As described in the previous week's talk by Mikhail Karpukhin, there is a rich interplay between isoperimetric problems for Laplace eigenvalues on surfaces and the study of harmonic maps and minimal surfaces in spheres. Over the last 10-15 years, a program initiated by Fraser and Schoen has revealed a similar relationship between isoperimetric problems for the…

Numerical simulations on infinite domains are challenging. In this talk, we will take geometric approaches to analyze the problems and provide new solutions.  One problem we tackle is the perfectly matched layer (PML) problem for computational waves on infinite domains. PML is a theoretical wave-absorbing medium attached to the truncated domain that generates no reflection…

In this talk we provide new Myers–Steenrod theorem based proofs to confirm that: travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to a boundary fixing diffeomorphism, which is the natural gauge for all of these problems. The method of the proof leads…

Neutrinos are very weakly interacting particles that have unique properties that allow indirect measurements that cannot be realized with any other particle or field. I will give an introduction to neutrino physics and describe two inverse problems with neutrinos. This is based on joint work with Gunther Uhlmann. https://ncsu.zoom.us/j/94955574178?pwd=cFpRUlZScXRmdGFscE5FZEhIb1NWdz09 Meeting ID: 949 5557 4178 Passcode: 301451

Shape-shifting materials from 2D thin sheets to 3D shapes are attractive for broad applications in programmable machines and robots, functional biomedical devices, and four-dimensional printing. Kirigami, the art of paper cutting, has recently emerged as a promising approach for shape morphing structures and materials due to its new properties such as auxeticity, stretchability, conformability, multistability, and optical chirality. In this talk,…

In this talk, we will discuss bordered aspects of the Heegaard Floer surgery formulas of  Ozsvath--Szabo and Ozsvath--Manolescu. In particular, we will explain how their theories naturally define bordered invariants for manifolds with toroidal boundary components. Time permitting, we will discuss applications of the theory. One application is a proof of the equivalence of lattice…

I will talk about the new construction of genus-zero free boundary minimal surfaces embedded in the unit ball in the Euclidean three-space which are compact and lie arbitrarily close to the boundary unit sphere with an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations…

I will discuss the travel time tomography problem for elastic media in the transversely isotropic setting. The mathematical study of this problem relates to X-ray tomography and boundary rigidity problems studied by de Hoop, Stefanov, Uhlmann, Vasy, et al., which reduce the inverse problems to the microlocal analysis of certain operators obtained from a pseudolinearization…

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of smooth, compactly supported perturbations of the metric, whereas all…

A free boundary minimal surface (FBMS) in a given three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. We will talk about the existence of FBMS, with particular emphasis on an equivariant min-max method and on the computation…

Let Σ be a closed surface (i.e. a 2-dimensional Riemannian manifold) satisfying the following condition: the first eigenvalue of the elliptic operator -Δ+βK is nonnegative, where K is the Gauss curvature and β is a positive constant . This condition was mainly motivated by the studies of positive scalar curvature in dimension three, and soon…

In this talk I will show that the Morse index of the critical Moebius band in the 4-dimensional Euclidean ball equals 5. This result makes use of the quartic Hopf differential technique and a comparison theorem between the index of a free boundary minimal surface in the Euclidean ball and its spectral index. The latter also enables us to reprove…

In this talk, we will survey G2-structures, which are cross product structures on 7-manifolds, and we will discuss recent developments on a natural geometric flow of G2-structures called Laplacian flow. The Laplacian flow was introduced by Robert Bryant as a tool to explore the geometry of G2-structures on 7-manifolds and to construct examples of G2-holonomy…

In this talk, I will present new results on the symmetry reduction of gas dynamic systems of PDEs following the general framework presented by  Lev Ovsyannikov in his article  "The “podmodeli” program. Gas dynamics" https://www.sciencedirect.com/science/article/pii/0021892894901376 The gas dynamics systems of equations, with an arbitrary state equation, has an 11-dimensional Lie algebra of symmetries which generates a group…

Disorder is a fact of life, and controlling it on the nanoscale is complex, expensive, and of limited use. On the other hand, disordered materials do offer a range of possible applications if we know how to identify their useful features. To this end, we propose an approach through Topological Learning  to quantify disorder and…

Many areas of interest within the Earth are anisotropic, meaning that the speed of sound is different in different directions. It turns out that pressure waves are far better behaved than shear waves, but fortunately the different polarizations are coupled together through algebraic geometry. I will explain the surprising power of algebraic geometry in the…

The natural question is: do structures satisfying given open relations (called the genuine solutions of the differential relation) exist on a given manifold? Replacing all derivatives appearing in a differential relation by the additional independent variables one obtains an open subset of the corresponding jet bundle. A formal solution of the differential relation is a…

In this talk, we will introduce the special Lagrangian and Lagrangian mean curvature type equations. We will derive a priori interior estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian angle. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data…

A theorem of Liouville asserts that the simplest conformal transformations on Euclidean space---translations, dilations, reflections, and inversions---generate all conformal transformations when the dimension is at least 3.  I will describe a new proof of this theorem which is shorter and more elementary than the argument, due to Nevanlinna, found in most modern textbooks.

We discuss a sharp area estimate for minimal submanifolds that pass through a prescribed point in a geodesic ball in a space form. The estimate in Euclidean space was first conjectured by Alexander, Hoffman, and Osserman in 1974 and proven in full generality by Brendle and Hung in 2017. We will show the sharp area estimate also holds…