A curve of constant width has the property that any two parallel supporting lines are the same distance apart in all directions.  A fundamental problem involving these curves is to find one which encloses the smallest amount of area for a given width. This problem was resolved long ago and has a few relatively simple solutions.…

An important problem in computer vision is to understand the space of  images that can be captured by an arrangement of cameras. A description of this space allows for statistical estimation methods to reconstruct  three-dimensional models of the scene that was imaged. The set of images captured by an arrangement of pinhole cameras is usually…

Many biological organisms are comprised of deformable porous media, with additional complexity of an embedded muscle. Using geometric variational methods, we derive the equations of motion for the dynamics of such active porous media. The use of variational methods allows to incorporate both the muscle action and incompressibility of the fluid and the elastic matrix…

In this “something for everyone” talk, we will place the matrix decompositions that are so valuable in all fields of computation in a historical abstract context.  It is well known that the singular value decomposition was discussed as far back as the 19th century by Beltrami and Jordan. Lesser known is that Cartan had a blueprint…

Affine Lie algebras, also sometimes called current algebras, are infinite-dimensional analogs of finite-dimensional semisimple Lie algebras. The representation theory of affine Lie algebras has applications in many areas of mathematics (number theory, combinatorics, group theory, geometry, topology, etc.) and physics (conformal field theory, integrable systems, statistical mechanics, etc.). To study the combinatorial properties of affine Lie algebra…

We review randomized algorithms for the numerical solution of least squares/regression problems, with a focus on algorithms that row-sketch from the left, or column-sketch from the right. These algorithms tend to be efficient and accurate on matrices that have many more rows than columns. We present probabilistic bounds for the amount of sampling required to…

Beginning with the solution of the classical Plateau problem—the problem of finding an area-minimizing disk whose boundary is a prescribed simple closed curve in Euclidean 3-space—we will survey some applications of  Calculus of Variations to solve geometric extremal problems. Particular emphasis will be placed on the problem of finding a smooth surface in 3-space with…

When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases…

I and many collaborators, postdocs, and students from many disciplines have explored lung mechanics and disease pathology for over 2 decades in a pan-university effort called the UNC Virtual Lung Project. In the last decade we have explored how viruses “traffic” in mucosal barriers, including the human respiratory tract (RT), in the presence of antibodies.…

 In this talk we will discuss the connection between invariant evolutions of polygons and completely integrable discrete systems via polygonal geometric invariants. We will give examples and show how some open problems for bi-Hamiltonian structures of discrete systems were made easier and solved using this correspondence. If time allows we will discuss some open problems.…

Synthetic aperture radars (SAR) use microwaves to obtain images of the Earth's surface from airplanes or satellites. SAR images can be taken during nighttime and prove insensitive to the clouds or dust in the atmosphere. Therefore, SAR complements the aerial or spaceborne photography, even though there are fundamental differences between the two technologies. For example,…

The vision of Isogeometric Analysis (IGA) was first presented in a paper published October 1, 2005 . Since then it has become a focus of research within both the fields of Finite Element Analysis (FEA) and Computer Aided Design (CAD) and has become a mainstream analysis methodology and provided a new paradigm for geometric design…

I will begin by probing into the past to discover the origins of the Finite Element Method (FEM), and then trace the evolution of those early developments to the present day in which the FEM is ubiquitous in science, engineering, mathematics, and medicine, and the most important discretization technology in Computational Mechanics. However, despite its…

In linear algebra we know that the Pfaffian of an antisymmetric matrix is a square root of the determinant of matrix. In this talk I will explain how one does the quantum linear algebra, a recent popular area that can be traced back to Gauss and is well connected with many areas of mathematics such as…

Congenital heart disease affects 1 in 100 infants and is the leading cause of infant mortality in the US. Among the most severe forms of congenital heart disease is single ventricle physiology, in which the heart develops with only one functional pumping chamber. These patients typically undergo three open chest surgeries, culminating in the Fontan…

We introduce a Swarm-Based Random Descent (SBRD) method for non-convex optimization. The swarm consists of agents, identified with positions, x, and masses, m.  There are three key aspects to the SBRD dynamics: (i) persistent transition of mass from high to lower ground; (ii) marching along the gradient descent: an m-dependent random choice of marching direction…

Simulating the time evolution of a Hamiltonian system on a classical computer is hard—the computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrating its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge. Assuming that a quantum…

The climate is changing due to the heat trapping caused by the rapid increase in greenhouse gases, mainly carbon dioxide, in the atmosphere. One way to state the issue is that we cannot, as a species, adapt to the new conditions quickly enough. This is an example of rate-induced tipping for which the mathematics has…

Spreading (diffusion) of new products is a classical problem. Traditionally, it has been analyzed using the compartmental Bass model, which implicitly assumes that all individuals are homogeneous and connected to each other. To relax these assumptions, research has gradually shifted to the more fundamental Bass model on networks, which is a particle model for the…

Minimal surfaces are fundamental geometric objects which have been studied intensively since the 1700's. Classes of minimal surfaces of particular interest are the complete embedded ones in Euclidean space, closed (compact boundaryless) embedded in the round three-sphere, free boundary compact embedded ones in the unit Euclidean three-ball, and self-shrinkers of the mean curvature flow. Since…