm's oscillation theory The classical Hardy inequality and its improvements Improved Hardy inequality with boundary singularity Weighted Hardy inequalities The Hardy inequality and second order nonlinear eigenvalue problems Hardy-Rellich type inequalities: Improved Hardy-Rellich inequalities on $H^2_0(\Omega)$ Weighted Hardy-Rellich inequalities on $H^2(\Omega)\cap H^1_0(\Omega)$ Critical dimensions for $4^{\textrm{th}}$ order nonlinear eigenvalue problems Hardy inequalities for general elliptic operators: General Hardy inequalities Improved Hardy inequalities for general elliptic operators Regularity and stability of solutions in non-self-adjoint problems Mass transport and optimal geometric inequalities: A general comparison principle for interacting gases Optimal Euclidean Sobolev inequalities Geometric inequalities Hardy-Rellich-Sobolev inequalities: The Hardy-Sobolev inequalities Domain curvature and best constants in the Hardy-Sobolev inequalities Aubin-Moser-Onofri inequalities: Log-Sobolev inequalities on the real line Trudinger-Moser-Onofri inequality on $\mathbb{S}^2$ Optimal Aubin-Moser-Onofri inequality on $\mathbb{S}^2$ Bibliography

Nassif Ghoussoub, University of British Columbia, Vancouver, BC, CanadaAmir Moradifam, Columbia University, New York, NY, USA

• Hardy type inequalities: Bessel pairs and Sturm's oscillation theory The classical Hardy inequality and its improvements Improved Hardy inequality with boundary singularity Weighted Hardy inequalities The Hardy inequality and second order nonlinear eigenvalue problems Hardy-Rellich type inequalities: Improved Hardy-Rellich inequalities on $H^2_0(\Omega)$ Weighted Hardy-Rellich inequalities on $H^2(\Omega)\cap H^1_0(\Omega)$ Critical dimensions for $4^{\textrm{th}}$ order nonlinear eigenvalue problems Hardy inequalities for general elliptic operators: General Hardy inequalities Improved Hardy inequalities for general elliptic operators Regularity and stability of solutions in non-self-adjoint problems Mass transport and optimal geometric inequalities: A general comparison principle for interacting gases Optimal Euclidean Sobolev inequalities Geometric inequalities Hardy-Rellich-Sobolev inequalities: The Hardy-Sobolev inequalities Domain curvature and best constants in the Hardy-Sobolev inequalities Aubin-Moser-Onofri inequalities: Log-Sobolev inequalities on the real line Trudinger-Moser-Onofri inequality on $\mathbb{S}^2$ Optimal Aubin-Moser-Onofri inequality on $\mathbb{S}^2$ Bibliography