Johns Hopkins

Title: Strichartz estimates for the Schrödinger equation on the sphere.

 Abstract: We will discuss Strichartz estimates for solutions of the Schrödinger equation on the standard round sphere, which is related to the results of Burq, Gérard and Tzvetkov (2004). The proof is based on the arithmetic properties of the spectrum of the Laplacian on the sphere along with some local bilinear estimate in harmonic analysis. We shall also discuss some related results on the flat tori and on manifolds with negative curvature. This is based on ongoing joint work with Christopher Sogge.

Title: The geometry and topology of lower bounds on scalar curvature

Abstract: In this talk, I will discuss some recent results about controlling the geometry and topology of manifolds from a (positive) lower bound on their scalar curvature. 

Title: On isolated singularities and generic regularity of min-max CMC hypersurfaces

Abstract: Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away. This talk will be accessible to graduate students with a light background in differential geometry and PDE. 

Tittle: Partial regularity for Lipschitz solutions to the minimal surface system 

Abstract: The minimal surface system is the Euler-Lagrange system for the area functional of a high codimension graph and reduces to the minimal surface equation in the case that the codimension is one. Though the regularity theory for the minimal surface equation is well understood, much less is known in the systems case. This is largely due to the examples provided by Lawson and Osserman in 1977 and the lack of a maximum principle for the system. In this talk, we discuss recent progress on the regularity theory for Lipschitz stationary solutions and Lipschitz viscosity solutions to the minimal surface system with an emphasis on partial regularity results. 

Title: The Lawson-Osserman conjecture for the minimal surface system

Abstract: In their seminal work on the minimal surface system, Lawson and Osserman conjectured that Lipschitz graphs that are critical points of the area functional with respect to outer variations are also critical with respect to domain variations. We will discuss the proof of this conjecture for two-dimensional graphs of arbitrary codimension. This is joint work with J. Hirsch and R. Tione. 

Title: The Nevo-Thangavelu spherical maximal function on two step nilpotent Lie groups.

Consider R^d \times R^m with the group structure of a 2-step Carnot  Lie group and natural parabolic dilations.  The maximal operator   originally introduced by Nevo and Thangavelu  in the setting of the Heisenberg groups is generated by (noncommutative)  convolution  associated with  measures on  spheres or generalized spheres in R^d.  We discuss a number of approaches that have been taken to prove L^p boundedness and then talk about  recent work with Jaehyeon Ryu in which we drop  the nondegeneracy condition in the known results on Métivier groups. The new results have the sharp L^p boundedness range for all  two step Carnot  groups with d\geq 3. 

Title: Optimal observability times for wave and Schrodinger equations on really simple domains

Abstract: Observability for evolution equations asks: if I take a partial measurement in a system, can it “see” some physical quantity, such as energy?  In a series of papers with E. Stafford, Z. Lu, and S. Carpenter, we showed that energy for the wave or Schrodinger equation can be observed from any one side of a simplex in any dimension.  For this talk we will consider the wave and Schrodinger equations on intervals and triangles and ask how long does it take to observe the energy.  The computations use only elementary integrations and multivariable calculus.  This is work in progress with Claire Isham.

Title: Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

Abstract: In 1989, Brian White conjectured that every Riemannian 3-sphere contains at least five embedded minimal tori. The number five is optimal, corresponding to the Lyusternik-Schnirelmann category of the space of Clifford tori. I will present recent joint work with Adrian Chu, where we confirm this conjecture for 3-spheres of positive Ricci curvature. Our proof is based on min-max theory, with heuristics largely inspired by mean curvature flow.

Title: The Saint Venant inequality and quantitative resolvent estimates for the Dirichlet Laplacian.

Abstract: Among all cylindrical beams of a given material, those with circular cross sections are the most resistant to twisting forces. The general dimensional analogue of this fact is the Saint Venant inequality, which says that balls have the largest “torsional rigidity” among subsets of Euclidean space with a fixed volume. We discuss recent results showing that for a given set E, the gap in the Saint Venant inequality quantitatively controls the L^2 difference between solutions of the Poisson equation on E and on the nearest ball, for any Holder continuous right-hand side. We additionally prove quantitative closeness of all eigenfunctions of the Dirichlet Laplacian. This talk is based on joint work with Mark Allen and Dennis Kriventsov.

Title: Restriction estimates for spectral projections

Abstract: Restriction estimates for spectral projections have been widely studied since the work of Burq, Gérard, and Tzvetkov as a method for investigating eigenfunction concentration. The problem of establishing the optimal $L^p$ bounds for the restriction of Laplace-Beltrami eigenfunctions remains open, particularly when the restriction submanifold is of codimension 1 or 2. This talk will explore how these optimal bounds are characterized by the geometry of the underlying manifold $M$ and its submanifold $H$, focusing on the case where $M$ is two-dimensional and $H$ is a smooth curve in $M$.