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Dissertation Information for Ignacio Dario Viglizzo

NAME:
- Ignacio Dario Viglizzo

DEGREE:
- Ph.D.

DISCIPLINE:
- Mathematics

SCHOOL:
- Indiana University (USA) (2005)

ADVISORS:
- Lawrence S. Moss

COMMITTEE MEMBERS:
- Enfandiar Haghverdi
- David McCarty
- Alberto Torchinski

MPACT Status: Incomplete - Not_Inspected

Title: Coalgebras on measurable spaces

Abstract: Given an endofunctor T in a category [Special characters omitted.] , a coalgebra is a pair ( X , c ) consisting of an object X and a morphism c : X [arrow right] T ( X ). X is called the carrier and the morphism c is called the structure map of the T -coalgebra.

The theory of coalgebras has been found to abstract common features of different areas like computer program semantics, modal logic, automata, non-wellfounded sets, etc. Most of the work on concrete examples, however, has been limited to the category Set . The work developed in this dissertation is concerned with the category Meas of measurable spaces and measurable functions.

Coalgebras of measurable spaces are of interest as a formalization of Markov Chains and can also be used to model probabilistic reasoning. We discuss some general facts related to the most interesting functor in Meas , Δ, that assigns to each measurable space, the space of all probability measures on it. We show that this functor does not preserve weak pullbacks or ω op -limits, conditions assumed in many theorems about coalgebras. The main result will be two constructions of final coalgebras for many interesting functors in Meas . The first construction (joint work with L. Moss), is based on a modal language that lets us build formulas that describe the elements of the final coalgebra. The second method makes use of a subset of the projective limit of the final sequence for the functor in question. That is, the sequence 1 [arrow left] T 1 [arrow left] T 2 1 [arrow left] ... obtained by iteratively applying the functor to the terminal element 1 of the category. Since these methods seem to be new, we also show how to use them in the category Set , where they provide some insight on how the structure map of the final coalgebra works.

We show as an application how to construct universal Type Spaces, an object of interest in Game Theory and Economics. We also compare our method with previously existing constructions.

MPACT Scores for Ignacio Dario Viglizzo

A = 0
C = 0
A+C = 0
T = 0
G = 0
W = 0
TD = 0
TA = 0
calculated 2010-09-30 18:54:00

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