In this talk, I will introduce a novel Riemannian geometric model for wave propagation in gas giant planets. Terrestrial planets and gas giants have one key difference: in gas, density goes to zero at the surface, and seismic waves come to a full stop. We model the sound speed in a planet by a Riemannian…

We start by giving a short introduction to geometric inverse problems. Then we present our recent results regarding inverse problems for the minimal surface equation. Minimal surfaces are solutions to a quasilinear elliptic equation. We determine a minimal surface up to an isometry from the corresponding Dirichlet-to-Neumann map in dimension 2. Applications of the results…

The structure of algebras of differential invariants, particularly their generators, is based on the symbolic invariant calculus provided by the method of equivariant moving frames.  I will discuss a new computational algorithm that will, in many cases, determine whether a given set of differential invariants is generating.  As an example, we establish a new result…

We show how to use continued fraction representations of 2-bridge knots to compute the average value of different invariants of the set of 2-bridge knots with fixed crossing number c. Examples include the Seifert genus, braid index, and the absolute value of the signature. We also mention other properties of the probability distributions of these…

A metric Lie algebra is a real Lie Algebra with an inner product. In this talk, we will outline a complete classification of all three-dimensional metric Lie algebras up to notions of equivalence determined by isomorphism and scaling and investigate the Ricci flow on the resulting parameter spaces of equivalence classes.

In this talk we discuss partial data inverse boundary problems for magnetic Schrödinger operators on bounded domains in the Euclidean space as well as some Riemannian manifolds with boundary. In particular, we show that the knowledge of the set of the Cauchy data on a portion of the boundary of a domain in the Euclidean…

In this talk, I will go through some old and new results concerning the rigidity and flexibility of scalar curvature.

The famous knot complement theorem of Gordon and Luecke states that two knots in the three-sphere are equivalent if and only if the complements are homeomorphic. This was proved more than 30 years ago using combinatorial methods. In this talk, we will prove some extended results using techniques from gauge theory. If there is time…

 Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications, in particular, to…

Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications, in particular, to…

In this talk we will discuss the differences in the methodology of determining time-dependent and time-independent coefficients appearing in a hyperbolic equation in a Riemannian manifold. The talk is based on two recent research projects: 1) We will prove that a local source-to-solution map of a hyperbolic partial differential operator on a complete Riemannian manifold (no…

In this talk, I will explore the relationship between a domain's shape and its first Laplace eigenvalue, with emphasis on the domains which minimize, or are more generally critical points, for this eigenvalue, for their given volume.

In recent years much attention has turned to rigidity, and almost-rigidity, problems involving lower scalar curvature bounds. In this talk, I'll discuss some contributions to this area, including some new stability theorems for spheres. Some of this is joint work with Davi Maximo, and some is joint with Paul Sweeney Jr.

Symplectic manifolds show up in many areas of mathematics, including topology, geometry, representation theory, and dynamics. In the talk I will give an introduction to symplectic manifolds and then discuss embeddings of 3-manifolds into symplectic 4-manifolds. This is joint work with Ali Daemi and Mike Miller Eismeier.

One of the main approaches to topological data analysis is to study discrete invariants of (multiparameter) persistence modules. In this talk, we discuss recent research which utilizes techniques from (relative) homological algebra to interpret classical examples of invariants and define new invariants. The Hilbert function and (generalizations of) the rank invariant serve as our main…

On a Riemannian manifold with boundary, the X-ray transform integrates a function or a tensor field along all geodesics through the manifold. The reconstruction of the integrand of interest from its X-ray transform is the basis of important inverse problems in seismology and medical imaging. For this transform, statements of injectivity and microlocal stability in the interior of the domain were established some…

The travel time map of a compact length space with a closed measurement set takes any point of the space to the function which measures the distance from this point to every point in the measurement set. The range of this map is called the travel time data, and it appears in the proofs of…

In 1966, Mark Kac posed the famous question “Can you hear the shape of a drum?” Mathematically, this amounts to recovering the geometry of a Riemannian manifold from knowledge of its Laplace spectrum. In the case of strictly convex and smooth bounded planar domains, the problem is very much open. One technique for studying the…

(Based on joint work with Yang Li). I will present an overview of Donaldson's program to extend the methods of gauge theory and Floer homology from 3- and 4-manifolds to higher dimensions, with a focus in this talk on Calabi-Yau 3-folds. After discussing the background material in gauge theory and Calabi-Yau geometry, I will highlight…