Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in…

Integer matrices are often characterized by the lattice of combinations of their rows or columns. This is captured nicely by the Smith canonical form, a diagonal matrix of invariant factors, to which any integer matrix can be transformed through left and right multiplication by unimodular matrices. Algorithms for computing Smith forms have seen dramatic improvements…

Multivalued matrix functions arise in solving various kinds of matrix equations. The matrix logarithm is the prototypical example. Another example is the Lambert W function of a matrix, which is much less well known but has been attracting recent interest. A theme of the talk is the importance of choosing appropriate principal values and making…

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n−d$ . The conjecture itself has been disproved by Klee-Walkup (1967) for unbounded polyhedra and by the speaker (2012) for bounded polytopes; but what we know about the underlying question is…

Categorification lifts natural numbers to vector spaces and integers to complexes. Natural number n becomes a vector space of dimension n, and an integer becomes the Euler characteristic of a complex of vector spaces. A well-known example of categorification is lifting the Euler characteristic of a topological space to its homology or cohomology groups. The…

In many applications one wishes to minimize an objective function that is not convex and is not differentiable at its minimizers.  We discuss two algorithms for minimization of nonsmooth, nonconvex functions.  Gradient Sampling is a simple method that, although computationally intensive, has a nice convergence theory.  The method is robust and the convergence theory has…

Through the actuation of elastic filaments, microorganisms can move through a viscous fluid or generate a flow within a complex environment. In this talk, we will focus on simple models of two such systems. First, as a step towards understanding the chemotactic behavior of bacteria within micropores, we consider a single elastic helical flagellum confined…

Public goods games involve interactions between players or organisms who produce a commonly available good (cooperators) and those who consume it without producing (cheaters, defectors, freeloaders) - several instances are known to occur in nature and economics. The solution that freeloading is a better individual choice is known as the “tragedy of the commons” -…

Classical control theory deals with controlled ODEs and PDEs. An interesting setting, where both ODE and PDE control appear naturally, is that of multi-agent systems and crowd dynamics. Recently, control problems involving measure-valued solutions attracted a lot of attention as a tool to deal with such systems. The measure can serve as the probabilistic representation…

In October 2017, I found myself testify for hours in a Federal court. I had not been arrested. Rather I was attempting to quantifying gerrymandering using analysis which grew from asking if a surprising 2012 election was in fact surprising. It hinged on probing the geopolitical structure of North Carolina using a Markov Chain Monte…

Time-dependent hyperbolic partial differential equations can be efficiently solved using adaptive mesh refinement, with a hierarchy of finer grid patches in regions where the solution is discontinuous or rapidly varying. These patches can be adjusted every few time steps to follow propagating waves. For many problems the primary interest is in tracking waves that reach…

I will discuss a new and somewhat mysterious connection between singularity theory and cluster algebras, more specifically between the topology of isolated singularities of plane curves and the mutation equivalence of quivers associated with their morsifications. The talk will assume no prior knowledge of any of these topics. This is joint work with Pavlo Pylyavskyy,…

The study of the structural properties of the set of points at which the viscosity solution of a first order Hamilton–Jacobi equation fails to be differentiable—in short, the singular set—started with the paper On the Singularities of Viscosity Solutions to Hamilton–Jacobi–Bellman Equations, Indiana Univ. Math. J. 36 (1987), 501–524 by Mete Soner and myself. These…

Homology 3-spheres, i.e. 3-dimensional manifolds with the same homology groups as the standard 3- sphere, play a central role in topology. Their study was initiated by Poincare in 1904, who constructed the first nontrivial example of a homology 3-sphere, and conjectured that the standard sphere is the only simply connected example. A century later, Poincare's…

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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…