MidWest Model Theory Day II

Tuesday, April 13th, 2010 at UIC


Speakers: Amador Martin Pizarro, Salih Azgin, Thanases Pheidas

Schedule: A map. All talks are about an hour long, in SEO 636. There will also be coffee&cookies in 636.
It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor.
Let me know if you are planning to come to lunch and/or dinner so we can make approximately correct reservations!

Abstracts:
Salih Azgin: Extremal Valued Fields
Amador Martin Pizarro: On Schanuel's conjecture. Zilber proved that the class of pseudo-exponentially closed fields satisfying Schanuel condition, and e.c. closed condition and having the countable closure property is excellent and hence it admits a unique model in each uncountable cardinal. Moreover, by a clever use of Ax's result on Schnauel's conjecture, he also proved that the complex numbers satisfy the countable closure property. In this talk we will concentrate on the study of predimension 0 extensions given by a single point, and show that the complex numbers are existentially closed for these extensions. This is joint work with A. Günaydin
Thanases Pheidas: Decision questions in Number Theory, Hilbert's tenth problem and the conjectures of Lang and Bombieri. We will survey the status of questions similar to Hilbert's tenth problem, i.e. whether there is an algorithm for determining the solvability of arbitrary polynomial equations over a ring A. We will focus on the similar problem of (existence of an algorithm to determine) whether a polynomial equation has infinitely many solutions in some finite extension over the field of rational numbers and the conjectural connection of that question to a decidability problem for the field of global meromorphic functions.



MidWest Model Theory Day I

February 16, 2010 at UIC



Schedule: All talks are about an hour long, in SEO 636. There will also be coffee&cookies in 636.
It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor.
Let me know if you are planning to come to lunch and/or dinner so we can make approximately correct reservations!

Abstracts:

Clifton Ealy: Thorn-Forking in Continuous Logic We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for classical real rosy theories. This is joint work with Isaac Goldbring.
Jana Marikova: Valuations on o-minimal fields. Let R be an o-minimal field and V a proper convex subring. We show that the o-minimality of the corresponding residue field with structure induced from R via the residue map is equivalent to (R,V) satisfying a first-order axiom scheme.
Chris Miller: A continuous extension property for o-minimal expansions of ordered groups. Let f:R^n-->R be bounded and definable in an o-minimal expansion of an ordered group R. Then there exist finitely many cells C_i in R^n of a special form such that their union has interior and, for each i, the frontier of C_i contains the origin and f extends continuously to the closure of C_i. This result has been used by Friedman, Kurdyka, Speissegger and me (to appear in JSL) to establish the existence of an open set U of real numbers such that, among other things, the expansion of the real exponential field by U Borel-interprets the real projective hierarchy, yet every definable set either has interior or is nowhere dense.