Mathematical Analysis of an Age-Structured Population Model Applicable to Early Humans
Min Gao, Vanderbilt University
Location: Stevenson 1307
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.
The Intersection Condition for Regular Polytopes
Marston Conder, University of Auckland, New Zealand
Location: Stevenson 1432
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.
Discrete Objects with Maximum Possible Symmetry
Marston Conder, New Zealand Institute of Mathematics & its Applications
Location: Stevenson 5211
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.
Talk Title TBA
Location: Stevenson 1210
Geodesically Tracking Quasi-Geodesic Paths for Coxeter Groups
Michael Mihalik, Vanderbilt University
Location: Stevenson 1310
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.
Wavelet Analysis Based on Algebraic Polynomial Identities
Johan De Villiers , Stellenbosch University
Location: Stevenson 1307
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.
Graduate Student Tea
Location: Stevenson 1425
Probability and March Madness
Michael Northington, Vanderbilt University
Location: Stevenson 1206
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.
Independent Sets in Computable Free Groups and Fields
Russell Miller, City University of New York
Location: Stevenson 1312
Kähler Geometry, On The Edge
Location: Stevenson Center 1431, 1210, 1432
Two day workshop, March 22-23, 2013. For a list of invited speakers, please visit: http://www.math.vanderbilt.edu/~suvaini/Workshop-2013/