Boolean networks, neural networks and reaction-diffusion automata share a common structure which we identify as a special class of a new notion of dynamical systems on graphs for which we present a Lyapunov function type concept which implies phase-locking of the dynamics.  For dynamical systems with 'time' being a monoid instead of the integers or the reals, we define a notion of chaos which extends Devaney's classical chaos notion and we prove a theorem that sensitive dependence of initial conditions is a consequence of the two other properties in the definition. The common theme of the two topics is the intention to push the limits of dynamical systems theory in order to investigate how coupling or feedback motifs influence macroscopic behavior and discuss the role of time being a line.

Two day workshop, March 22-23, 2013. For a list of invited speakers, please visit: http://www.math.vanderbilt.edu/~suvaini/Workshop-2013/

Eighty years after the beginning of the general theory of computability, ideas from the "asymptotic point of view" prevalent in several areas of mathematics have begun to interact with computability theory. This will be a very general talk, developing the necessary ideas from scratch. I will try to give an idea of how this point of view leads to new questions and new answers. Tea at 3:30 pm in SC 1425

After reviewing some results on genericity and generic computability in group theory, I will discuss the rich interaction of generic computability and the general theory of computation and then discuss coarse computability and coarse degrees.

In 1980s, Yau conjectured that the existence of Kaehler Einstein metric on Fano manifold is related to an algebraic geometric  condition of ``stability''.  The recent work with Donaldson, Sun Song confirmed this conjecture.  In the talk, we will  review history of this problems as well as this subject, and we also will review earlier work of  G. Tian and others on this problems.  We will outline the strategy of proof, which involves deforming through metrics  with cone singularities.  If time permits, we will give more details about various aspects of the proof. Tea at 3:30 pm in SC 1425.

Spectral triple is the fundamental object of the metric aspects of Connes' noncommutative geometry. A spectral metric space is a spectral triple (A, H, D) with additional properties guaranteeing that the Connes metric on the state space of A induces the weak*-topology. It is, in fact, the noncommutative analog of a complete metric space. Let (A,H,D) be a spectral metric space and G be a group of automorphisms of A. In this talk I will consider the problem of whether there is a natural spectral triple for the crossed product algebra C*(G,A) that can characterize the metric properties of the dynamical system (G,A). I will discuss a solution to this problem when a single automorphism of A generates G as an equicontinuous family of quasi-isometries. I will also address the converse problem, namely, when a spectral metric space for the crossed product gives rise to one for A. When the action is not equicontinuous (e.g., when the action is uniformly hyperbolic), following the philosophy of Diffeomorphism-Invariant Geometry of Connes and Moscovici, we suggest replacing the dynamical system (G,A) by a dynamical system (G,B), where G acts isometrically. The algebra B is called the metric bundle associated with (G,A). Some candidates for the metric bundle B will be introduced. This talk is based on a joint work with Jean Bellissard and Matilde Marcolli.

A partial differential equation model is provided for the two-slit experiment of quantum mechanics. The state variable of the equation is the probability density function of particle positions. The equation has a local diffusion term corresponding to stochastic variation of particles, and a nonlocal dispersion term corresponding to oscillation of particles in the transverse direction perpendicular to their forward motion. The model supports the ensemble interpretation of quantum mechanics and gives descriptive agreement with the Schrodinger equation model of the experiment.

Perhaps the most outstanding open problem in mathematical relativity is the so called Cosmic Censorship conjecture, which roughly asserts that singularities in the evolution of spacetime must always be hidden inside black holes, and moreover that spacetime must eventually settle down to a stationary final state. Based on heurisitc physical arguments, R. Penrose derived a series of geometric inequalities relating total mass, area of the event horizon, electricromagnetic charge, and angular momentum, all of which serve as necessary conditions for the validity of Cosmic Censorship. In this talk, we will detail recent advances in the rigorous mathematical formulation and proofs of some of these inequalities. Tea at 3:30 pm in SC 1425

A group is called topologizable if it admits a non-discrete Hausdorff group topology. In this talk I will discuss some recent results (joint with A. Klyachko and A. Olshanskii)  and open questions about topologizable and non-topologizable groups.

The American Heritage Dictionary defines gerrymandering as the act of “dividing a geographic area into voting districts so as to give unfair advantage to one party.” The problem of gerrymandering has led to the development of several mathematical measures of shape compactness, some of which have been used in court cases to argue for or against the legality of congressional redistricting plans. In this talk, we will show how the notion of convexity can be used to detect irregularly shaped districts. We will explore both theoretical and empirical aspects of this convexity-based measure of shape compactness.