Control systems, typically nonlinear,  abound in our daily experience of the world. Mathematically, they can be described by differential equations endowed with functional parameters. The prototypical example is balancing a broomstick on the tip of your finger. A basic problem in control theory is the following.  Suppose a control system ${\displaystyle \omega }$ is in state ${\displaystyle A}$ and you want to ``steer" it to state ${\displaystyle B}$. How do you choose the (functional) parameters in the system --  the controls --  in order to achieve this? One oft-used technique is to find a special type of transformation, ${\displaystyle T}$, that maps ${\displaystyle \omega }$ to a certain type of universal linear control system called a Brunovsky normal form, ${\displaystyle \beta }$. The trajectories of ${\displaystyle \beta }$ can easily be written down in terms of arbitrary functions of time and their time derivatives after which ${\displaystyle T^{-1}}$ maps  the trajectories of ${\displaystyle \beta }$  to those of ${\displaystyle \omega }$, from which the steering problem can be solved. If we can construct such a solution, ${\displaystyle s}$ of ${\displaystyle \omega }$, we will say that it is {\em explicitly integrable}. This settles the steering problem for linearizable control systems which form a small subset of control systems of interest. But while every linearizable system is explicitly integrable, the converse is false and for the past few decades there has been a research program underway to characterize the phenomenon of explicit integrability beyond the linearizable case. However it turns out that the Brunovsky normal forms are familiar objects in differential geometry. They correspond to the contact systems on a special class of jet spaces over the real line. In this talk we will explore this geometry to show how to establish explicit integrability for control systems which are intrinsically nonlinear - those that have no linearizing diffeomorphisms.  A key ingredient is the invariance of ${\displaystyle \omega }$ under the free and regular action of a Lie group.

Symbolic dynamics (in one dimension) is the study of subshifts: subsets of bi-infinite sequences of symbols from a finite alphabet which are invariant under a shift to the left.  We say two subshifts are conjugate (isomorphic) if there is a homeomorphism between the underlying subsets which commutes with the shift map.  In particular, we will focus on vertex shifts, which are given by bi-infinite walks on a directed graph.  A decomposition theorem due to Williams states that every conjugacy between vertex shifts can be written as a sequence of state (vertex) splittings followed by state amalgamations.  In this talk, I will introduce the basic definitions of subshifts, and discuss a fascinating problem which has been open for some time: the decidability of conjugacy between vertex shifts.  Barring an extreme and unprecedented stroke of insight, I will not solve this problem, but instead focus on a simpler version where only state amalgamations are allowed.  We will see that this state amalgamation problem is NP-hard, which has implications for the simplification of Markov shifts.

We consider the problem of finding a homomorphism from an input digraph G to a fixed digraph H (HOM(H)). A homomorphism from G to H is a mapping from vertices in G to H that preserve the adjacency (a generalization of coloring problem). A polymorphism of H is just a homomorphism from H^k (for some integer k>1)  to H. The algebraic work of several researchers such as Bulatov, Jeavons, Krokhin, Maroti and McKenzie led to discovery of weak-near-unanimity polymorphism and the proof of NP-completeness of HOM(H) when H does not admit any weak-near-unanimity polymorphism.

We show that if H admits a weak-near-unanimity polymorphism ${\displaystyle \varphi }$ then HOM(H) is polynomial time solvable. Our algorithm implies the validity of the dichotomy conjecture due to Feder and Vardi (1993). We transform the problem into an instance of the list homomorphism problem where initially all the lists are full (contain all the vertices of H). We use the polymorphism ${\displaystyle \varphi }$ as a guide to reduce the lists to singleton lists, which yields a homomorphism if one exists.

This is based on joint work with Tomas Feder and Jeff Kinne and many useful discussions with Victor Dalmau and Pavol Hell.

Data arranged as a high-dimensional grid of numbers is a "tensor".
Questions from Hilbert to modern day P vs. NP have been cast as asking:
what properties of a tensor are unchanged when we change a basis? We
explore tensors with an underlying Lie algebra that upends a century of
convention of tensor products with rings.  The resulting tensor spaces
are highly compressed, display nuanced structure, and some can even be
classified. I will show the problems, the method, and the many things we
don't yet understand but might soon.

We provide conditions for when quotients of AF algebras are quasi-Leibniz quantum compact metric spaces, building on our previous work with F. Latrémolière. Given a ${\displaystyle C^{*}}$-algebra, its ideal space may be equipped with natural topologies. Next, we impart criteria for when convergence of ideals in the Fell topology of an AF algebra can provide convergence of quotients in quantum propinquity. We then apply these findings to a certain class of ideals of the Boca-Mundici AF algebra by providing a continuous map from this class of ideals equipped with various topologies, including the Jacobson and Fell topologies, to the space of quotients with the quantum propinquity topology.