This Week in Logic at CUNY

This Week in Logic at CUNY:

– – – – Monday, May 7, 2012 – – – –

– – – – Tuesday, May 8, 2012 – – – –

Computational Logic Seminar
Time 2:00 – 4:00 PM, May 8, Room 3209.
Speaker: Sergei Artemov (Graduate Center)
Title: Joining provability and explicit proofs.

Abstract: We will study the system GLA that combines provability logic
and the logic of proofs. The principal results of this talk were
obtained by E. Nogina and we will follow her work closely.

– – – – Wednesday, May 09, 2012 – – – –

Models of Peano Arithmetic
Wednesday, May 9, 2012 6:30 pm Room 4214-03
Professor Leszek Kolodziejczyk (Institute of Mathematics, University
of Warsaw)
Open Induction-style methods in the study of fragments of bounded arithmetic

Abstract. Sam Buss’ bounded arithmetic S_2, defined in the mid-80’s,
has been studied mostly because of its connections with computational
complexity and propositional proof complexity. The main goal in the
area, to separate natural fragments of S_2 from the full theory, is
apparently out of reach except for some very weak fragments. For this
reason, attention has largely shifted to somewhat different questions,
but a better understanding of the border between “very weak”
subtheories of S_2 and those “too strong for current methods” is
still desirable. In this talk, I will discuss some separations
obtained by techniques inspired by open induction, first introduced
into bounded arithmetic by Boughattas and Ressayre.

– – – – Thursday, May 10, 2012 – – – –

– – – – Friday, May 11, 2012 – – – –

Model Theory Seminar
Friday, May 11, 2012 12:30 pm GC 6417
Professor Philipp Rothmaler (The City University of New York, BCC)
More examples: definable subgroups of modules and Ziegler spectra of rings

Abstract. This talk is first of all intended as illustration to the
previous talk on modules in that I will present more examples of pp
definable subgroups and more pictures. I will then introduce the
Ziegler spectrum of a ring, a central concept of the model theory of
modules, and compute it for the integers. Some other examples of
Ziegler spectra will be mentioned as well.

Logic Workshop
Friday, May 11, 2012 2:00 pm GC 6417
Professor Philipp Rothmaler (The City University of New York, BCC)
Definable subcategories and Mittag-Leffler modules

Abstract. Given an associative ring R, a definable subcategory is a
subcategory of R-Mod, the category of all left R-modules, that is
closed under pure submodule, direct sum (or product), and direct limit
(=colimit). It can be shown that these are indeed definable, namely,
they are exactly the subcategories axiomatized by pp implications
(i.e. by axioms of the form F–>G, where F and G are (1-place) pp
formulas), and they are in bijective correspondence with the closed
subsets of the Ziegler spectrum (cf. preceding talk in Model Theory
Seminar). Following Raynaud and Gruson (1971), a left R-module M is
called Mittag-Leffler if the canonical map (\prod K_i) \tensor M –>
\prod (K_i \tensor M) is injective for all collections of K_i from
K=Mod-R. In old work (1994), I extended this to arbitrary subclasses K
of Mod-R in the obvious way and proved that M is K-Mittag-Leffler iff
M is, in the model-theoretic sense, positively atomic with respect to
the dual class DK of K in R-Mod. Here D is Prest’s elementary duality
(as introduced in Alves’ earlier talk, cf. today’s talk in the Model
Theory Seminar). As a consequence one sees that it suffices to
consider definable subcategories K (and DK) in the above. An
interesting special case first investigated by Goodearl (1972) is when
K consists of the single module R (considered as a right R-module) or,
equivalently, when K consists of all flat right R-modules. Then DK is
the class of all absolutely pure left R-modules. It is interesting to
note that these classes are different from the definable categories
they generate precisely when the ring is not left coherent.
I will explain all of the above and prove some recent structure results.

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