Marcello LUCIA

November 14th (in person):
Chao Li (NYU-Courant Institute)
TBA

October 31st (in person):
Yanyang Li (University of Chicago)
Existence and regularity of anisotropic minimal hypersurfaces

October 17th (in person):
Paula Burkhardt-Guim (NYU-Courant Institute)
TBA

Yi Li (John Jay College of Criminal Justice at CUNY)
Monotone properties of the eigenfunctions of Neumann problems
Abstract: In this talk we prove the hot spots conjecture for long rotationally symmetric domains of the Euclidean space by the continuity method. More precisely, we show that the odd Neumann eigenfunction in xn associated with lowest nonzero eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili’s conjecture 8.3 holds true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue.

September 12th (in person):
Junming Xie (Rutgers university)
Four-dimensional gradient Ricci solitons with nonnegative (or half nonnegative) isotropic curvature

Friedemann Schuricht (Technische Universitat Dresden, Germany)
Theory of Traces and the Divergence Theorem
Abstract: We introduce a general approach to traces that we consider as linear continuous functionals on some function space. Here we focus on special choices and obtain an integral calculus for traces based on finitely additive measures. This allows the computation of the precise representative of an integrable function and of the trace of a Sobolev or BV function by integrals instead of the usual limit of averages. For integrable vector fields where the distributional divergence is a measure, we also derive Gauss-Green formulas on arbitrary Borel sets. It turns out that a second boundary integral is needed to treat singularities that had not been accessible before. The advantage of the integral calculus is that neither a normal field nor a trace function on the boundary is needed. Also inner boundaries and concentrations on the boundary can be treated this way. The Gauss-Green formulas are also available for Sobolev and BV functions. As application the existence of a weak solution of a boundary value problem containing the p-Laplace operator can be shown.

TUESDAY September 10th at 2:45pm (in person):
Undergraduate Simons Lectures at CSI (Bulding 1P, Lecture Hall)
Bernd Kawohl (University of Cologne, Germany)
On buttons and balls that cannot run away (Convex sets of constant width)
Abstract

July 11 at 2:00pm (in person):
Yihong Du (University of New England, Australia)
On the KPP equation with nonlocal diffusion and free boundaries
Abstract: A new phenomenon in nonlocal diffusion models is that accelerated propagation may happen, that is, the propagation speed could be infinite, which never occurs in the corresponding local diffusion model with compactly supported initial data. In this talk, we will first briefly review the history of the KPP model used to describe the propagation of biological/chemical species, and then look at some very recent results on the KPP equation with nonlocal diffusion and free boundaries. For several natural classes of kernel functions appearing in the nonlocal diffusion term, we will show how the exact rate of acceleration can be determined. The talk is based on joint works with Dr Wenjie Ni.

April 18 (in person):
Jianxiong Wang (University of Connecticut)
Higher order conformal equations on hyperbolic spaces and the symmetry of solutions
Abstract: The classification of solutions for semilinear PDEs, as well as the classification of critical points of the corresponding functionals, have wide applications in the study of partial differential equations and differential geometry. The classical moving plane method and the moving sphere method in Euclidean space provide an effective approach to capturing the symmetry of solutions. In this talk, we develop a moving sphere approach for integral equations in the hyperbolic space, to obtain the symmetry property as well as a characterization result towards positive solutions for nonlinear problems involving the GJMS operators (a generalization of the Paneitz operator). Our methods also rely on Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic spaces together with a Kelvin transform.

March 28 (in person):
Jesse Ratzkin (Universität Würzburg)
TBA
Abstract: TBA

March 14 (in person):
Lorenzo Sarnataro (Princeton University)
TBA
Abstract: TBA