Title: The Lawson-Osserman conjecture for the minimal surface systemAbstract: 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 […]
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 […]
Title: Optimal observability times for wave and Schrodinger equations on really simple domainsAbstract: 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 […]
Title: Existence of 5 minimal tori in 3-spheres of positive Ricci curvatureAbstract: 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 […]
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 […]
Title: Restriction estimates for spectral projectionsAbstract: 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 […]
Title: Positive scalar curvature and exotic structures on simply connected four manifolds.Abstract: We address Gromov’s band width inequality and Rosenberg’s S1-stability conjecture for smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless, we show that both of these results hold upon considering simply […]
Title: Arnold-Thom conjecture for the arrival time of surfacesAbstract: Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. In this talk, I will explain the proof of Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time […]
Title: Some recent developments on the fully nonlinear Yamabe problems Abstract: In recent joint work with YanYan Li and Zongyuan Li, we broaden the scope of fully nonlinear Yamabe problems by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions […]
Title: Finite time explosion for SPDEs Abstract: The classical existence and uniqueness theorems for SPDEs prove that, under appropriate assumptions, SPDEs with globally Lipschitz continuous forcing terms have unique global solutions. This talk outlines recent results about SPDEs exposed to superlinearly growing deterministic and stochastic forcing terms. I describe sufficient conditions that guarantee that, despite […]
Title: Restriction and local smoothing estimates using decoupling and two-ends inequalities.Abstract: I will discuss restriction estimates and local smoothing estimates derived from decoupling theorems and two-ends inequalities, with a focus on the differences between these two problems. Specifically, I will introduce the wave packet density, which is the key to the induction on scales argument […]
Title: Triple junction solution for the Allen-Cahn systemAbstract: In this talk, I will discuss recent results on the vector-valued Allen-Cahn system with a triple-well potential. We establish the existence of an entire minimizing solution that asymptotically converges to a triple junction, corresponding to a planar minimal cone, at infinity. Furthermore, we prove the uniqueness of […]