The Department of Mathematics and Statistics offers two master's degrees, one in Mathematics and the other in Statistics, and two doctoral degrees, one in Mathematics and one in Statistics. Areas of study for Mathematics include: algebra, algebraic geometry, real and complex analysis, differential geometry, and topology. The areas of study for Statistics are: mathematical statistics, survival analysis, modeling, statistical computing for massive data, Bayesian regulation, bioinformatics, longitudinal and functional data analysis, statistical computation, asymptotic theory, objective Bayes, bootstrap, post-selection inference, and application of statistics to medicine. Because it is difficult to make up coherent programs for students entering in the middle of the year, students are ordinarily admitted only in the fall.

Mathematics and Statistics

In 1954 Alfred Tarski [210] announced that ‘a new branch of metamathematics’ had appeared under the name of the theory of models. The subject grew fast: the Omega Group bibliography of model theory in 1987 [148] ran to 617 pages. By the mid 1980s there were already too many dialects of model theory for anybody to be expert in more than a fraction. For example very few model theorists could claim to understand both the work of Zilber and Hrushovski at the edge of algebraic geometry, and the studies by Immerman and Vardi of classifications of finite structures. And neither of these lines of research had much contact with what English-speaking philosophers and European computational linguists had come to refer to as ‘model-theoretic’ methods or concepts. Nevertheless all these brands of model theory had common origins and important family resemblances. Some other things called models definitely lie outside the family. For example this chapter has nothing to say about ‘modelling’, which means constructing a formal theory to describe or explain some phenomena. Likewise in cognitive science the ‘mental models’ of Gentner and Stephens [67] or Johnson-Laird [100] lie outside our topic. All the flavours of model theory rest on one fundamental notion, and that is the notion of a formula φ being true under an interpretation I. The classic treatment is Tarski’s paper [202] from 1933. In this paper Tarski supposes that we have a language L with a precisely defined syntax. Ignoring punctuation, the symbols of L are of two kinds: constants and variables. The constants have fixed meanings; they will usually include logical expressions such as ‘and’ and ‘equals’. The variables have no meaning, but (to shortcircuit Tarski’s very careful formulation a little) we can assign an object to each variable, and ask whether a given formula φ of L becomes true when each variable is regarded as a name of its assigned object. The grammatical categories of the variables determine what kinds of object can be assigned to them; for example we can assign individuals to individual variables, classes

Model Theory: Combinatorics

Model theory is an important area of mathematical logic which has deep philosophical roots, many philosophical applications, and great philosophical interest in itself. The aim of this book is to introduce, organise, survey, and develop these connections between philosophy and model theory, for the benefit of philosophers and logicians alike.

Philosophy and Model Theory

We introduce Black Boxes, the intention being provably in ZFC, we have guesses of extra structure The guesses are pairwise quite far but together are "dense" We first deal with the simplest case, were the existence comes from winning a game by just writing down the opponent's moves We show how it helps when instead orders we have tress with boundedly many levels, having freedom in the last After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them Chapter IV of upcoming book "Nonstructure Theory"

Black Boxes

Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let \mu be the Hanf number of K. Suppose K is tame. 
MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^\mu)^+} then K is categorical in all cardinals greater than \beth_{(2^\mu)^+}. 
This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).

Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes

Categoricity of Uncountable Theories

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Categoricity of Uncountable Theories