A series of informal talks, following the book by Peter J. Olver.

A group has Property FW if every action on a set commensurating a subset fixes a subset at bounded distance. This is a combinatorial weakening of Kazhdan Property T (and strengthening of Serre's Property FA), which was characterized in a similar (measurable) fashion by Robertson and Steger. I will discuss Property FW in various contexts, and notably for lattices in Lie groups.

Given a graph G, the hitting time from u to v is the expected number of steps in a simple random walk u v_1 v_2 v_3 ... v where no v_i equals v. The average hitting time (AHT) of u is the arithmetic mean of the hitting times from u to every other vertex in G. When examining the spread of information through a social network represented as a connected digraph, the AHT can be seen as a proxy measure for a vertex's centrality in this diffusion process.  Since social networks are best represented as small-world graphs, we used the graph construction methods described by Watts and Strogatz and by Barabasi and Albert, comparing them to simple random graphs built using the Viger-Latapy algorithm. The AHT distribution varies widely between different graph structures and generation methods, although in every case is extremely closely tied to vertex degree.  Varying the order or average degree of a graph has consistent and predictable effects on the parameters of the AHT distribution, as does varying the control variables for the generation method in question.  In particular, graphs with strong preferential attachment behavior demonstrated an isolated group of vertices with extremely low AHT, as well as greater clustering in discrete "layers" defined by similar AHT values.

 

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.

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.

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

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.

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.

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.

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

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

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/

We consider maximal independent sets within various sorts of groups and fields freely generated by countably many generators. The simplest example is the free divisible abelian group, which is just an infinite-dimensional rational vector space. As one moves up to free abelian groups, free groups, and free fields; (i.e. purely transcendental field extensions), maximal independent sets and independent generating sets both become more complicated, from the point of view of computable model theory, but sometimes in unpredictable ways, and certain questions remain open. We present the topic partly for its own sake, but also with the intention of introducing the techniques of computable model theory and illustrating some of its possible uses for an audience to which it may be unfamiliar. This is joint work with Charles McCoy.

Probability is one of the most important and, often, most misunderstood areas of mathematics.  Applications of probability theory span from the most basic examples of flipping coins, to real world statistics used in everyday life, and even to the mechanics of the smallest particles that make up are universe.  In this talk, we will cover some of the basic rules of probability theory and look at a few non intuitive results.  Also, we will look at an interesting application where a probabilistic object called a Markov chain is used to predict the results of the NCAA tournament.  As it turns out, this method, developed by researchers at Georgia Tech, has been overwhelmingly more successful than any other ranking system (such as RPI, AP poll, ESPN poll, Sagarin rankings, etc.) in predicting the outcome of NCAA tournament games.  We will discuss the mathematics behind this model and some basics about the theory of Markov chains.

By starting out from a given refinable function, and relying on a corresponding space decomposition which is not necessarily orthogonal, we present a general wavelet construction method based on the solution of a system of algebraic polynomial identities. The resulting decomposition sequences are finite, and, for any given vanishing moment order, the wavelets thus constructed are minimally supported, and possess robust- stable integer shifts. The special case of cardinal B-splines are given special attention.

If (W,S) is a finitely generated Coxeter system we classify the quasi-geodesic paths (rays or lines) in the corresponding Cayley graph that are tracked by geodesics. The main corollary is that if W acts geometrically on a CAT(0) space X, then geometric geodesics in X are tracked by Cayley geodesics in X. This allows one to effectively transfer the group theory and combinatorics of (W,S) to help analyze the (local and asymptotic) geometry of X.

Symmetry is pervasive in both nature and human culture. The notion of chirality (or `handedness') is similarly pervasive, but less well understood. In this lecture, I will talk about a number of situations involving discrete objects that have maximum possible symmetry in their class, or maximum possible rotational symmetry while being chiral. Examples include geometric solids, combinatorial graphs (networks), maps on surfaces, dessins d'enfants, abstract polytopes, and even compact Riemann surfaces (from a certain perspective). I will describe some recent discoveries about such objects with maximum symmetry, illustrated by pictures as much as possible. Tea at 3:30 pm in SC 1425.

An abstract polytope is a generalised form of a geometric polytope, and may be viewed as a partially-ordered set (endowed with a rank function) that satisfies certain properties motivated by the geometry. A polytope is called `regular' if its automorphism group is transitive (and hence sharply-transitive) on the set of all flags -- which are the maximal chains in the poset. The automorphism group of a regular polytope is a smooth quotient of a 'string' Coxeter group (with a linear Dynkin diagram). Conversely, any finite smooth quotient of such a group is the automorphism group of a regular polytope, provided that it satisfies a condition known as the `intersection condition'. In this talk I will explain these things, and describe some recent discoveries about the intersection condition, including its application to find the smallest regular polytopes of any given rank.

The age structure of human populations is exceptional among animal species. Unlike most species, human juvenility is extremely extended and death is not coincident with the end of the reproductive period. Recently, a mathematical model was developed to examine the age structure of early humans, which reveals an extraordinary balance of human fertility and mortality. This model has two types of nonlinear mortalities, one term corresponding to the effects of crowding and the other term corresponding to the senescent burden on the juvenile population. We study this semilinear partial differential equation with a nonlinear boundary condition. We analyze the existence, uniqueness and regularity of solutions to the model equations. An intrinsic growth constant is obtained and linked to the existence and the stability of the trivial or the positive equilibrium.  The model supports the hypothesis that the age structure of early humans was robust in its balance of juvenile, reproductive, and senescent classes.